Convex Optimization & Euclidean Distance Geometry

DATTORRO

CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY Mεβοο

Dattorro

CONVEX

OPTIMIZATION

&

EUCLIDEAN

DISTANCE

GEOMETRY

Meboo

Convex Optimization & Euclidean Distance Geometry

Jon Dattorro

Mεβoo Publishing

Meboo Publishing USA PO Box 12 Palo Alto, California 94302 Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo, 2005, v2009.09.22. ISBN 0976401304 (English)

ISBN 9780615193687 (International II)

This is version 2009.09.22: available in print, as conceived, in color. cybersearch: I. II. III. IV. V. VI.

convex optimization semidefinite program rank constraint convex geometry distance matrix convex cones programs by Matlab

typesetting by with donations from SIAM and AMS. This searchable electronic color pdfBook is click-navigable within the text by page, section, subsection, chapter, theorem, example, definition, cross reference, citation, equation, figure, table, and hyperlink. A pdfBook has no electronic copy protection and can be read and printed by most computers. The publisher hereby grants the right to reproduce this work in any format but limited to personal use.

© 2001-2009 Meboo All rights reserved.

for Jennie Columba ♦

♦

Antonio

♦ & Sze Wan

¡ ¢ EDM = Sh ∩ S⊥ c − S+

Prelude The constant demands of my department and university and the ever increasing work needed to obtain funding have stolen much of my precious thinking time, and I sometimes yearn for the halcyon days of Bell Labs. −Steven Chu, Nobel laureate [77]

Convex Analysis is the calculus of inequalities while Convex Optimization is its application. Analysis is inherently the domain of a mathematician while Optimization belongs to the engineer. Practitioners in the art of Convex Optimization engage themselves with discovery of which hard problems, perhaps previously believed nonconvex, can be transformed into convex equivalents; because once convex form of a problem is found, then a globally optimal solution is close at hand − the hard work is finished: Finding convex expression of a problem is itself, in a very real sense, the solution.

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

7

There is a great race under way to determine which important problems can be posed in a convex setting. Yet, that skill acquired by understanding the geometry and application of Convex Optimization will remain more an art for some time to come; the reason being, there is generally no unique transformation of a given problem to its convex equivalent. This means, two researchers pondering the same problem are likely to formulate the convex equivalent differently; hence, one solution is likely different from the other for the same problem, and any convex combination of those two solutions remains optimal. Any presumption of only one right or correct solution becomes nebulous. Study of equivalence & sameness, uniqueness, and duality therefore pervade study of Optimization. Tremendous benefit accrues when an optimization problem can be transformed to its convex equivalent, primarily because any locally optimal solution is then guaranteed globally optimal. Solving a nonlinear system, for example, by instead solving an equivalent convex optimization problem is therefore highly preferable.0.1 Yet it can be difficult for the engineer to apply theory without an understanding of Analysis. These pages comprise my journal over an eight year period bridging gaps between engineer and mathematician; they constitute a translation, unification, and cohering of about three hundred papers, books, and reports from several different fields of mathematics and engineering. Beacons of historical accomplishment are cited throughout. Much of what is written here will not be found elsewhere. Care to detail, clarity, accuracy, consistency, and typography accompanies removal of ambiguity and verbosity out of respect for the reader. Consequently there is much cross-referencing and background material provided in the text, footnotes, and appendices so as to be self-contained and to provide understanding of fundamental concepts. −Jon Dattorro Stanford, California 2009

0.1

That is what motivates a convex optimization known as geometric programming, invented in 1960s, [56] [75] which has driven great advances in the electronic circuit design industry. [32, §4.7] [231] [358] [361] [92] [166] [176] [177] [178] [179] [180] [246] [247] [287]

8

Convex Optimization & Euclidean Distance Geometry 1 Overview

21

2 Convex geometry 2.1 Convex set . . . . . . . . . . . . . . 2.2 Vectorized-matrix inner product . . 2.3 Hulls . . . . . . . . . . . . . . . . . 2.4 Halfspace, Hyperplane . . . . . . . 2.5 Subspace representations . . . . . . 2.6 Extreme, Exposed . . . . . . . . . . 2.7 Cones . . . . . . . . . . . . . . . . 2.8 Cone boundary . . . . . . . . . . . 2.9 Positive semidefinite (PSD) cone . . 2.10 Conic independence (c.i.) . . . . . . 2.11 When extreme means exposed . . . 2.12 Convex polyhedra . . . . . . . . . . 2.13 Dual cone & generalized inequality 3 Geometry of convex functions 3.1 Convex function . . . . . . . . . . 3.2 Conic origins of Lagrangian . . . 3.3 Practical norm functions, absolute 3.4 Inverted functions and roots . . . 3.5 Affine function . . . . . . . . . . 3.6 Epigraph, Sublevel set . . . . . . 9

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . value . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

35 36 49 59 71 83 88 93 102 109 132 138 138 147

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . .

209 . 210 . 214 . 214 . 223 . 226 . 230

10 CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 3.7 3.8 3.9 3.10

Gradient . . . . . . . Matrix-valued convex Quasiconvex . . . . . Salient properties . .

. . . . . function . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

4 Semidefinite programming 4.1 Conic problem . . . . . . . . . . . . . . . 4.2 Framework . . . . . . . . . . . . . . . . . 4.3 Rank reduction . . . . . . . . . . . . . . 4.4 Rank-constrained semidefinite program . 4.5 Constraining cardinality . . . . . . . . . 4.6 Cardinality and rank constraint examples 4.7 Convex Iteration rank-1 . . . . . . . . . 5 Euclidean Distance Matrix 5.1 EDM . . . . . . . . . . . . . . . . . . . . 5.2 First metric properties . . . . . . . . . . 5.3 ∃ fifth Euclidean metric property . . . . 5.4 EDM definition . . . . . . . . . . . . . . 5.5 Invariance . . . . . . . . . . . . . . . . . 5.6 Injectivity of D & unique reconstruction 5.7 Embedding in affine hull . . . . . . . . . 5.8 Euclidean metric versus matrix criteria . 5.9 Bridge: Convex polyhedra to EDMs . . . 5.10 EDM-entry composition . . . . . . . . . 5.11 EDM indefiniteness . . . . . . . . . . . . 5.12 List reconstruction . . . . . . . . . . . . 5.13 Reconstruction examples . . . . . . . . . 5.14 Fifth property of Euclidean metric . . . 6 Cone of distance matrices 6.1 Defining EDM cone . . . . . . . . . . . . 6.2 √ Polyhedral bounds . . . . . . . . . . . . 6.3 EDM cone is not convex . . . . . . . . 6.4 A geometry of completion . . . . . . . . 6.5 EDM definition in 11T . . . . . . . . . . −1 6.6 Correspondence to PSD cone SN . . . + 6.7 Vectorization & projection interpretation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

238 250 256 259

. . . . . . .

261 . 262 . 271 . 285 . 294 . 315 . 332 . 367

. . . . . . . . . . . . . .

373 . 374 . 375 . 375 . 380 . 412 . 417 . 423 . 428 . 436 . 443 . 446 . 454 . 458 . 464

. . . . . . .

475 . 477 . 479 . 481 . 482 . 488 . 496 . 502

CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 11 6.8 Dual EDM cone . . . . . . . . . . . . . . . . . . . . . . . . . . 507 6.9 Theorem of the alternative . . . . . . . . . . . . . . . . . . . . 521 6.10 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 7 Proximity problems 7.1 First prevalent problem: . 7.2 Second prevalent problem: 7.3 Third prevalent problem: . 7.4 Conclusion . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

A Linear algebra A.1 Main-diagonal δ operator, λ , trace, vec A.2 Semidefiniteness: domain of test . . . . . A.3 Proper statements . . . . . . . . . . . . . A.4 Schur complement . . . . . . . . . . . . A.5 Eigen decomposition . . . . . . . . . . . A.6 Singular value decomposition, SVD . . . A.7 Zeros . . . . . . . . . . . . . . . . . . . . B Simple matrices B.1 Rank-one matrix (dyad) B.2 Doublet . . . . . . . . . B.3 Elementary matrix . . . B.4 Auxiliary V -matrices . . B.5 Orthogonal matrix . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

C Some analytical optimal results C.1 Properties of infima . . . . . . . . C.2 Trace, singular and eigen values . C.3 Orthogonal Procrustes problem . C.4 Two-sided orthogonal Procrustes

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . .

525 . 533 . 544 . 555 . 566

. . . . . . .

569 . 569 . 574 . 577 . 589 . 594 . 597 . 602

. . . . .

609 . 610 . 615 . 616 . 618 . 623

. . . .

627 . 627 . 628 . 634 . 636

D Matrix calculus 641 D.1 Directional derivative, Taylor series . . . . . . . . . . . . . . . 641 D.2 Tables of gradients and derivatives . . . . . . . . . . . . . . . 662 E Projection 671 E.1 Idempotent matrices . . . . . . . . . . . . . . . . . . . . . . . 675 E.2 I − P , Projection on algebraic complement . . . . . . . . . . 680

12 CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY E.3 E.4 E.5 E.6 E.7 E.8 E.9 E.10

Symmetric idempotent matrices . . . . Algebra of projection on affine subsets Projection examples . . . . . . . . . . Vectorization interpretation, . . . . . . Projection on matrix subspaces . . . . Range/Rowspace interpretation . . . . Projection on convex set . . . . . . . . Alternating projection . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

681 686 687 693 701 705 705 719

F Notation and a few definitions

739

Bibliography

755

Index

783

List of Figures 1 Overview 1 Orion nebula . . . . . . . . . . . . . . . 2 Application of trilateration is localization 3 Molecular conformation . . . . . . . . . . 4 Face recognition . . . . . . . . . . . . . . 5 Swiss roll . . . . . . . . . . . . . . . . . 6 USA map reconstruction . . . . . . . . . 7 Honeycomb . . . . . . . . . . . . . . . . 8 Robotic vehicles . . . . . . . . . . . . . . 9 Reconstruction of David . . . . . . . . . 10 David by distance geometry . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

2 Convex geometry 11 Slab . . . . . . . . . . . . . . . . . . . . . . . . 12 Open, closed, convex sets . . . . . . . . . . . . . 13 Intersection of line with boundary . . . . . . . . 14 Tangentials . . . . . . . . . . . . . . . . . . . . 15 Inverse image . . . . . . . . . . . . . . . . . . . 16 Inverse image under linear map . . . . . . . . . 17 Tesseract . . . . . . . . . . . . . . . . . . . . . 18 Linear injective mapping of Euclidean body . . 19 Linear noninjective mapping of Euclidean body 20 Convex hull of a random list of points . . . . . . 21 Hulls . . . . . . . . . . . . . . . . . . . . . . . . 22 Two Fantopes . . . . . . . . . . . . . . . . . . . 23 Convex hull of rank-1 matrices . . . . . . . . . . 13

. . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . .

21 22 23 24 25 26 28 30 31 34 34

. . . . . . . . . . . . .

35 38 40 41 44 47 48 51 53 54 60 62 65 67

14

LIST OF FIGURES 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

A simplicial cone . . . . . . . . . . . . . . . . . . . . . . . . Hyperplane illustrated ∂H is a partially bounding line . . . Hyperplanes in R2 . . . . . . . . . . . . . . . . . . . . . . . Affine independence . . . . . . . . . . . . . . . . . . . . . . . {z ∈ C | aTz = κi } . . . . . . . . . . . . . . . . . . . . . . . . Hyperplane supporting closed set . . . . . . . . . . . . . . . Minimizing hyperplane over affine set in nonnegative orthant Closed convex set illustrating exposed and extreme points . Two-dimensional nonconvex cone . . . . . . . . . . . . . . . Nonconvex cone made from lines . . . . . . . . . . . . . . . Nonconvex cone is convex cone boundary . . . . . . . . . . . Cone exterior is convex cone . . . . . . . . . . . . . . . . . . Truncated nonconvex cone X . . . . . . . . . . . . . . . . . Not a cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum element. Minimal element . . . . . . . . . . . . . K is a pointed polyhedral cone having empty interior . . . . Exposed and extreme directions . . . . . . . . . . . . . . . . Positive semidefinite cone . . . . . . . . . . . . . . . . . . . Convex Schur-form set . . . . . . . . . . . . . . . . . . . . . Projection of truncated PSD cone . . . . . . . . . . . . . . . Circular cone showing axis of revolution . . . . . . . . . . . Circular section . . . . . . . . . . . . . . . . . . . . . . . . . Polyhedral inscription . . . . . . . . . . . . . . . . . . . . . Conically (in)dependent vectors . . . . . . . . . . . . . . . . Pointed six-faceted polyhedral cone and its dual . . . . . . . Minimal set of generators for halfspace about origin . . . . . Venn diagram for cones and polyhedra . . . . . . . . . . . . Unit simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . Two views of a simplicial cone and its dual . . . . . . . . . . Two equivalent constructions of dual cone . . . . . . . . . . Dual polyhedral cone construction by right angle . . . . . . K is a halfspace about the origin . . . . . . . . . . . . . . . Iconic primal and dual objective functions . . . . . . . . . . Orthogonal cones . . . . . . . . . . . . . . . . . . . . . . . . ∗ Blades K and K . . . . . . . . . . . . . . . . . . . . . . . . Shrouded polyhedral cone . . . . . . . . . . . . . . . . . . . Simplicial cone K in R2 and its dual . . . . . . . . . . . . . . Monotone nonnegative cone KM+ and its dual . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70 72 74 77 79 80 87 90 94 94 95 95 96 97 100 104 108 111 113 116 121 123 125 133 134 137 140 143 145 149 150 151 152 157 159 172 178 188

LIST OF FIGURES 62 63 64

15

Monotone cone KM and its dual . . . . . . . . . . . . . . . . . 190 Two views of monotone cone KM and its dual . . . . . . . . . 191 First-order optimality condition . . . . . . . . . . . . . . . . . 195

3 Geometry of convex functions 209 65 Convex functions having unique minimizer . . . . . . . . . . . 211 66 Minimum/Minimal element, dual cone characterization . . . . 213 67 1-norm ball B1 from compressed sensing/compressive sampling 217 68 Cardinality minimization, signed versus unsigned variable . . 219 69 1-norm variants . . . . . . . . . . . . . . . . . . . . . . . . . . 219 70 Affine function . . . . . . . . . . . . . . . . . . . . . . . . . . 227 71 Supremum of affine functions . . . . . . . . . . . . . . . . . . 229 72 Epigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 73 Gradient in R2 evaluated on grid . . . . . . . . . . . . . . . . 238 74 Quadratic function convexity in terms of its gradient . . . . . 246 75 Contour plot of real convex function at selected levels . . . . . 247 76 Iconic quasiconvex function . . . . . . . . . . . . . . . . . . . 257 4 Semidefinite programming 77 Venn diagram of convex problem classes . . . . . . . . 78 Visualizing positive semidefinite cone in high dimension 79 Primal/Dual transformations . . . . . . . . . . . . . . 80 Convex iteration versus projection . . . . . . . . . . . 81 Trace heuristic . . . . . . . . . . . . . . . . . . . . . . 82 Sensor-network localization . . . . . . . . . . . . . . . . 83 2-lattice of sensors and anchors for localization example 84 3-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Uncertainty ellipsoids orientation and eccentricity . . . 88 2-lattice localization solution . . . . . . . . . . . . . . . 89 3-lattice localization solution . . . . . . . . . . . . . . . 90 4-lattice localization solution . . . . . . . . . . . . . . . 91 5-lattice localization solution . . . . . . . . . . . . . . . 92 Sparse sampling theorem . . . . . . . . . . . . . . . . . 93 Signal dropout . . . . . . . . . . . . . . . . . . . . . . 94 Signal dropout reconstruction . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

261 . 265 . 266 . 276 . 297 . 298 . 300 . 304 . 305 . 306 . 307 . 308 . 313 . 313 . 314 . 314 . 318 . 322 . 323

16

LIST OF FIGURES 95 96 97 98 99 100 101 102 103 104 105 106

Simplex with intersecting line problem in compressed sensing . Geometric interpretations of sparse sampling . . . . . . . . . . Permutation matrix column-norm and column-sum constraint max cut problem . . . . . . . . . . . . . . . . . . . . . . . . Shepp-Logan phantom . . . . . . . . . . . . . . . . . . . . . . MRI radial sampling pattern in Fourier domain . . . . . . . . Aliased phantom . . . . . . . . . . . . . . . . . . . . . . . . . Neighboring-pixel stencil on Cartesian grid . . . . . . . . . . . Differentiable almost everywhere . . . . . . . . . . . . . . . . . MIT logo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-pixel camera . . . . . . . . . . . . . . . . . . . . . . . . . One-pixel camera - compression estimates . . . . . . . . . . .

5 Euclidean Distance Matrix 107 Convex hull of three points . . . . . . . . . . . . . . . . . . 108 Complete dimensionless EDM graph . . . . . . . . . . . . 109 Fifth Euclidean metric property . . . . . . . . . . . . . . . 110 Fermat point . . . . . . . . . . . . . . . . . . . . . . . . . 111 Arbitrary hexagon in R3 . . . . . . . . . . . . . . . . . . . 112 Kissing number . . . . . . . . . . . . . . . . . . . . . . . . 113 Trilateration . . . . . . . . . . . . . . . . . . . . . . . . . . 114 This EDM graph provides unique isometric reconstruction 115 Two sensors • and three anchors ◦ . . . . . . . . . . . . . 116 Two discrete linear trajectories of sensors . . . . . . . . . . 117 Coverage in cellular telephone network . . . . . . . . . . . 118 Contours of equal signal power . . . . . . . . . . . . . . . . 119 Depiction of molecular conformation . . . . . . . . . . . . 120 Square diamond . . . . . . . . . . . . . . . . . . . . . . . . 121 Orthogonal complements in SN abstractly oriented . . . . . 122 Elliptope E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Elliptope E 2 interior to S2+ . . . . . . . . . . . . . . . . . . 124 Smallest eigenvalue of −VNTDVN . . . . . . . . . . . . . . 125 Some entrywise EDM compositions . . . . . . . . . . . . . 126 Map of United States of America . . . . . . . . . . . . . . 127 Largest ten eigenvalues of −VNT OVN . . . . . . . . . . . . 128 Relative-angle inequality tetrahedron . . . . . . . . . . . . 129 Nonsimplicial pyramid in R3 . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

325 328 336 345 350 355 357 359 361 363 364 365 373 374 377 379 388 390 391 396 400 400 401 404 404 406 415 418 437 438 444 444 457 460 467 471

LIST OF FIGURES

17

6 Cone of distance matrices 130 Relative boundary of cone of Euclidean distance matrices 131 Intersection of EDM cone with hyperplane . . . . . . . . 132 Neighborhood graph . . . . . . . . . . . . . . . . . . . . 133 Trefoil knot untied . . . . . . . . . . . . . . . . . . . . . 134 Trefoil ribbon . . . . . . . . . . . . . . . . . . . . . . . . 135 Example of VX selection to make an EDM . . . . . . . . 136 Vector VX spirals . . . . . . . . . . . . . . . . . . . . . . 137 Three views of translated negated elliptope . . . . . . . . 138 Halfline T on PSD cone boundary . . . . . . . . . . . . . 139 Vectorization and projection interpretation example . . . 140 Orthogonal complement of geometric center subspace . . 141 EDM cone construction by flipping PSD cone . . . . . . 142 Decomposing a member of polar EDM cone . . . . . . . 143 Ordinary dual EDM cone projected on S3h . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

475 . 478 . 480 . 483 . 485 . 487 . 490 . 492 . 500 . 504 . 505 . 510 . 511 . 516 . 522

7 Proximity problems 525 144 Pseudo-Venn diagram . . . . . . . . . . . . . . . . . . . . . . 528 145 Elbow placed in path of projection . . . . . . . . . . . . . . . 529 146 Convex envelope . . . . . . . . . . . . . . . . . . . . . . . . . 549 A Linear algebra 569 147 Geometrical interpretation of full SVD . . . . . . . . . . . . . 601 B Simple matrices 148 Four fundamental 149 Four fundamental 150 Four fundamental 151 Gimbal . . . . . .

subspaces of any dyad . . . . . subspaces of a doublet . . . . . subspaces of elementary matrix . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

609 611 615 616 625

D Matrix calculus 641 152 Convex quadratic bowl in R2 × R . . . . . . . . . . . . . . . . 652 E Projection 671 153 Action of pseudoinverse . . . . . . . . . . . . . . . . . . . . . . 672

18

LIST OF FIGURES 154 155 156 157 158 159 160 161 162 163 164 165 166

Nonorthogonal projection of x ∈ R3 on R(U ) = R2 . . . . . . . Biorthogonal expansion of point x ∈ aff K . . . . . . . . . . . . Dual interpretation of projection on convex set . . . . . . . . . Projection product on convex set in subspace . . . . . . . . . von Neumann-style projection of point b . . . . . . . . . . . . Alternating projection on two halfspaces . . . . . . . . . . . . Distance, optimization, feasibility . . . . . . . . . . . . . . . . Alternating projection on nonnegative orthant and hyperplane Geometric convergence of iterates in norm . . . . . . . . . . . Distance between PSD cone and iterate in A . . . . . . . . . . Dykstra’s alternating projection algorithm . . . . . . . . . . . Polyhedral normal cones . . . . . . . . . . . . . . . . . . . . . Normal cone to elliptope . . . . . . . . . . . . . . . . . . . . .

678 689 708 717 720 721 723 726 726 731 732 734 735

List of Tables 2 Convex geometry Table 2.9.2.3.1, rank versus dimension of S3+ faces

118

Table 2.10.0.0.1, maximum number of c.i. directions

133

Cone Table 1

185

Cone Table S

186

Cone Table A

187

Cone Table 1*

192

4 Semidefinite programming Faces of S3+ corresponding to faces of S+3

268

5 Euclidean Distance Matrix Pr´ecis 5.7.2: affine dimension in terms of rank

426

B Simple matrices Auxiliary V -matrix Table B.4.4

622

D Matrix calculus Table D.2.1, algebraic gradients and derivatives

663

Table D.2.2, trace Kronecker gradients

664

Table D.2.3, trace gradients and derivatives

665

Table D.2.4, logarithmic determinant gradients, derivatives

667

Table D.2.5, determinant gradients and derivatives

668

Table D.2.6, logarithmic derivatives

668

Table D.2.7, exponential gradients and derivatives

669

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

19

20

Chapter 1 Overview Convex Optimization Euclidean Distance Geometry People are so afraid of convex analysis. −Claude Lemar´echal, 2003 In layman’s terms, the mathematical science of Optimization is the study of how to make a good choice when confronted with conflicting requirements. The qualifier convex means: when an optimal solution is found, then it is guaranteed to be a best solution; there is no better choice. Any convex optimization problem has geometric interpretation. If a given optimization problem can be transformed to a convex equivalent, then this interpretive benefit is acquired. That is a powerful attraction: the ability to visualize geometry of an optimization problem. Conversely, recent advances in geometry and in graph theory hold convex optimization within their proofs’ core. [369] [297] This book is about convex optimization, convex geometry (with particular attention to distance geometry), and nonconvex, combinatorial, and geometrical problems that can be relaxed or transformed into convex problems. A virtual flood of new applications follows by epiphany that many problems, presumed nonconvex, can be so transformed. [10] [11] [55] [88] [136] [138] [257] [276] [284] [336] [337] [366] [369] [32, 4.3, p.316-322]

§

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

21

22

CHAPTER 1. OVERVIEW

Figure 1: Orion nebula. (astrophotography by Massimo Robberto)

Euclidean distance geometry is, fundamentally, a determination of point conformation (configuration, relative position or location) by inference from interpoint distance information. By inference we mean: e.g., given only distance information, determine whether there corresponds a realizable conformation of points; a list of points in some dimension that attains the given interpoint distances. Each point may represent simply location or, abstractly, any entity expressible as a vector in finite-dimensional Euclidean space; e.g., distance geometry of music [98]. It is a common misconception to presume that some desired point conformation cannot be recovered in absence of complete interpoint distance information. We might, for example, want to realize a constellation given only interstellar distance (or, equivalently, distance from Earth and relative angular measurement; the Earth as vertex to two stars). At first it may seem O(N 2 ) data is required, yet there are many circumstances where this can be reduced to O(N ). If we agree that a set of points can have a shape (three points can form a triangle and its interior, for example, four points a tetrahedron), then

xˇ4

xˇ2

xˇ3

§

Figure 2: Application of trilateration ( 5.4.2.3.5) is localization (determining position) of a radio signal source in 2 dimensions; more commonly known by radio engineers as the process “triangulation”. In this scenario, anchors xˇ2 , xˇ3 , xˇ4 are illustrated as fixed antennae. [194] The radio signal source (a sensor • x1 ) anywhere in affine hull of three antenna bases can be uniquely localized by measuring distance to each (dashed white arrowed line segments). Ambiguity of lone distance measurement to sensor is represented by circle about each antenna. Trilateration is expressible as a semidefinite program; hence, a convex optimization problem. [298]

we can ascribe shape of a set of points to their convex hull. It should be apparent: from distance, these shapes can be determined only to within a rigid transformation (rotation, reflection, translation). Absolute position information is generally lost, given only distance information, but we can determine the smallest possible dimension in which an unknown list of points can exist; that attribute is their affine dimension (a triangle in any ambient space has affine dimension 2, for example). In circumstances where stationary reference points are also provided, it becomes possible to determine absolute position or location; e.g., Figure 2. 23

24

CHAPTER 1. OVERVIEW

Figure 3: [175] [106] Distance data collected via nuclear magnetic resonance (NMR) helped render this 3-dimensional depiction of a protein molecule. At the beginning of the 1980s, Kurt W¨ uthrich [Nobel laureate] developed an idea about how NMR could be extended to cover biological molecules such as proteins. He invented a systematic method of pairing each NMR signal with the right hydrogen nucleus (proton) in the macromolecule. The method is called sequential assignment and is today a cornerstone of all NMR structural investigations. He also showed how it was subsequently possible to determine pairwise distances between a large number of hydrogen nuclei and use this information with a mathematical method based on distance-geometry to calculate a three-dimensional structure for the molecule. −[260] [353] [169]

Geometric problems involving distance between points can sometimes be reduced to convex optimization problems. Mathematics of this combined study of geometry and optimization is rich and deep. Its application has already proven invaluable discerning organic molecular conformation by measuring interatomic distance along covalent bonds; e.g., Figure 3. [83] [326] [129] [43] Many disciplines have already benefitted and simplified consequent to this theory; e.g., distance-based pattern recognition (Figure 4), localization in wireless sensor networks by measurement of intersensor distance along channels of communication, [44, 5] [364] [42] wireless location of a radio-signal source such as a cell phone by multiple measurements of signal strength, the global positioning system (GPS), and multidimensional scaling which is a numerical representation of qualitative data by finding a low-dimensional scale.

§

25

Figure 4: This coarsely discretized triangulated algorithmically flattened human face (made by Kimmel & the Bronsteins [210]) represents a stage in machine recognition of human identity; called face recognition. Distance geometry is applied to determine discriminating-features.

Euclidean distance geometry together with convex optimization have also found application in artificial intelligence:

ˆ to machine learning by discerning naturally occurring manifolds in: – Euclidean bodies (Figure 5, §6.4.0.0.1), – Fourier spectra of kindred utterances [197],

ˆ to

– and photographic image sequences [347],

robotics; e.g., automatic control of vehicles maneuvering in formation. (Figure 8)

by chapter We study pervasive convex Euclidean bodies; their many manifestations and representations. In particular, we make convex polyhedra, cones, and dual cones visceral through illustration in chapter 2 Convex geometry where geometric relation of polyhedral cones to nonorthogonal bases

26

CHAPTER 1. OVERVIEW

O

Figure 5: Swiss roll from Weinberger & Saul [347]. The problem of manifold learning, illustrated for N = 800 data points sampled from a “Swiss roll” 1 . A discretized manifold is revealed by connecting each data point and its k =6 nearest neighbors 2 . An unsupervised learning algorithm unfolds the Swiss roll while preserving the local geometry of nearby data points 3 . Finally, the data points are projected onto the two dimensional subspace that maximizes their variance, yielding a faithful embedding of the original manifold 4 .

O

O

O

27 (biorthogonal expansion) is examined. It is shown that coordinates are unique in any conic system whose basis cardinality equals or exceeds spatial dimension. The conic analogue to linear independence, called conic independence, is introduced as a tool for study, analysis, and manipulation of cones; a natural extension and next logical step in progression: linear, affine, conic. We explain conversion between halfspace- and vertex-description of a convex cone, we motivate the dual cone and provide formulae for finding it, and we show how first-order optimality conditions or alternative systems of linear inequality or linear matrix inequality can be explained by dual generalized inequalities with respect to convex cones. Arcane theorems of alternative generalized inequality are, in fact, simply derived from cone membership relations; generalizations of algebraic Farkas’ lemma translated to geometry of convex cones. Any convex optimization problem can be visualized geometrically. Desire to visualize in high dimension [Sagan, Cosmos −The Edge of Forever, 22:55′ ] is deeply embedded in the mathematical psyche. [1] Chapter 2 provides tools to make visualization easier, and we teach how to visualize in high dimension. The concepts of face, extreme point, and extreme direction of a convex Euclidean body are explained here; crucial to understanding convex optimization. How to find the smallest face of any pointed closed convex cone containing convex set C , for example, is divulged; later shown to have practical application to presolving convex programs. The convex cone of positive semidefinite matrices, in particular, is studied in depth:

ˆ We interpret, for example, inverse image of the positive semidefinite cone under affine transformation. (Example 2.9.1.0.2) ˆ Subsets of the positive semidefinite cone, discriminated by rank exceeding some lower bound, are convex. In other words, high-rank subsets of the positive semidefinite cone boundary united with its interior are convex. (Theorem 2.9.2.6.3) There is a closed form for projection on those convex subsets.

ˆ The positive semidefinite cone is a circular cone in low dimension,

while Gerˇsgorin discs specify inscription of a polyhedral cone into that positive semidefinite cone. (Figure 46)

Chapter 3 Geometry of convex functions observes Fenchel’s analogy between convex sets and functions: We explain, for example, how the real affine function relates to convex functions as the hyperplane relates to convex

28

CHAPTER 1. OVERVIEW

original

reconstruction

Figure 6: About five thousand points along borders constituting United States were used to create an exhaustive matrix of interpoint distance for each and every pair of points in an ordered set (a list); called Euclidean distance matrix. From that noiseless distance information, it is easy to reconstruct map exactly via Schoenberg criterion (856). ( 5.13.1.0.1, confer Figure 126) Map reconstruction is exact (to within a rigid transformation) given any number of interpoint distances; the greater the number of distances, the greater the detail (as it is for all conventional map preparation).

§

sets. Partly a toolbox of practical useful convex functions and a cookbook for optimization problems, methods are drawn from the appendices about matrix calculus for determining convexity and discerning geometry. Semidefinite programming is reviewed in chapter 4 with particular attention to optimality conditions for prototypical primal and dual problems, their interplay, and the perturbation method of rank reduction of optimal solutions (extant but not well known). Positive definite Farkas’ lemma is derived, and we also show how to determine if a feasible set belongs exclusively to a positive semidefinite cone boundary. An arguably good three-dimensional polyhedral analogue to the positive semidefinite cone of 3 × 3 symmetric matrices is introduced: a new tool for visualizing coexistence of low- and high-rank optimal solutions in six isomorphic dimensions and a mnemonic aid for understanding semidefinite programs. We find a

29 minimum-cardinality Boolean solution to an instance of Ax = b : minimize x

kxk0

subject to Ax = b xi ∈ {0, 1} ,

(643) i=1 . . . n

The sensor-network localization problem is solved in any dimension in this chapter. We introduce a method of convex iteration for constraining rank in the form rank G ≤ ρ and cardinality in the form card x ≤ k . Cardinality minimization is applied to a discrete image-gradient of the Shepp-Logan phantom, from Magnetic Resonance Imaging (MRI) in the field of medical imaging, for which we find a new lower bound of 1.9% cardinality. We show how to handle polynomial constraints, and how to transform a rank constrained problem to a rank-1 problem. The EDM is studied in chapter 5 Euclidean Distance Matrix; its properties and relationship to both positive semidefinite and Gram matrices. We relate the EDM to the four classical properties of Euclidean metric; thereby, observing existence of an infinity of properties of the Euclidean metric beyond triangle inequality. We proceed by deriving the fifth Euclidean metric property and then explain why furthering this endeavor is inefficient because the ensuing criteria (while describing polyhedra in angle or area, volume, content, and so on ad infinitum) grow linearly in complexity and number with problem size. Reconstruction methods are explained and applied to a map of the United States; e.g., Figure 6. We also experimentally test a conjecture of Borg & Groenen by reconstructing a distorted but recognizable isotonic map of the USA using only ordinal (comparative) distance data: Figure 126(e)(f). We demonstrate an elegant method for including dihedral (or torsion) angle constraints into a molecular conformation problem. We explain why trilateration (a.k.a, localization) is a convex optimization problem. We show how to recover relative position given incomplete interpoint distance information, and how to pose EDM problems or transform geometrical problems to convex optimizations; e.g., kissing number of packed spheres about a central sphere (solved in R3 by Isaac Newton). The set of all Euclidean distance matrices forms a pointed closed convex cone called the EDM cone: EDMN . We offer a new proof of Schoenberg’s

30

CHAPTER 1. OVERVIEW

§

Figure 7: These bees construct a honeycomb by solving a convex optimization problem ( 5.4.2.3.4). The most dense packing of identical spheres about a central sphere in 2 dimensions is 6. Sphere centers describe a regular lattice.

seminal characterization of EDMs: D ∈ EDM

N

⇔

(

−VNTDVN º 0 D ∈ SN h

(856)

Our proof relies on fundamental geometry; assuming, any EDM must correspond to a list of points contained in some polyhedron (possibly at its vertices) and vice versa. It is known, but not obvious, this Schoenberg criterion implies nonnegativity of the EDM entries; proved herein. We characterize the eigenvalue spectrum of an EDM, and then devise a polyhedral spectral cone for determining membership of a given matrix (in Cayley-Menger form) to the convex cone of Euclidean distance matrices; id est, a matrix is an EDM if and only if its nonincreasingly ordered vector of eigenvalues belongs to a polyhedral spectral cone for EDMN ;  µ· ¸¶ · N ¸ 0 1T R+   λ ∈ ∩ ∂H N 1 −D R− (1071) D ∈ EDM ⇔   D ∈ SN h We will see: spectral cones are not unique. In chapter 6 Cone of distance matrices we explain a geometric relationship between the cone of Euclidean distance matrices, two positive semidefinite cones, and the elliptope. We illustrate geometric requirements, in particular, for projection of a given matrix on a positive semidefinite cone that establish its membership to the EDM cone. The faces of the EDM cone are described, but still open is the question whether all its faces are exposed as they are for the positive semidefinite cone.

31

Figure 8: Robotic vehicles in concert can move larger objects, guard a perimeter, perform surveillance, find land mines, or localize a plume of gas, liquid, or radio waves. [128]

The Schoenberg criterion (856) for identifying a Euclidean distance matrix is revealed to be a discretized membership relation (dual generalized inequalities, a new Farkas’-like lemma) between the EDM cone and its ∗ ordinary dual: EDMN . A matrix criterion for membership to the dual EDM cone is derived that is simpler than the Schoenberg criterion: ∗

D∗ ∈ EDMN ⇔ δ(D∗ 1) − D∗ º 0

(1220)

There is a concise equality, relating the convex cone of Euclidean distance matrices to the positive semidefinite cone, apparently overlooked in the literature; an equality between two large convex Euclidean bodies: ¢ ¡ N⊥ N (1214) EDMN = SN ∩ S − S h c +

In chapter 7 Proximity problems we explore methods of solution to a few fundamental and prevalent Euclidean distance matrix proximity problems; the problem of finding that distance matrix closest, in some sense,

32

CHAPTER 1. OVERVIEW

to a given matrix H : minimize D

k−V (D − H)V k2F

subject to rank V D V ≤ ρ D ∈ EDMN minimize D

kD −

Hk2F

subject to rank V D V ≤ ρ D ∈ EDMN

minimize √ ◦ D

√ k ◦ D − Hk2F

subject to rank V D V ≤ ρ p √ ◦ D ∈ EDMN

minimize √ ◦ D

(1256) √ ◦ 2 k−V ( D − H)V kF

subject to rank V D V ≤ ρ p √ ◦ D ∈ EDMN

We apply a convex iteration method for constraining rank. Known heuristics for rank minimization are also explained. We offer new geometrical proof, of a famous discovery by Eckart & Young in 1936 [119], with particular regard to Euclidean projection of a point on that generally nonconvex subset of the positive semidefinite cone boundary comprising all semidefinite matrices having rank not exceeding a prescribed bound ρ . We explain how this problem is transformed to a convex optimization for any rank ρ .

appendices Provided so as to be more self-contained:

ˆ linear

algebra (appendix A is primarily concerned with proper statements of semidefiniteness for square matrices),

ˆ simple matrices (dyad, doublet, elementary, Householder, Schoenberg, orthogonal, etcetera, in appendix B),

ˆa

collection of known analytical solutions to some important optimization problems (appendix C),

ˆ matrix calculus remains somewhat unsystematized when compared

to ordinary calculus (appendix D concerns matrix-valued functions, matrix differentiation and directional derivatives, Taylor series, and tables of first- and second-order gradients and matrix derivatives),

33

ˆ an

elaborate exposition offering insight into orthogonal and nonorthogonal projection on convex sets (the connection between projection and positive semidefiniteness, for example, or between projection and a linear objective function in appendix E),

ˆ Matlab code on Wıκımization to discriminate EDMs, to determine

conic independence, to reduce or constrain rank of an optimal solution to a semidefinite program, compressed sensing (compressive sampling) for digital image and audio signal processing, and two distinct methods of reconstructing a map of the United States: one given only distance data, the other given only comparative distance data.

34

CHAPTER 1. OVERVIEW

Figure 9: Three-dimensional reconstruction of David from distance data.

Figure 10: Digital Michelangelo Project, Stanford University. Measuring distance to David by laser rangefinder. Spatial resolution is 0.29mm.

Chapter 2 Convex geometry Convexity has an immensely rich structure and numerous applications. On the other hand, almost every “convex” idea can be explained by a two-dimensional picture. −Alexander Barvinok [26, p.vii]

As convex geometry and linear algebra are inextricably bonded by asymmetry (linear inequality), we provide much background material on linear algebra (especially in the appendices) although a reader is assumed comfortable with [309] [311] [188] or any other intermediate-level text. The essential references to convex analysis are [185] [285]. The reader is referred to [308] [26] [346] [39] [57] [282] [333] for a comprehensive treatment of convexity. There is relatively less published pertaining to matrix-valued convex functions. [200] [189, 6.6] [273]

§

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

35

36

2.1

CHAPTER 2. CONVEX GEOMETRY

Convex set

A set C is convex iff for all Y , Z ∈ C and 0 ≤ µ ≤ 1 µ Y + (1 − µ)Z ∈ C

(1)

Under that defining condition on µ , the linear sum in (1) is called a convex combination of Y and Z . If Y and Z are points in real finite-dimensional Euclidean vector space Rn or Rm×n (matrices), then (1) represents the closed line segment joining them. All line segments are thereby convex sets. Apparent from the definition, a convex set is a connected set. [240, 3.4, 3.5] [39, p.2] A convex set can, but does not necessarily, contain the origin 0. An ellipsoid centered at x = a (Figure 13, p.41), given matrix C ∈ Rm×n

§ §

{x ∈ Rn | kC(x − a)k2 = (x − a)T C TC(x − a) ≤ 1}

(2)

is a good icon for a convex set.2.1

2.1.1

subspace

A nonempty subset R of real Euclidean vector space Rn is called a subspace ( 2.5) if every vector2.2 of the form αx + βy , for α , β ∈ R , is in R whenever vectors x and y are. [232, 2.3] A subspace is a convex set containing the origin, by definition. [285, p.4] Any subspace is therefore open in the sense that it contains no boundary, but closed in the sense [240, 2]

§

§

§

R+R=R

(3)

R = −R

(4)

It is not difficult to show as is true for any subspace R , because x ∈ R ⇔ −x ∈ R . The intersection of an arbitrary collection of subspaces remains a subspace. Any subspace not constituting the entire ambient vector space Rn is a proper subspace; e.g.,2.3 any line through the origin in two-dimensional Euclidean space R2 . The vector space Rn is itself a conventional subspace, inclusively, [211, 2.1] although not proper.

§

This particular definition is slablike (Figure 11) in Rn when C has nontrivial nullspace. A vector is assumed, throughout, to be a column vector. 2.3 We substitute abbreviation e.g. in place of the Latin exempli gratia.

2.1

2.2

2.1. CONVEX SET

2.1.2

37

linear independence

Arbitrary given vectors in Euclidean space {Γi ∈ Rn , i = 1 . . . N } are linearly independent (l.i.) if and only if, for all ζ ∈ RN Γ1 ζ1 + · · · + ΓN −1 ζN −1 + ΓN ζN = 0

(5)

has only the trivial solution ζ = 0 ; in other words, iff no vector from the given set can be expressed as a linear combination of those remaining. 2.1.2.1

§

preservation

§

(confer 2.4.2.4, 2.10.1) Linear transformation preserves linear dependence. [211, p.86] Conversely, linear independence can be preserved under linear transformation. Given Y = [ y1 y2 · · · yN ] ∈ RN ×N , consider the mapping T (Γ) : Rn×N → Rn×N , Γ Y

(6)

whose domain is the set of all matrices Γ ∈ Rn×N holding a linearly independent set columnar. Linear independence of {Γyi ∈ Rn , i = 1 . . . N } demands, by definition, there exist no nontrivial solution ζ ∈ RN to Γy1 ζi + · · · + ΓyN −1 ζN −1 + ΓyN ζN = 0

(7)

By factoring out Γ , we see that triviality is ensured by linear independence of {yi ∈ RN }.

2.1.3

Orthant:

name given to a closed convex set that is the higher-dimensional generalization of quadrant from the classical Cartesian partition of R2 ; a Cartesian cone. The most common is the nonnegative orthant Rn+ or Rn×n + (analogue to quadrant I) to which membership denotes nonnegative vectoror matrix-entries respectively; e.g., n R+ , {x ∈ Rn | xi ≥ 0 ∀ i}

(8)

The nonpositive orthant Rn− or Rn×n (analogue to quadrant III) denotes − negative and 0 entries. Orthant convexity2.4 is easily verified by definition (1). 2.4

All orthants are self-dual simplicial cones. (§2.13.5.1, §2.12.3.1.1)

38

CHAPTER 2. CONVEX GEOMETRY

Figure 11: A slab is a convex Euclidean body infinite in extent but not affine. Illustrated in R2 , it may be constructed by intersecting two opposing halfspaces whose bounding hyperplanes are parallel but not coincident. Because number of halfspaces used in its construction is finite, slab is a polyhedron ( 2.12). (Cartesian axes and vector inward-normal to each halfspace-boundary are drawn for reference.)

§

2.1.4

affine set

A nonempty affine set (from the word affinity) is any subset of Rn that is a translation of some subspace. Any affine set is convex and open so contains no boundary: e.g., empty set ∅ , point, line, plane, hyperplane ( 2.4.2), subspace, etcetera. For some parallel 2.5 subspace M and any point x ∈ A

§

A is affine ⇔ A = x + M = {y | y − x ∈ M}

(9)

§

The intersection of an arbitrary collection of affine sets remains affine. The affine hull of a set C ⊆ Rn ( 2.3.1) is the smallest affine set containing it.

2.1.5

dimension

Dimension of an arbitrary set S is the dimension of its affine hull; [346, p.14] dim S , dim aff S = dim aff(S − s) ,

s∈ S

(10)

the same as dimension of the subspace parallel to that affine set aff S when nonempty. Hence dimension (of a set) is synonymous with affine dimension. [185, A.2.1] 2.5

Two affine sets are said to be parallel when one is a translation of the other. [285, p.4]

2.1. CONVEX SET

2.1.6

39

empty set versus empty interior

Emptiness ∅ of a set is handled differently than interior in the classical literature. It is common for a nonempty convex set to have empty interior; e.g., paper in the real world:

ˆ An ordinary flat sheet of paper is a nonempty convex set having empty interior in R3 but nonempty interior relative to its affine hull.

2.1.6.1

relative interior

§

Although it is always possible to pass to a smaller ambient Euclidean space where a nonempty set acquires an interior [26, II.2.3], we prefer the qualifier relative which is the conventional fix to this ambiguous terminology.2.6 So we distinguish interior from relative interior throughout: [308] [346] [333]

ˆ Classical interior int C is defined as a union of points: x is an interior

point of C ⊆ Rn if there exists an open ball of dimension n and nonzero radius centered at x that is contained in C .

ˆ Relative interior rel int C of a convex set C ⊆ R

n

is interior relative to

2.7

its affine hull.

Thus defined, it is common (though confusing) for int C the interior of C to be empty while its relative interior is not: this happens whenever dimension of its affine hull is less than dimension of the ambient space (dim aff C < n ; e.g., were C paper) or in the exception when C is a single point; [240, 2.2.1]

§

rel int{x} , aff{x} = {x} ,

int{x} = ∅ ,

x ∈ Rn

(11)

In any case, closure of the relative interior of a convex set C always yields closure of the set itself; rel int C = C (12) Closure is invariant to translation. If C is convex then rel int C and C are convex. [185, p.24] If C has nonempty interior, then rel int C = int C 2.6

(13)

Superfluous mingling of terms as in relatively nonempty set would be an unfortunate consequence. From the opposite perspective, some authors use the term full or full-dimensional to describe a set having nonempty interior. 2.7 Likewise for relative boundary (§2.6.1.3), although relative closure is superfluous. [185, §A.2.1]

40

CHAPTER 2. CONVEX GEOMETRY

(a) R2 (b)

(c) Figure 12: (a) Closed convex set. (b) Neither open, closed, or convex. Yet PSD cone can remain convex in absence of certain boundary components ( 2.9.2.6.3). Nonnegative orthant with origin excluded ( 2.6) and positive orthant with origin adjoined [285, p.49] are convex. (c) Open convex set.

§

§

Given the intersection of any convex set C with an affine set A rel int(C ∩ A) = rel int(C) ∩ A

2.1.7

§

(14)

classical boundary

(confer 2.6.1.3) Boundary of a set C is the closure of C less its interior;

§

∂ C = C \ int C

(15)

[51, 1.1] which follows from the fact int C = C and presumption of nonempty interior.2.8 Implications are: 2.8

Otherwise, for x ∈ Rn as in (11), [240, §2.1- §2.3] int{x} = ∅ = ∅

the empty set is both open and closed.

(16)

2.1. CONVEX SET

41 R

(a)

R2

(b)

R3

(c)

Figure 13: (a) Ellipsoid in R is a line segment whose boundary comprises two points. Intersection of line with ellipsoid in R , (b) in R2 , (c) in R3 . Each ellipsoid illustrated has entire boundary constituted by zero-dimensional faces; in fact, by vertices ( 2.6.1.0.1). Intersection of line with boundary is a point at entry to interior. These same facts hold in higher dimension.

§

ˆ int C = C \∂ C ˆ an open set has boundary defined although not contained in the set ˆ interior of an open set is equivalent to the set itself; from which an open set is defined: [240, p.109] C is open ⇔ int C = C

(17)

The set illustrated in Figure 12b is not open because it is not equivalent to its interior, for example, it is not closed because it does not contain its boundary, and it is not convex because it does not contain all convex combinations of its boundary points. 2.1.7.1

Line intersection with boundary

A line can intersect the boundary of a convex set in any dimension at a point demarcating the line’s entry to the set interior. On one side of that

42

CHAPTER 2. CONVEX GEOMETRY

entry-point along the line is the exterior of the set, on the other side is the set interior. In other words,

ˆ starting from any point of a convex set, a move toward the interior is an immediate entry into the interior. [26, §II.2]

When a line intersects the interior of a convex body in any dimension, the boundary appears to the line to be as thin as a point. This is intuitively plausible because, for example, a line intersects the boundary of the ellipsoids in Figure 13 at a point in R , R2 , and R3 . Such thinness is a remarkable fact when pondering visualization of convex polyhedra ( 2.12, 5.14.3) in four dimensions, for example, having boundaries constructed from other three-dimensional convex polyhedra called faces. We formally define face in ( 2.6). For now, we observe the boundary of a convex body to be entirely constituted by all its faces of dimension lower than the body itself. For example: The ellipsoids in Figure 13 have boundaries composed only of zero-dimensional faces. The two-dimensional slab in Figure 11 is an unbounded polyhedron having one-dimensional faces making its boundary. The three-dimensional bounded polyhedron in Figure 20 has zero-, one-, and two-dimensional polygonal faces constituting its boundary.

§

§

§

2.1.7.1.1 Example. Intersection of line with boundary in R6 . The convex cone of positive semidefinite matrices S3+ ( 2.9) in the ambient subspace of symmetric matrices S3 ( 2.2.2.0.1) is a six-dimensional Euclidean body in isometrically isomorphic R6 ( 2.2.1). The boundary of the positive semidefinite cone in this dimension comprises faces having only the dimensions 0, 1, and 3 ; id est, {ρ(ρ + 1)/2, ρ = 0, 1, 2}. Unique minimum-distance projection P X ( E.9) of any point X ∈ S3 on that cone S3+ is known in closed form ( 7.1.2). Given, for example, λ ∈ int R3+ and diagonalization ( A.5.1) of exterior point   λ1 0  λ2 (18) X = QΛQT ∈ S3 , Λ ,  0 −λ3

§

§

§

§

§

§

where Q ∈ R3×3 is an orthogonal matrix, then the projection on S3+ in R6 is   λ1 0 QT ∈ S3+ λ2 P X = Q (19) 0 0

2.1. CONVEX SET

43

This positive semidefinite matrix P X nearest X thus has rank 2, found by discarding all negative eigenvalues. The line connecting these two points is {X + (P X − X)t | t ∈ R} where t = 0 ⇔ X and t = 1 ⇔ P X . Because this line intersects the boundary of the positive semidefinite cone S3+ at point P X and passes through its interior (by assumption), then the matrix corresponding to an infinitesimally positive perturbation of t there should reside interior to the cone (rank 3). Indeed, for ε an arbitrarily small positive constant,   λ1 0 QT ∈ int S3+ λ2 X + (P X − X)t|t=1+ε = Q(Λ+(P Λ−Λ)(1+ε))QT = Q 0 ελ3 (20) 2 2.1.7.2

Tangential line intersection with boundary

A higher-dimensional boundary ∂ C of a convex Euclidean body C is simply a dimensionally larger set through which a line can pass when it does not intersect the body’s interior. Still, for example, a line existing in five or more dimensions may pass tangentially (intersecting no point interior to C [204, 15.3]) through a single point relatively interior to a three-dimensional face on ∂ C . Let’s understand why by inductive reasoning. Figure 14(a) shows a vertical line-segment whose boundary comprises its two endpoints. For a line to pass through the boundary tangentially (intersecting no point relatively interior to the line-segment), it must exist in an ambient space of at least two dimensions. Otherwise, the line is confined to the same one-dimensional space as the line-segment and must pass along the segment to reach the end points. Figure 14(b) illustrates a two-dimensional ellipsoid whose boundary is constituted entirely by zero-dimensional faces. Again, a line must exist in at least two dimensions to tangentially pass through any single arbitrarily chosen point on the boundary (without intersecting the ellipsoid interior). Now let’s move to an ambient space of three dimensions. Figure 14(c) shows a polygon rotated into three dimensions. For a line to pass through its zero-dimensional boundary (one of its vertices) tangentially, it must exist in at least the two dimensions of the polygon. But for a line to pass tangentially through a single arbitrarily chosen point in the relative interior

§

44

CHAPTER 2. CONVEX GEOMETRY

(a)

(b) 2

R

R3

(c)

(d)

Figure 14: Line tangential: (a) (b) to relative interior of a zero-dimensional face in R2 , (c) (d) to relative interior of a one-dimensional face in R3 .

2.1. CONVEX SET

45

of a one-dimensional face on the boundary as illustrated, it must exist in at least three dimensions. Figure 14(d) illustrates a solid circular cone (drawn truncated) whose one-dimensional faces are halflines emanating from its pointed end (vertex ). This cone’s boundary is constituted solely by those one-dimensional halflines. A line may pass through the boundary tangentially, striking only one arbitrarily chosen point relatively interior to a one-dimensional face, if it exists in at least the three-dimensional ambient space of the cone. From these few examples, way deduce a general rule (without proof):

ˆ A line may pass tangentially through a single arbitrarily chosen point

relatively interior to a k-dimensional face on the boundary of a convex Euclidean body if the line exists in dimension at least equal to k + 2.

Now the interesting part, with regard to Figure 20 showing a bounded polyhedron in R3 ; call it P : A line existing in at least four dimensions is required in order to pass tangentially (without hitting int P ) through a single arbitrary point in the relative interior of any two-dimensional polygonal face on the boundary of polyhedron P . Now imagine that polyhedron P is itself a three-dimensional face of some other polyhedron in R4 . To pass a line tangentially through polyhedron P itself, striking only one point from its relative interior rel int P as claimed, requires a line existing in at least five dimensions.2.9 It is not too difficult to deduce:

ˆ A line may pass through a single arbitrarily chosen point interior to a

k-dimensional convex Euclidean body (hitting no other interior point) if that line exists in dimension at least equal to k + 1.

In layman’s terms, this means: a being capable of navigating four spatial dimensions (one dimension beyond our physical reality) could see inside three-dimensional objects. 2.9

This rule can help determine whether there exists unique solution to a convex optimization problem whose feasible set is an intersecting line; e.g., the trilateration problem (§5.4.2.3.5).

46

2.1.8

CHAPTER 2. CONVEX GEOMETRY

intersection, sum, difference, product

§

2.1.8.0.1 Theorem. Intersection. [285, 2, thm.6.5] Intersection of an arbitrary collection of convex sets {Ci } is convex. For a finite collection of N sets, a T necessarily nonempty intersection of relative TN interior i=1 rel int Ci = rel int N i=1 Ci equals relative interior of intersection. T T ⋄ And for a possibly infinite collection, Ci = Ci .

In converse this theorem is implicitly false in so far as a convex set can be formed by the intersection of sets that are not. Unions of convex sets are generally not convex. [185, p.22] Vector sum of two convex sets C1 and C2 is convex [185, p.24] C1 + C2 = {x + y | x ∈ C1 , y ∈ C2 }

(21)

but not necessarily closed unless at least one set is closed and bounded. By additive inverse, we can similarly define vector difference of two convex sets C1 − C2 = {x − y | x ∈ C1 , y ∈ C2 } (22)

which is convex. Applying this definition to nonempty convex set C1 self-difference C1 − C1 is generally nonempty, nontrivial, and convex; for any convex cone K , ( 2.7.2) the set K − K constitutes its affine [285, p.15] Cartesian product of convex sets ¸ ¾ · ½· ¸ C1 x C1 × C2 = | x ∈ C1 , y ∈ C2 = C2 y

§

, its e.g., hull.

(23)

remains convex. The converse also holds; id est, a Cartesian product is convex iff each set is. [185, p.23] Convex results are also obtained for scaling κ C of a convex set C , rotation/reflection Q C , or translation C + α ; each similarly defined. Given any operator T and convex set C , we are prone to write T (C) meaning T (C) , {T (x) | x ∈ C} (24) Given linear operator T , it therefore follows from (21),

T (C1 + C2 ) = {T (x + y) | x ∈ C1 , y ∈ C2 }

= {T (x) + T (y) | x ∈ C1 , y ∈ C2 } = T (C1 ) + T (C2 )

(25)

2.1. CONVEX SET

47

f (C) f (a)

C

f F

f −1 (F )

(b)

Figure 15: (a) Image of convex set in domain of any convex function f is convex, but there is no converse. (b) Inverse image under convex function f .

2.1.9

inverse image

§

While epigraph ( 3.6) of a convex function must be convex, it generally holds that inverse image (Figure 15) of a convex function is not. The most prominent examples to the contrary are affine functions ( 3.5):

§

2.1.9.0.1 Theorem. Inverse image. Let f be a mapping from Rp×k to Rm×n .

§

[285, 3]

ˆ The image of a convex set C under any affine function f (C) = {f (X) | X ∈ C} ⊆ Rm×n

(26)

is convex.

ˆ Inverse image of a convex set F , f −1 (F ) = {X | f (X) ∈ F} ⊆ Rp×k

(27)

a single- or many-valued mapping, under any affine function f is convex. ⋄

48

CHAPTER 2. CONVEX GEOMETRY A†Ax = xp

Rn R(AT )

xp {x}

xp = A† b

Axp = b

Rm R(A) b

Ax = b

0 x = xp + η η N (A)

0

Aη = 0

{b}

N (AT )

Figure 16:(confer Figure 153) Action of linear map represented by A ∈ Rm×n : Component of vector x in nullspace N (A) maps to origin while component in rowspace R(AT ) maps to range R(A). For any A ∈ Rm×n , AA†Ax = b ( E), and inverse image of b ∈ R(A) is a nonempty affine set: xp + N (A).

§

In particular, any affine transformation of an affine set remains affine. [285, p.8] Ellipsoids are invariant to any [sic] affine transformation. Although not precluded, this inverse image theorem does not require a uniquely invertible mapping f . Figure 16, for example, mechanizes inverse image under a general linear map. Example 2.9.1.0.2 and Example 3.6.0.0.2 offer further applications. Each converse of this two-part theorem is generally false; id est, given f affine, a convex image f (C) does not imply that set C is convex, and neither does a convex inverse image f −1 (F ) imply set F is convex. A counter-example, invalidating a converse, is easy to visualize when the affine function is an orthogonal projector [309] [232]:

§

2.1.9.0.2 Corollary. Projection on subspace.2.10 (1857) [285, 3] Orthogonal projection of a convex set on a subspace or nonempty affine set is another convex set. ⋄ 2.10

For hyperplane representations see §2.4.2. For projection of convex sets on hyperplanes see [346, §6.6]. A nonempty affine set is called an affine subset (§2.3.1.0.1). Orthogonal projection of points on affine subsets is reviewed in §E.4.

2.2. VECTORIZED-MATRIX INNER PRODUCT

49

Again, the converse is false. Shadows, for example, are umbral projections that can be convex when the body providing the shade is not.

2.2

Vectorized-matrix inner product

Euclidean space Rn comes equipped with a linear vector inner-product hy , zi , y Tz

(28)

§

We prefer those angle brackets to connote a geometric rather than algebraic perspective; e.g., vector y might represent a hyperplane normal ( 2.4.2). Two vectors are orthogonal (perpendicular ) to one another if and only if their inner product vanishes; y ⊥ z ⇔ hy , zi = 0

(29)

When orthogonal vectors each have unit norm, then they are orthonormal. A vector inner-product defines Euclidean norm (vector 2-norm) p kyk = 0 ⇔ y = 0 (30) kyk2 = kyk , y T y ,

§

For linear operation A on a vector, represented by a real matrix, the adjoint operation AT is transposition and defined for matrix A by [211, 3.10] hy , ATzi , hAy , zi

(31)

The vector inner-product for matrices is calculated just as it is for vectors; by first transforming a matrix in Rp×k to a vector in Rpk by concatenating its columns in the natural order. For lack of a better term, we shall call that linear bijective (one-to-one and onto [211, App.A1.2]) transformation vectorization. For example, the vectorization of Y = [ y1 y2 · · · yk ] ∈ Rp×k [152] [305] is   y1  y2  pk  vec Y ,  (32)  ...  ∈ R yk

§

§

§

§

Then the vectorized-matrix inner-product is trace of matrix inner-product; for Z ∈ Rp×k , [57, 2.6.1] [185, 0.3.1] [357, 8] [340, 2.2] hY , Z i , tr(Y TZ) = vec(Y )T vec Z

(33)

50

CHAPTER 2. CONVEX GEOMETRY

§

where ( A.1.1) tr(Y TZ) = tr(Z Y T ) = tr(Y Z T ) = tr(Z T Y ) = 1T (Y ◦ Z)1

§

(34)

and where ◦ denotes the Hadamard product 2.11 of matrices [146, 1.1.4]. The adjoint operation AT on a matrix can therefore be defined in like manner: hY , ATZi , hAY , Z i

(35)

Take any element C1 from a matrix-valued set in Rp×k , for example, and consider any particular dimensionally compatible real vectors v and w . Then vector inner-product of C1 with vwT is ¡ ¢ hvwT , C1 i = hv , C1 wi = v TC1 w = tr(wv T C1 ) = 1T (vwT )◦ C1 1 (36)

Further, linear bijective vectorization is distributive with respect to Hadamard product; id est, vec(Y ◦ Z) = vec(Y ) ◦ vec(Z )

(37)

2.2.0.0.1 Example. Application of inverse image theorem. Suppose set C ⊆ Rp×k were convex. Then for any particular vectors v ∈ Rp and w ∈ Rk , the set of vector inner-products Y , v TCw = hvwT , Ci ⊆ R

(38)

is convex. It is easy to show directly that convex combination of elements from Y remains an element of Y .2.12 Instead given convex set Y , C must be convex consequent to inverse image theorem 2.1.9.0.1. 2.11

Hadamard product is a simple entrywise product of corresponding entries from two matrices of like size; id est, not necessarily square. A commutative operation, the Hadamard product can be extracted from within a Kronecker product. [188, p.475] 2.12 To verify that, take any two elements C1 and C2 from the convex matrix-valued set C , and then form the vector inner-products (38) that are two elements of Y by definition. Now make a convex combination of those inner products; videlicet, for 0 ≤ µ ≤ 1 µ hvwT , C1 i + (1 − µ) hvwT , C2 i = hvwT , µ C1 + (1 − µ)C2 i The two sides are equivalent by linearity of inner product. The right-hand side remains a vector inner-product of vwT with an element µ C1 + (1 − µ)C2 from the convex set C ; hence, it belongs to Y . Since that holds true for any two elements from Y , then it must be a convex set. ¨

2.2. VECTORIZED-MATRIX INNER PRODUCT R2

51 R3

(a)

(b)

Figure 17: (a) Cube in R3 projected on paper-plane R2 . Subspace projection operator is not an isomorphism because new adjacencies are introduced. (b) Tesseract is a projection of hypercube in R4 on R3 .

More generally, vwT in (38) may be replaced with any particular matrix Z ∈ Rp×k while convexity of set hZ , Ci ⊆ R persists. Further, by replacing v and w with any particular respective matrices U and W of dimension compatible with all elements of convex set C , then set U TCW is convex by the inverse image theorem because it is a linear mapping of C . 2

2.2.1

Frobenius’

2.2.1.0.1 Definition. Isomorphic. An isomorphism of a vector space is a transformation equivalent to a linear bijective mapping. Image and inverse image under the transformation operator are then called isomorphic vector spaces. △ Isomorphic vector spaces are characterized by preservation of adjacency; id est, if v and w are points connected by a line segment in one vector space, then their images will be connected by a line segment in the other. Two Euclidean bodies may be considered isomorphic if there exists an isomorphism, of their vector spaces, under which the bodies correspond. [342, I.1] Projection ( E) is not an isomorphism, Figure 17 for example; hence, perfect reconstruction (inverse projection) is generally impossible without additional information. When Z = Y ∈ Rp×k in (33), Frobenius’ norm is resultant from vector

§

§

52

CHAPTER 2. CONVEX GEOMETRY

inner-product; (confer (1613)) kY k2F = k vec Y k22 = hY , Y i = tr(Y T Y ) P P P 2 σ(Y )2i λ(Y T Y )i = Yij = = i, j

(39)

i

i

§

where λ(Y T Y )i is the i th eigenvalue of Y T Y , and σ(Y )i the i th singular value of Y . Were Y a normal matrix ( A.5.1), then σ(Y ) = |λ(Y )| [368, 8.1] thus P λ(Y )2i = kλ(Y )k22 = hλ(Y ) , λ(Y )i = hY , Y i kY k2F = (40)

§

§

i

The converse (40) ⇒ normal matrix Y also holds. [188, 2.5.4] Frobenius norm is the Euclidean norm of vectorized matrices. Because the metrics are equivalent, for X ∈ Rp×k k vec X − vec Y k2 = kX −Y kF

(41)

and because vectorization (32) is a linear bijective map, then vector space Rp×k is isometrically isomorphic with vector space Rpk in the Euclidean sense and vec is an isometric isomorphism2.13 of Rp×k . Because of this Euclidean structure, all the known results from convex analysis in Euclidean space Rn carry over directly to the space of real matrices Rp×k . 2.2.1.1

Injective linear operators

Injective mapping (transformation) means one-to-one mapping; synonymous with uniquely invertible linear mapping on Euclidean space. Linear injective mappings are fully characterized by lack of a nontrivial nullspace. 2.2.1.1.1 Definition. Isometric isomorphism. An isometric isomorphism of a vector space having a metric defined on it is a linear bijective mapping T that preserves distance; id est, for all x, y ∈ dom T kT x − T yk = kx − yk Then the isometric isomorphism T is a bijective isometry. 2.13

(42) △

Given matrix A , its range R(A) (§2.5) is isometrically isomorphic with its vectorized range vec R(A) but not with R(vec A).

2.2. VECTORIZED-MATRIX INNER PRODUCT

53

T

R3

R2

dim dom T = dim R(T ) Figure 18: Linear injective mapping T x = Ax : R2 → R3 of Euclidean body remains two-dimensional under mapping represented by skinny full-rank matrix A ∈ R3×2 ; two bodies are isomorphic by Definition 2.2.1.0.1. Unitary linear operator Q : Rk → Rk , represented by orthogonal matrix Q ∈ Rk×k ( B.5), is an isometric isomorphism; e.g., discrete Fourier transform via (784). Suppose T (X) = U XQ , for example. Then we say the Frobenius norm is orthogonally invariant; meaning, for X, Y ∈ Rp×k and dimensionally compatible orthonormal matrix 2.14 U and orthogonal matrix Q

§

kU (X −Y )QkF = kX −Y kF

(43)

 1 0 Yet isometric operator T : R2 → R3 , represented by A =  0 1  on R2 , 0 0 3 is injective but not a surjective map to R . [211, 1.6, 2.6] This operator T can therefore be a bijective isometry only with respect to its range. Any linear injective transformation on Euclidean space is uniquely invertible on its range. In fact, any linear injective transformation has a range whose dimension equals that of its domain. In other words, for any invertible linear transformation T [ibidem]

§ §

dim dom(T ) = dim R(T )



(44)

Any matrix U whose columns are orthonormal with respect to each other (U T U = I ); these include the orthogonal matrices. 2.14

54

CHAPTER 2. CONVEX GEOMETRY PT 3

R3

R

B

P T (B)

x

PTx

Figure 19: Linear noninjective mapping P T x = A†Ax : R3 → R3 of three-dimensional Euclidean body B has affine dimension 2 under projection on rowspace of fat full-rank matrix A ∈ R2×3 . Set of coefficients of orthogonal projection T B = {Ax | x ∈ B} is isomorphic with projection P (T B) [sic]. e.g., T represented by skinny-or-square full-rank matrices. (Figure 18) An important consequence of this fact is:

ˆ Affine dimension, of any n-dimensional Euclidean body in domain of operator T , is invariant to linear injective transformation.

2.2.1.2

Noninjective linear operators

§

Mappings in Euclidean space created by noninjective linear operators can be characterized in terms of an orthogonal projector ( E). Consider noninjective linear operator T x = Ax : Rn → Rm represented by fat matrix A ∈ Rm×n (m < n). What can be said about the nature of this m-dimensional mapping? Concurrently, consider injective linear operator P y = A† y : Rm → Rn where R(A† ) = R(AT ). P (Ax) = P T x achieves projection of vector x on the row space R(AT ). ( E.3.1) This means vector Ax can be succinctly interpreted as coefficients of orthogonal projection. Pseudoinverse matrix A† is skinny and full-rank, so operator P y is a linear bijection with respect to its range R(A† ). By Definition 2.2.1.0.1, image P (T B) of projection P T (B) on R(AT ) in Rn must therefore be isomorphic

§

2.2. VECTORIZED-MATRIX INNER PRODUCT

55

with the set of projection coefficients T B = {Ax | x ∈ B} in Rm and have the same affine dimension by (44). To illustrate, we present a three-dimensional Euclidean body B in Figure 19 where any point x in the nullspace N (A) maps to the origin.

2.2.2

Symmetric matrices

2.2.2.0.1 Definition. Symmetric matrix subspace. Define a subspace of RM ×M : the convex set of all symmetric M×M matrices; © ª SM , A ∈ RM ×M | A = AT ⊆ RM ×M (45)

This subspace comprising symmetric matrices SM is isomorphic with the vector space RM (M +1)/2 whose dimension is the number of free variables in a symmetric M ×M matrix. The orthogonal complement [309] [232] of SM is © ª SM ⊥ , A ∈ RM ×M | A = −AT ⊂ RM ×M (46) the subspace of antisymmetric matrices in RM ×M ; id est, SM ⊕ SM ⊥ = RM ×M where unique vector sum ⊕ is defined on page 742.

(47) △

All antisymmetric matrices are hollow by definition (have 0 main diagonal). Any square matrix A ∈ RM ×M can be written as a sum of its symmetric and antisymmetric parts: respectively, 1 1 A = (A + AT ) + (A − AT ) 2 2

(48)

2

The symmetric part is orthogonal in RM to the antisymmetric part; videlicet, ¡ ¢ tr (A + AT )(A − AT ) = 0 (49) In the ambient space of real matrices, the antisymmetric matrix subspace can be described ½ ¾ 1 M⊥ M ×M T S = ⊂ RM ×M (50) (A − A ) | A ∈ R 2

because any matrix in SM is orthogonal to any matrix in SM ⊥ . Further confined to the ambient subspace of symmetric matrices, because of antisymmetry, SM ⊥ would become trivial.

56 2.2.2.1

CHAPTER 2. CONVEX GEOMETRY Isomorphism of symmetric matrix subspace

When a matrix is symmetric in SM , we may still employ the vectorization 2 transformation (32) to RM ; vec , an isometric isomorphism. We might instead choose to realize in the lower-dimensional subspace RM (M +1)/2 by ignoring redundant entries (below the main diagonal) during transformation. Such a realization would remain isomorphic but not isometric. Lack of 2 isometry is a spatial distortion due now to disparity in metric between RM and RM (M +1)/2 . To realize isometrically in RM (M +1)/2 , we must make a correction: For Y = [Yij ] ∈ SM we introduce the symmetric vectorization 

Y11  √2Y12   Y22 √  2Y 13 svec Y ,   √2Y  23  Y  33 ..  . YMM



      ∈ RM (M +1)/2     

(51)

where all entries off the main diagonal have been scaled. Now for Z ∈ SM hY , Zi , tr(Y TZ) = vec(Y )T vec Z = svec(Y )T svec Z

(52)

Then because the metrics become equivalent, for X ∈ SM k svec X − svec Y k2 = kX − Y kF

(53)

and because symmetric vectorization (51) is a linear bijective mapping, then svec is an isometric isomorphism of the symmetric matrix subspace. In other words, SM is isometrically isomorphic with RM (M +1)/2 in the Euclidean sense under transformation svec . The set of all symmetric matrices SM forms a proper subspace in RM ×M , so for it there exists a standard orthonormal basis in isometrically isomorphic RM (M +1)/2    ei eT i = j = 1...M  i , ´ ³ {Eij ∈ SM } = (54) T  √1 ei eT , 1≤i<j≤M  + e e j j i 2

2.2. VECTORIZED-MATRIX INNER PRODUCT

57

where M (M + 1)/2 standard basis matrices Eij are formed from the standard basis vectors ·½ ¸ 1, i = j ei = , j = 1 . . . M ∈ RM (55) 0 , i 6= j Thus we have a basic orthogonal expansion for Y ∈ SM Y =

j M X X j=1 i=1

hEij , Y i Eij

(56)

whose coefficients hEij , Y i =

(

Yii , i = 1...M √ Yij 2 , 1 ≤ i < j ≤ M

(57)

correspond to entries of the symmetric vectorization (51).

2.2.3

Symmetric hollow subspace

2.2.3.0.1 Definition. Hollow subspaces. [327] M ×M Define a subspace of R : the convex set of all (real) symmetric M ×M matrices having 0 main diagonal; © ª ×M RM , A ∈ RM ×M | A = AT , δ(A) = 0 ⊂ RM ×M (58) h

§

where the main diagonal of A ∈ RM ×M is denoted ( A.1) δ(A) ∈ RM

(1359)

Operating on a vector, linear operator δ naturally returns a diagonal matrix; δ(δ(A)) is a diagonal matrix. Operating recursively on a vector Λ ∈ RN or diagonal matrix Λ ∈ SN , operator δ(δ(Λ)) returns Λ itself; δ 2(Λ) ≡ δ(δ(Λ)) = Λ

(1361)

×M The subspace RM (58) comprising (real) symmetric hollow matrices is h isomorphic with subspace RM (M −1)/2 ; its orthogonal complement is © ª RhM ×M ⊥ , A ∈ RM ×M | A = −AT + 2δ 2(A) ⊆ RM ×M (59)

58

CHAPTER 2. CONVEX GEOMETRY

the subspace of antisymmetric antihollow matrices in RM ×M ; id est, ×M RM ⊕ RhM ×M ⊥ = RM ×M h

Yet defined instead as a proper subspace of ambient SM © ª ×M SM A ∈ SM | δ(A) = 0 ≡ RM ⊂ SM h , h

⊥ the orthogonal complement SM of symmetric hollow subspace SM h h , © ª ⊥ SM , A ∈ SM | A = δ 2(A) ⊆ SM h

(60)

(61)

(62)

called symmetric antihollow subspace, is simply the subspace of diagonal matrices; id est, M⊥ SM = SM (63) h ⊕ Sh △ Any matrix A ∈ RM ×M can be written as a sum of its symmetric hollow and antisymmetric antihollow parts: respectively, ¶ µ ¶ µ 1 1 T 2 T 2 (A + A ) − δ (A) + (A − A ) + δ (A) (64) A= 2 2 The symmetric hollow part is orthogonal to the antisymmetric antihollow 2 part in RM ; videlicet, µµ ¶µ ¶¶ 1 1 T 2 T 2 tr (A + A ) − δ (A) (A − A ) + δ (A) = 0 (65) 2 2 ×M because any matrix in subspace RM is orthogonal to any matrix in the h antisymmetric antihollow subspace ¾ ½ 1 M ×M M ×M ⊥ T 2 (A − A ) + δ (A) | A ∈ R ⊆ RM ×M (66) Rh = 2

of the ambient space of real matrices; which reduces to the diagonal matrices in the ambient space of symmetric matrices © ª © ª ⊥ SM = δ 2(A) | A ∈ SM = δ(u) | u ∈ RM ⊆ SM (67) h

§

In anticipation of their utility with Euclidean distance matrices (EDMs) in 5, for symmetric hollow matrices we introduce the linear bijective

2.3. HULLS

59

vectorization dvec that is the natural analogue to symmetric matrix vectorization svec (51): for Y = [Yij ] ∈ SM h   Y12  Y13     Y23    √  Y  14  ∈ RM (M −1)/2 dvec Y , 2 (68)  Y  24    Y  34   .   .. YM −1,M

Like svec (51), dvec is an isometric isomorphism on the symmetric hollow subspace. For X ∈ SM h k dvec X − dvec Y k2 = kX − Y kF

(69)

The set of all symmetric hollow matrices SM h forms a proper subspace M ×M in R , so for it there must be a standard orthonormal basis in isometrically isomorphic RM (M −1)/2 ½ ¾ ¢ 1 ¡ T M T {Eij ∈ Sh } = √ ei ej + ej ei , 1 ≤ i < j ≤ M (70) 2

where M (M − 1)/2 standard basis matrices Eij are formed from the standard basis vectors ei ∈ RM . The symmetric hollow majorization corollary A.1.2.0.2 characterizes eigenvalues of symmetric hollow matrices.

2.3

Hulls

We focus on the affine, convex, and conic hulls: convex sets that may be regarded as kinds of Euclidean container or vessel united with its interior.

2.3.1

Affine hull, affine dimension

Affine dimension of any set in Rn is the dimension of the smallest affine set (empty set, point, line, plane, hyperplane ( 2.4.2), translated subspace, Rn )

§

60

CHAPTER 2. CONVEX GEOMETRY

Figure 20: Convex hull of a random list of points in R3 . Some points from that generating list reside interior to this convex polyhedron ( 2.12). [348, Convex Polyhedron] (Avis-Fukuda-Mizukoshi)

§

§

§

that contains it. For nonempty sets, affine dimension is the same as dimension of the subspace parallel to that affine set. [285, 1] [185, A.2.1] Ascribe the points in a list {xℓ ∈ Rn , ℓ = 1 . . . N } to the columns of matrix X : X = [ x1 · · · xN ] ∈ Rn×N (71) In particular, we define affine dimension r of the N -point list X as dimension of the smallest affine set in Euclidean space Rn that contains X ; r , dim aff X

(72)

Affine dimension r is a lower bound sometimes called embedding dimension. [327] [171] That affine set A in which those points are embedded is unique and called the affine hull [308, 2.1];

§

A , aff {xℓ ∈ Rn , ℓ = 1 . . . N } = aff X = x1 + R{xℓ − x1 , ℓ = 2 . . . N } = {Xa | aT 1 = 1} ⊆ Rn

(73)

for which we call list X a set of generators. Hull A is parallel to subspace R{xℓ − x1 , ℓ = 2 . . . N } = R(X − x1 1T ) ⊆ Rn

where

R(A) = {Ax | ∀ x}

(74)

(135)

Given some arbitrary set C and any x ∈ C

aff C = x + aff(C − x) where aff(C −x) is a subspace.

(75)

2.3. HULLS

61

2.3.1.0.1 Definition. Affine subset. We analogize affine subset to subspace,2.15 defining it to be any nonempty △ affine set ( 2.1.4).

§

aff ∅ , ∅

(76)

The affine hull of a point x is that point itself; aff{x} = {x}

(77)

The affine hull of two distinct points is the unique line through them. (Figure 21) The affine hull of three noncollinear points in any dimension is that unique plane containing the points, and so on. The subspace of symmetric matrices Sm is the affine hull of the cone of positive semidefinite matrices; ( 2.9)

§

m aff Sm + = S

(78)

2.3.1.0.2 Example. Affine hull of rank-1 correlation matrices. [203] The set of all m × m rank-1 correlation matrices is defined by all the binary vectors y in Rm (confer 5.9.1.0.1)

§

T {yy T ∈ Sm + | δ(yy ) = 1}

(79)

Affine hull of the rank-1 correlation matrices is equal to the set of normalized symmetric matrices; id est, m T aff{yy T ∈ Sm + | δ(yy ) = 1} = {A ∈ S | δ(A) = 1}

(80) 2

2.3.1.0.3 Exercise. Affine hull of correlation matrices. Prove (80) via definition of affine hull. Find the convex hull instead. H 2.15

The popular term affine subspace is an oxymoron.

62

CHAPTER 2. CONVEX GEOMETRY

A

affine hull (drawn truncated)

C convex hull

K conic hull (truncated)

range or span is a plane (truncated) R

Figure 21: Given two points in Euclidean vector space of any dimension, their various hulls are illustrated. Each hull is a subset of range; generally, A , C, K ⊆ R ∋ 0. (Cartesian axes drawn for reference.)

2.3. HULLS 2.3.1.1

63

M Partial order induced by RN + and S+

Notation a º 0 means vector a belongs to the nonnegative orthant RN + , while a ≻ 0 means vector a belongs to the nonnegative orthant’s interior int RN + , whereas a º b denotes comparison of vector a to vector b N on R with respect to the nonnegative orthant; id est, a º b means a − b belongs to the nonnegative orthant, but neither a or b necessarily belongs to that orthant. With particular respect to the nonnegative orthant, a º b ⇔ a i ≥ b i ∀ i (343). More generally, a ºK b or a ≻K b denotes comparison with respect to pointed closed convex cone K , but equivalence with entrywise comparison does not hold. ( 2.7.2.2) The symbol ≥ is reserved for scalar comparison on the real line R with respect to the nonnegative real line R+ as in aTy ≥ b . Comparison of matrices with respect to the positive semidefinite cone SM + , like I º A º 0 in Example 2.3.2.0.1, is explained in 2.9.0.1.

§

§

2.3.2

Convex hull

§

§

The convex hull [185, A.1.4] [285] of any bounded2.16 list or set of N points X ∈ Rn×N forms a unique bounded convex polyhedron (confer 2.12.0.0.1) whose vertices constitute some subset of that list; P , conv {xℓ , ℓ = 1 . . . N } = conv X = {Xa | aT 1 = 1, a º 0} ⊆ Rn (81) Union of relative interior and relative boundary ( 2.6.1.3) of the polyhedron comprise its convex hull P , the smallest closed convex set that contains the list X ; e.g., Figure 20. Given P , the generating list {xℓ } is not unique. But because every bounded polyhedron is the convex hull of its vertices, [308, 2.12.2] the vertices of P comprise a minimal set of generators. Given some arbitrary set C ⊆ Rn , its convex hull conv C is equivalent to the smallest convex set containing it. (confer 2.4.1.1.1) The convex hull is a subset of the affine hull;

§

§

§

conv C ⊆ aff C = aff C = aff C = aff conv C

(82)

An arbitrary set C in Rn is bounded iff it can be contained in a Euclidean ball having finite radius. [103, §2.2] (confer §5.7.3.0.1) The smallest ball containing C has radius inf supkx − yk and center x⋆ whose determination is a convex problem because supkx − yk 2.16

x y∈C

is a convex function of x ; but the supremum may be difficult to ascertain.

y∈C

64

CHAPTER 2. CONVEX GEOMETRY

Any closed bounded convex set C is equal to the convex hull of its boundary;

§

C = conv ∂ C

(83)

conv ∅ , ∅

(84)

§

§

2.3.2.0.1 Example. Hull of projection matrices. [131] [267] [11, 4.1] [271, 3] [222, 2.4] [223] Convex hull of the set comprising outer product of orthonormal matrices has equivalent expression: for 1 ≤ k ≤ N ( 2.9.0.1)

§

© ª © ª conv U U T | U ∈ RN ×k , U T U = I = A ∈ SN | I º A º 0 , hI , Ai = k ⊂ SN + (85) This important convex body we call Fantope (after mathematician Ky Fan). In case k = 1, there is slight simplification: ((1550), Example 2.9.2.4.1) © ª © ª conv U U T | U ∈ RN , U T U = 1 = A ∈ SN | A º 0 , hI , Ai = 1

(86)

In case k = N , the Fantope is identity matrix I . More generally, the set {U U T | U ∈ RN ×k , U T U = I }

(87)

§

comprises the extreme points ( 2.6.0.0.1) of its convex hull. By (1407), each and every extreme point U U T has only k nonzero eigenvalues λ and they all equal 1 ; id est, λ(U U T )1:k = λ(U T U ) = 1. So the Frobenius norm of each and every extreme point equals the same constant kU U T k2F = k

(88)

§

Each extreme point simultaneously lies on the boundary of the positive semidefinite cone (when k < N , 2.9) and qon the boundary of a hypersphere k 2

1 ) 2

of dimension k(N − + k(1 − Nk ) centered at Nk I along and radius the ray (base 0) through the identity matrix I in isomorphic vector space RN (N +1)/2 ( 2.2.2.1). Figure 22 illustrates extreme points (87) comprising the boundary of a Fantope, the boundary of a disc corresponding to k = 1, N = 2 ; but that 2 circumscription does not hold in higher dimension. ( 2.9.2.5)

§

§

2.3. HULLS

65

svec ∂ S2+ ·

I γ

α

√

2β

√ Figure 22: Two Fantopes. Circle, (radius 1/ 2) shown here on boundary of positive semidefinite cone S2+ in isometrically isomorphic R3 from Figure 41, comprises boundary of a Fantope (85) in this dimension (k = 1, N = 2). Lone point illustrated is identity matrix I , interior to PSD cone, and is that Fantope corresponding to k = 2, N = 2. (View is from inside PSD cone looking toward origin.)

α β β γ

¸

66

CHAPTER 2. CONVEX GEOMETRY

2.3.2.0.2 Example. Nuclear norm ball : convex hull of rank-1 matrices. From (86), in Example 2.3.2.0.1, we learn that the convex hull of normalized symmetric rank-1 matrices is a slice of the positive semidefinite cone. In 2.9.2.4 we find the convex hull of all symmetric rank-1 matrices to be the entire positive semidefinite cone. In the present example we abandon symmetry; instead posing, what is the convex hull of bounded nonsymmetric rank-1 matrices: X conv{uv T | kuv T k ≤ 1 , u ∈ Rm , v ∈ Rn } = {X ∈ Rm×n | σ(X)i ≤ 1} (89)

§

i

§

where σ(X) is a vector of singular values. (Since kuv T k = kukkvk (1542), norm of each vector constituting a dyad uv T ( B.1) in the hull is effectively bounded above by 1.)

§

P Proof. (⇐) Suppose σ(X)i ≤ 1. As in A.6, define a singular value decomposition: X = U ΣV T where U = [u1 . . . umin{m, n} ] ∈ Rm×min{m, n} , n×min{m, n} V , and whose sum singular values is P= [v1 . . . vmin{m, n} ] ∈ R P σiof √ √ T σ(X)i = tr Σ = κ ≤ 1. Then we may write X = κui κvi which is a κ convex combination of dyads each of whose norm does not exceed 1. (Srebro) (⇒) Now suppose we P are given a convex combination of dyads P T X = αi ui vi such that αi = 1, αi ≥ 0 ∀ i , and kui viT k ≤ 1 ∀ i . Then by Ptriangle inequality for P singular values [189, cor.3.4.3] P P T T σ(X)i ≤ σ(αi ui vi ) = αi kui vi k ≤ αi . ¨

Given any particular dyad uvpT in the convex hull, because its polar −uvpT and every convex combination of the two belong to that hull, then the unique line containing those two points ±uvpT (their affine combination (73)) must intersect the hull’s boundary at the normalized dyads {±uv T | kuv T k = 1}. Any point formed by convex combination of dyads in the hull must therefore be expressible as a convex combination of dyads on the boundary: Figure 23.

conv{uv T | kuv T k ≤ 1 , u ∈ Rm , v ∈ Rn } ≡ conv{uv T | kuv T k = 1 , u ∈ Rm , v ∈ Rn } (90) id est, dyads may be normalized and the hull’s boundary contains them; X ∂{X ∈ Rm×n | σ(X)i ≤ 1} ⊇ {uv T | kuv T k = 1 , u ∈ Rm , v ∈ Rn } (91) i

§

Normalized dyads constitute the set of extreme points ( 2.6.0.0.1) of this nuclear norm ball which is, therefore, their convex hull. 2

2.3. HULLS

67

kuv T k = 1

kxy T k = 1

uvpT

xypT kuv T k ≤ 1

{X ∈ Rm×n |

P i

σ(X)i ≤ 1}

0

−uvpT k−uv T k = 1

−xypT k−xy T k = 1

Figure 23: uvpT is a convex combination of normalized dyads k±uv T k = 1 ; similarly for xypT . Any point in line segment joining xypT to uvpT is expressible as a convex combination of two to four points indicated on boundary. Affine dimension of vectorized nuclear norm ball in X∈ R2×2 is 2 because (1542) σ1 = kukkvk = 1 by (90).

68

CHAPTER 2. CONVEX GEOMETRY

2.3.2.0.3 Exercise. Convex hull of outer product. Describe the interior of a Fantope. Find the convex hull of nonorthogonal projection matrices ( E.1.1):

§

{U V T | U ∈ RN ×k , V ∈ RN ×k , V T U = I }

(92)

Find the convex hull of nonsymmetric matrices bounded under some norm: {U V T | U ∈ Rm×k , V ∈ Rn×k , kU V T k ≤ 1}

(93) H

2.3.2.0.4 Example. Permutation polyhedron. [187] [295] [243] A permutation matrix Ξ is formed by interchanging rows and columns of identity matrix I . Since Ξ is square and ΞT Ξ = I , the set of all permutation matrices is a proper subset of the nonconvex manifold of orthogonal matrices ( B.5). In fact, the only orthogonal matrices having all nonnegative entries are permutations of the identity. Regarding the permutation matrices as a set of points in Euclidean space, its convex hull is a bounded polyhedron ( 2.12) described (Birkhoff, 1946)

§

§

conv{Ξ = Π i (I ∈ S n ) ∈ Rn×n , i = 1 . . . n!} = {X ∈ Rn×n | X T 1 = 1, X1 = 1, X ≥ 0} (94) th where Π i is a linear operator here representing the i permutation. This polyhedral hull, whose n! vertices are the permutation matrices, is also known as the set of doubly stochastic matrices. The permutation matrices are the minimum cardinality (fewest nonzero entries) doubly stochastic matrices. The only orthogonal matrices belonging to this polyhedron are the permutation matrices. It is remarkable that n! permutation matrices can be described as the extreme points ( 2.6.0.0.1) of a bounded polyhedron, of affine dimension (n−1)2 , that is itself described by only 2n equalities (2n−1 linearly independent equality constraints in nonnegative variables). By Carath´eodory’s theorem, conversely, any doubly stochastic matrix can be described as a convex combination of at most (n−1)2 + 1 permutation matrices. [188, 8.7] This polyhedron, then, can be a device for relaxing an integer, combinatorial, or Boolean optimization problem.2.17 [62] [262, 3.1] 2

§

§

2.17

§

Relaxation replaces an objective function with its convex envelope or expands a feasible set to one that is convex. Dantzig first showed in 1951 that, by this device, the so-called assignment problem can be formulated as a linear program. [294] [26, §II.5]

2.3. HULLS

69

§

2.3.2.0.5 Example. Convex hull of orthonormal matrices. [27, 1.2] n×k T Consider rank-k matrices U ∈ R such that U U = I . These are the orthonormal matrices; a closed bounded submanifold, of all orthogonal matrices, having dimension nk − 12 k(k + 1) [48]. Their convex hull is expressed, for 1 ≤ k ≤ n conv{U ∈ Rn×k | U T U = I } = {X ∈ Rn×k | kXk2 ≤ 1} = {X ∈ Rn×k | kX Tak ≤ kak ∀ a ∈ Rn }

(95)

When k = n , matrices U are orthogonal and the convex hull is called the spectral norm ball which is the set of all contractions. [189, p.158] [307, p.313] The orthogonal matrices then constitute the extreme points ( 2.6.0.0.1) of this hull. By Schur complement ( A.4), the spectral norm kXk2 constraining largest singular value σ(X)1 can be expressed as a semidefinite constraint ¸ · I X º0 (96) XT I

§

§

because of equivalence X TX ¹ I ⇔ σ(X) ¹ 1 with singular values. (1498) 2 (1394) (1395)

2.3.3

Conic hull

In terms of a finite-length point list (or set) arranged columnar in X ∈ Rn×N (71), its conic hull is expressed K , cone {xℓ , ℓ = 1 . . . N } = cone X = {Xa | a º 0} ⊆ Rn

§

(97)

§

id est, every nonnegative combination of points from the list. Conic hull of any finite-length list forms a polyhedral cone [185, A.4.3] ( 2.12.1.0.1; e.g., Figure 48a); the smallest closed convex cone ( 2.7.2) that contains the list. By convention, the aberration [308, 2.1]

§

cone ∅ , {0}

§

(98)

Given some arbitrary set C , it is apparent conv C ⊆ cone C

(99)

70

CHAPTER 2. CONVEX GEOMETRY

§

Figure 24: A simplicial cone ( 2.12.3.1.1) in R3 whose boundary is drawn truncated; constructed using A ∈ R3×3 and C = 0 in (261). By the most fundamental definition of a cone ( 2.7.1), entire boundary can be constructed from an aggregate of rays emanating exclusively from the origin. Each of three extreme directions corresponds to an edge ( 2.6.0.0.3); they are conically, affinely, and linearly independent for this cone. Because this set is polyhedral, exposed directions are in one-to-one correspondence with extreme directions; there are only three. Its extreme directions give rise to what is called a vertex-description of this polyhedral cone; simply, the conic hull of extreme directions. Obviously this cone can also be constructed by intersection of three halfspaces; hence the equivalent halfspace-description.

§

§

2.4. HALFSPACE, HYPERPLANE

2.3.4

71

Vertex-description

The conditions in (73), (81), and (97) respectively define an affine combination, convex combination, and conic combination of elements from the set or list. Whenever a Euclidean body can be described as some hull or span of a set of points, then that representation is loosely called a vertex-description and those points are called generators.

2.4

Halfspace, Hyperplane

A two-dimensional affine subset is called a plane. An (n − 1)-dimensional affine subset in Rn is called a hyperplane. [285] [185] Every hyperplane partially bounds a halfspace (which is convex but not affine).

2.4.1

Halfspaces H+ and H−

Euclidean space Rn is partitioned in two by any hyperplane ∂H ; id est, H− + H+ = Rn . The resulting (closed convex) halfspaces, both partially bounded by ∂H , may be described as an asymmetry H− = {y | aTy ≤ b} = {y | aT(y − yp ) ≤ 0} ⊂ Rn H+ = {y | aTy ≥ b} = {y | aT(y − yp ) ≥ 0} ⊂ Rn

(100) (101)

where nonzero vector a ∈ Rn is an outward-normal to the hyperplane partially bounding H− while an inward-normal with respect to H+ . For any vector y − yp that makes an obtuse angle with normal a , vector y will lie in the halfspace H− on one side (shaded in Figure 25) of the hyperplane while acute angles denote y in H+ on the other side. An equivalent more intuitive representation of a halfspace comes about when we consider all the points in Rn closer to point d than to point c or equidistant, in the Euclidean sense; from Figure 25, H− = {y | ky − dk ≤ ky − ck}

(102)

This representation, in terms of proximity, is resolved with the more conventional representation of a halfspace (100) by squaring both sides of the inequality in (102); ½ ¾ ½ ¶ ¾ µ kck2 − kdk2 c+d T T H− = y | (c − d) y ≤ = y | (c − d) y − ≤0 2 2 (103)

72

CHAPTER 2. CONVEX GEOMETRY

H+ a

c

yp y

d ∂H = {y | aT(y − yp ) = 0} = N (aT ) + yp

∆

H−

N (aT ) = {y | aTy = 0}

Figure 25: Hyperplane illustrated ∂H is a line partially bounding halfspaces H− = {y | aT(y − yp ) ≤ 0} and H+ = {y | aT(y − yp ) ≥ 0} in R2 . Shaded is a rectangular piece of semiinfinite H− with respect to which vector a is outward-normal to bounding hyperplane; vector a is inward-normal with respect to H+ . Halfspace H− contains nullspace N (aT ) (dashed line through origin) because aTyp > 0. Hyperplane, halfspace, and nullspace are each drawn truncated. Points c and d are equidistant from hyperplane, and vector c − d is normal to it. ∆ is distance from origin to hyperplane.

2.4. HALFSPACE, HYPERPLANE 2.4.1.1

73

PRINCIPLE 1: Halfspace-description of convex sets

§

§

§

The most fundamental principle in convex geometry follows from the geometric Hahn-Banach theorem [232, 5.12] [18, 1] [122, I.1.2] which guarantees any closed convex set to be an intersection of halfspaces.

§

§

2.4.1.1.1 Theorem. Halfspaces. [185, A.4.2(b)] [39, 2.4] A closed convex set in Rn is equivalent to the intersection of all halfspaces that contain it. ⋄ Intersection of multiple halfspaces in Rn may be represented using a matrix constant A \ Hi − = {y | ATy ¹ b} = {y | AT(y − yp ) ¹ 0} (104) i \ Hi + = {y | ATy º b} = {y | AT(y − yp ) º 0} (105) i

where b is now a vector, and the i th column of A is normal to a hyperplane ∂Hi partially bounding Hi . By the halfspaces theorem, intersections like this can describe interesting convex Euclidean bodies such as polyhedra and cones, giving rise to the term halfspace-description.

2.4.2

Hyperplane ∂H representations

Every hyperplane ∂H is an affine set parallel to an (n − 1)-dimensional subspace of Rn ; it is itself a subspace if and only if it contains the origin. dim ∂H = n − 1

(106)

so a hyperplane is a point in R , a line in R2 , a plane in R3 , and so on. Every hyperplane can be described as the intersection of complementary halfspaces; [285, 19]

§

∂H = H− ∩ H+ = {y | aTy ≤ b , aTy ≥ b} = {y | aTy = b}

(107)

a halfspace-description. Assuming normal a ∈ Rn to be nonzero, then any hyperplane in Rn can be described as the solution set to vector equation aTy = b (illustrated in Figure 25 and Figure 26 for R2 ); ∂H , {y | aTy = b} = {y | aT(y−yp ) = 0} = {Z ξ+yp | ξ ∈ Rn−1 } ⊂ Rn (108)

74

CHAPTER 2. CONVEX GEOMETRY

a=

1 −1

(a)

1

·

1 1

¸

1

b=

−1

·

−1 −1

¸

−1

1

−1

{y | bTy = −1}

{y | aTy = 1}

{y | bTy = 1}

{y | aTy = −1}

c= (c)

·

−1 1

¸

1

1

−1

(b)

−1

1

1 −1

−1

{y | d Ty = −1}

{y | cTy = 1} {y | cTy = −1}

d=

(d) ·

1 −1

¸

{y | d Ty = 1} e= −1

{y | eTy = −1}

·

1 0

1

¸ (e)

{y | eTy = 1}

Figure 26: (a)-(d) Hyperplanes in R2 (truncated). Movement in normal direction increases vector inner-product. This visual concept is exploited to attain analytical solution of linear programs; e.g., Example 2.4.2.6.2, Exercise 2.5.1.2.2, Example 3.5.0.0.2, [57, exer.4.8-exer.4.20]. Each graph is also interpretable as a contour plot of a real affine function of two variables as in Figure 70. (e) Ratio |β|/kαk from {x | αTx = β} represents radius of hypersphere about 0 supported by hyperplane whose normal is α .

2.4. HALFSPACE, HYPERPLANE

75

All solutions y constituting the hyperplane are offset from the nullspace of aT by the same constant vector yp ∈ Rn that is any particular solution to aTy = b ; id est, y = Z ξ + yp (109) where the columns of Z ∈ Rn×n−1 constitute a basis for the nullspace N (aT ) = {x ∈ Rn | aTx = 0} .2.18 Conversely, given any point yp in Rn , the unique hyperplane containing it having normal a is the affine set ∂H (108) where b equals aTyp and where a basis for N (aT ) is arranged in Z columnar. Hyperplane dimension is apparent from the dimensions of Z ; that hyperplane is parallel to the span of its columns. 2.4.2.0.1 Exercise. Hyperplane scaling. Given normal y , draw a hyperplane {x ∈ R2 | xTy = 1}. Suppose z = 12 y . On the same plot, draw the hyperplane {x ∈ R2 | xTz = 1}. Now suppose z = 2y , then draw the last hyperplane again with this new z . What is the apparent effect of scaling normal y ? H 2.4.2.0.2 Example. Distance from origin to hyperplane. Given the (shortest) distance ∆ ∈ R+ from the origin to a hyperplane having normal vector a , we can find its representation ∂H by dropping a perpendicular. The point thus found is the orthogonal projection of the origin on ∂H ( E.5.0.0.5), equal to a∆/kak if the origin is known a priori to belong to halfspace H− (Figure 25), or equal to −a∆/kak if the origin belongs to halfspace H+ ; id est, when H− ∋ 0 © ª © ª ∂H = y | aT (y − a∆/kak) = 0 = y | aTy = kak∆ (110)

§

or when H+ ∋ 0 © ª © ª ∂H = y | aT (y + a∆/kak) = 0 = y | aTy = −kak∆

(111)

Knowledge of only distance ∆ and normal a thus introduces ambiguity into the hyperplane representation. 2

We will later find this expression for y in terms of nullspace of aT (more generally, of matrix AT (137)) to be a useful device for eliminating affine equality constraints, much as we did here. 2.18

76

CHAPTER 2. CONVEX GEOMETRY

2.4.2.1

Matrix variable

Any halfspace in Rmn may be represented using a matrix variable. For variable Y ∈ Rm×n , given constants A ∈ Rm×n and b = hA , Yp i ∈ R H− = {Y ∈ Rmn | hA , Y i ≤ b} = {Y ∈ Rmn | hA , Y − Yp i ≤ 0} H+ = {Y ∈ Rmn | hA , Y i ≥ b} = {Y ∈ Rmn | hA , Y − Yp i ≥ 0}

§

(112) (113)

Recall vector inner-product from 2.2: hA , Y i = tr(AT Y ) = vec(A)T vec(Y ). Hyperplanes in Rmn may, of course, also be represented using matrix variables. ∂H = {Y | hA , Y i = b} = {Y | hA , Y − Yp i = 0} ⊂ Rmn

(114)

Vector a from Figure 25 is normal to the hyperplane illustrated. Likewise, nonzero vectorized matrix A is normal to hyperplane ∂H ; A ⊥ ∂H in Rmn 2.4.2.2

(115)

Vertex-description of hyperplane

Any hyperplane in Rn may be described as affine hull of a minimal set of points {xℓ ∈ Rn , ℓ = 1 . . . n} arranged columnar in a matrix X ∈ Rn×n : (73) ∂H = aff{xℓ ∈ Rn , ℓ = 1 . . . n} , = aff X ,

dim aff{xℓ ∀ ℓ} = n−1

(116)

dim aff X = n−1

= x1 + R{xℓ − x1 , ℓ = 2 . . . n} , dim R{xℓ − x1 , ℓ = 2 . . . n} = n−1 = x1 + R(X − x1 1T ) ,

dim R(X − x1 1T ) = n−1

where R(A) = {Ax | ∀ x} 2.4.2.3

(135)

Affine independence, minimal set

For any particular affine set, a minimal set of points constituting its vertex-description is an affinely independent generating set and vice versa. Arbitrary given points {xi ∈ Rn , i = 1 . . . N } are affinely independent (a.i.) if and only if, over all ζ ∈ RN Ä ζ T 1 = 1, ζ k = 0 (confer 2.1.2)

§

xi ζi + · · · + xj ζj − xk = 0 ,

i 6= · · · = 6 j 6= k = 1 . . . N

(117)

2.4. HALFSPACE, HYPERPLANE

77 A3

A2

A1

0 Figure 27: Any one particular point of three points illustrated does not belong to affine hull A i (i ∈ 1, 2, 3, each drawn truncated) of points remaining. Three corresponding vectors in R2 are, therefore, affinely independent (but neither linearly or conically independent).

has no solution ζ ; in words, iff no point from the given set can be expressed as an affine combination of those remaining. We deduce l.i. ⇒ a.i.

(118)

§

Consequently, {xi , i = 1 . . . N } is an affinely independent set if and only if {xi − x1 , i = 2 . . . N } is a linearly independent (l.i.) set.· [191,¸ 3] (Figure 27) X (for X ∈ Rn×N This is equivalent to the property that the columns of 1T as in (71)) form a linearly independent set. [185, A.1.3]

§

2.4.2.4

Preservation of affine independence

§

§

Independence in the linear ( 2.1.2.1), affine, and conic ( 2.10.1) senses can be preserved under linear transformation. Suppose a matrix X ∈ Rn×N (71) holds an affinely independent set in its columns. Consider a transformation on the domain of such matrices T (X) : Rn×N → Rn×N , XY

(119)

where fixed matrix Y , [ y1 y2 · · · yN ] ∈ RN ×N represents linear operator T . Affine independence of {Xyi ∈ Rn , i = 1 . . . N } demands (by definition (117))

78

CHAPTER 2. CONVEX GEOMETRY

there exist no solution ζ ∈ RN Ä ζ T 1 = 1, ζ k = 0, to Xyi ζi + · · · + Xyj ζj − Xyk = 0 ,

i 6= · · · = 6 j 6= k = 1 . . . N

(120)

By factoring out X , we see that is ensured by affine independence of {yi ∈ RN } and by R(Y )∩ N (X) = 0 where N (A) = {x | Ax = 0} 2.4.2.5

(136)

affine maps

Affine transformations preserve affine hulls. Given any affine mapping T of vector spaces and some arbitrary set C [285, p.8] aff(T C) = T (aff C) 2.4.2.6

(121)

PRINCIPLE 2: Supporting hyperplane

§

§

The second most fundamental principle of convex geometry also follows from the geometric Hahn-Banach theorem [232, 5.12] [18, 1] that guarantees existence of at least one hyperplane in Rn supporting a convex set (having nonempty interior)2.19 at each point on its boundary. The partial boundary ∂H of a halfspace that contains arbitrary set Y is called a supporting hyperplane ∂H to Y when the hyperplane contains at least one point of Y . [285, 11]

§

2.4.2.6.1 Definition. Supporting hyperplane ∂H . Assuming set Y and some normal a 6= 0 reside in opposite halfspaces2.20 (Figure 29(a)), then a hyperplane supporting Y at yp ∈ ∂Y is © ª ∂H− = y | aT(y − yp ) = 0 , yp ∈ Y , aT(z − yp ) ≤ 0 ∀ z ∈ Y (122) Given normal a , instead, the supporting hyperplane is © ª ∂H− = y | aTy = sup{aTz | z ∈ Y} 2.19

(123)

It is customary to speak of a hyperplane supporting set C but not containing C ; called nontrivial support. [285, p.100] Hyperplanes in support of lower-dimensional bodies are admitted. 2.20 Normal a belongs to H+ by definition.

2.4. HALFSPACE, HYPERPLANE

79

H−

C H+

{z ∈ R2 | aTz = κ1 } a 0 > κ3 > κ2 > κ1

{z ∈ R2 | aTz = κ2 } {z ∈ R2 | aTz = κ3 }

Figure 28: (confer Figure 70) Each linear contour, of equal inner product in vector z with normal a , represents i th hyperplane in R2 parametrized by scalar κi . Inner product κi increases in direction of normal a . i th line segment {z ∈ C | aTz = κi } in convex set C ⊂ R2 represents intersection with hyperplane. (Cartesian axes drawn for reference.)

80

CHAPTER 2. CONVEX GEOMETRY

yp tradition

(a)

Y H−

yp nontraditional

(b)

Y

a

H+

∂H−

H−

a ˜ H+

∂H+

Figure 29: (a) Hyperplane ∂H− (122) supporting closed set Y ∈ R2 . Vector a is inward-normal to hyperplane with respect to halfspace H+ , but outward-normal with respect to set Y . A supporting hyperplane can be considered the limit of an increasing sequence in the normal-direction like that in Figure 28. (b) Hyperplane ∂H+ nontraditionally supporting Y . Vector a ˜ is inward-normal to hyperplane now with respect to both halfspace H+ and set Y . Tradition [185] [285] recognizes only positive normal polarity in support function σY as in (123); id est, normal a , figure (a). But both interpretations of supporting hyperplane are useful.

2.4. HALFSPACE, HYPERPLANE

81

where real function σY (a) = sup{aTz | z ∈ Y}

(519)

is called the support function for Y . An equivalent but nontraditional representation2.21 for a supporting hyperplane is obtained by reversing polarity of normal a ; (1607) © ª ˜T(y − yp ) = 0 , yp ∈ Y , a ˜T(z − yp ) ≥ 0 ∀ z ∈ Y ∂H+ = y | a © ª (124) = y|a ˜Ty = − inf{˜ aTz | z ∈ Y} = sup{−˜ aTz | z ∈ Y}

where normal a ˜ and set Y both now reside in H+ . (Figure 29(b)) When a supporting hyperplane contains only a single point of Y , that hyperplane is termed strictly supporting.2.22 △ A set of nonempty interior that has a supporting hyperplane at every point on its boundary, conversely, is convex. A convex set C ⊂ Rn , for example, can be expressed as the intersection of all halfspaces partially bounded by hyperplanes supporting it; videlicet, [232, p.135] \© ª y | aTy ≤ σC (a) (125) C=

§

a∈R

n

by the halfspaces theorem ( 2.4.1.1.1). There is no geometric difference between supporting hyperplane ∂H+ or ∂H− or ∂H and2.23 an ordinary hyperplane ∂H coincident with them. 2.4.2.6.2 Example. Minimization on a hypercube. Consider minimization of a linear function on a hypercube, given vector c minimize x

cTx

subject to −1 ¹ x ¹ 1 2.21

(126)

useful for constructing the dual cone; e.g., Figure 53(b). Tradition would instead have us construct the polar cone; which is, the negative dual cone. 2.22 Rockafellar terms a strictly supporting hyperplane tangent to Y if it is unique there; [285, §18, p.169] a definition we do not adopt because our only criterion for tangency is intersection exclusively with a relative boundary. Hiriart-Urruty & Lemar´echal [185, p.44] (confer [285, p.100]) do not demand any tangency of a supporting hyperplane. 2.23 If vector-normal polarity is unimportant, we may instead signify a supporting hyperplane by ∂H .

82

CHAPTER 2. CONVEX GEOMETRY

This convex optimization problem is called a linear program 2.24 because the objective of minimization is linear and the constraints describe a polyhedron (intersection of a finite number of halfspaces and hyperplanes). Applying graphical concepts from Figure 26, Figure 28, and Figure 29, an optimal solution can be shown to be x⋆ = − sgn(c) but is not necessarily unique. Because an optimal solution always exists at a hypercube vertex ( 2.6.1.0.1) regardless of the value of nonzero vector c [89], mathematicians see this geometry as a means to relax a discrete problem (whose desired solution is integer or combinatorial, confer Example 4.2.3.1.1). [222, 3.1] [223] 2

§

§

2.4.2.6.3 Exercise. Unbounded below. Suppose instead we minimize over the unit hypersphere in Example 2.4.2.6.2; kxk ≤ 1. What is an expression for optimal solution now? Is that program still linear? Now suppose minimization of absolute value in (126). Are the following programs equivalent for some arbitrary real convex set C ? (confer (487)) minimize x∈R

|x|

minimize α , β

subject to −1 ≤ x ≤ 1 x∈C

≡

α+β

subject to 1 ≥ β ≥ 0 1≥α≥0 α−β ∈ C

(127)

Many optimization problems of interest and some methods of solution require nonnegative variables. The method illustrated below splits a variable into parts; x = α − β (extensible to vectors). Under what conditions on vector a and scalar b is an optimal solution x⋆ negative infinity? minimize α∈R , β∈ R

α−β

subject to β ≥ 0 α≥0 · ¸ T α a = b β Minimization of the objective function 2.25 entails maximization of β . 2.24

(128)

H

The term program has its roots in economics. It was originally meant with regard to a plan or to efficient organization of some industrial process. [89, §2] 2.25 The objective is the function that is argument to minimization or maximization.

2.5. SUBSPACE REPRESENTATIONS 2.4.2.7

83

PRINCIPLE 3: Separating hyperplane

§

§

§

The third most fundamental principle of convex geometry again follows from the geometric Hahn-Banach theorem [232, 5.12] [18, 1] [122, I.1.2] that guarantees existence of a hyperplane separating two nonempty convex sets in Rn whose relative interiors are nonintersecting. Separation intuitively means each set belongs to a halfspace on an opposing side of the hyperplane. There are two cases of interest:

§

1) If the two sets intersect only at their relative boundaries ( 2.6.1.3), then there exists a separating hyperplane ∂H containing the intersection but containing no points relatively interior to either set. If at least one of the two sets is open, conversely, then the existence of a separating hyperplane implies the two sets are nonintersecting. [57, 2.5.1]

§

2) A strictly separating hyperplane ∂H intersects the closure of neither set; its existence is guaranteed when intersection of the closures is empty and at least one set is bounded. [185, A.4.1]

§

2.4.3

Angle between hyperspaces

Given halfspace-descriptions, dihedral angle between hyperplanes or halfspaces is defined as the angle between their defining normals. Given normals a and b respectively describing ∂Ha and ∂Hb , for example ¶ µ ha , bi radians (129) Á(∂Ha , ∂Hb ) , arccos kak kbk

2.5

Subspace representations

There are two common forms of expression for Euclidean subspaces, both coming from elementary linear algebra: range form R and nullspace form N ; a.k.a, vertex-description and halfspace-description respectively. The fundamental vector subspaces associated with a matrix A ∈ Rm×n [309, 3.1] are ordinarily related by orthogonal complement

§

R(AT ) ⊥ N (A) , R(AT ) ⊕ N (A) = Rn ,

N (AT ) ⊥ R(A)

(130)

N (AT ) ⊕ R(A) = Rm

(131)

84

CHAPTER 2. CONVEX GEOMETRY

and of dimension: dim R(AT ) = dim R(A) = rank A ≤ min{m , n}

(132)

with complementarity dim N (A) = n − rank A ,

dim N (AT ) = m − rank A

(133)

These equations (130)-(133) comprise the fundamental theorem of linear algebra. [309, p.95, p.138] From these four fundamental subspaces, the rowspace and range identify one form of subspace description (range form or vertex-description ( 2.3.4))

§

R(AT ) , span AT = {ATy | y ∈ Rm } = {x ∈ Rn | ATy = x , y ∈ R(A)} (134) R(A) , span A = {Ax | x ∈ Rn } = {y ∈ Rm | Ax = y , x ∈ R(AT )} (135) while the nullspaces identify the second common form (nullspace form or halfspace-description (107)) N (A) , {x ∈ Rn | Ax = 0} N (AT ) , {y ∈ Rm | ATy = 0}

(136) (137)

Range forms (134) (135) are realized as the respective span of the column vectors in matrices AT and A , whereas nullspace form (136) or (137) is the solution set to a linear equation similar to hyperplane definition (108). Yet because matrix A generally has multiple rows, halfspace-description N (A) is actually the intersection of as many hyperplanes through the origin; for (136), each row of A is normal to a hyperplane while each row of AT is a normal for (137).

2.5.1

Subspace or affine subset . . .

Any particular vector subspace Rp can be described as N (A) the nullspace of some matrix A or as R(B) the range of some matrix B . More generally, we have the choice of expressing an n − m-dimensional affine subset in Rn as the intersection of m hyperplanes, or as the offset span of n − m vectors:

2.5. SUBSPACE REPRESENTATIONS 2.5.1.1

85

. . . as hyperplane intersection

Any affine subset A of dimension n − m can be described as an intersection of m hyperplanes in Rn ; given fat (m ≤ n) full-rank (rank = min{m , n}) matrix  T  a1 ..   A, ∈ Rm×n (138) . T am and vector b ∈ Rm ,

n

A , {x ∈ R | Ax = b} =

m \ ©

i=1

x | aTi x = b i

ª

(139)

a halfspace-description. (107) For example: The intersection of any two independent2.26 hyperplanes in R3 is a line, whereas three independent hyperplanes intersect at a point. In R4 , the intersection of two independent hyperplanes is a plane (Example 2.5.1.2.1), whereas three hyperplanes intersect at a line, four at a point, and so on. A describes a subspace whenever b = 0 in (139). For n > k m \ ª © k n k A ∩ R = {x ∈ R | Ax = b} ∩ R = x ∈ Rk | a i (1 : k)Tx = b i (140)

§

i=1

The result in 2.4.2.2 is extensible; id est, any affine subset A also has a vertex-description: 2.5.1.2

. . . as span of nullspace basis

Alternatively, we may compute a basis for nullspace of matrix A in (138) and then equivalently express affine subset A as its span plus an offset: Define Z , basis N (A) ∈ Rn×n−m

so AZ = 0. Then we have the vertex-description in Z , © ª A = {x ∈ Rn | Ax = b} = Z ξ + xp | ξ ∈ Rn−m ⊆ Rn

(141)

(142)

the offset span of n − m column vectors, where xp is any particular solution to Ax = b . For example, A describes a subspace whenever xp = 0. 2.26

Hyperplanes are said to be independent iff the normals defining them are linearly independent.

86

CHAPTER 2. CONVEX GEOMETRY

2.5.1.2.1 Example. Intersecting planes in 4-space. Two planes can intersect at a point in four-dimensional Euclidean vector space. It is easy to visualize intersection of two planes in three dimensions; a line can be formed. In four dimensions it is harder to visualize. So let’s resort to the tools acquired. Suppose an intersection of two hyperplanes in four dimensions is specified by a fat full-rank matrix A1 ∈ R2×4 (m = 2, n = 4) as in (139): ½ A1 , x ∈ R4

¯ · ¾ ¸ ¯ a11 a12 a13 a14 ¯ ¯ a21 a22 a23 a24 x = b1

(143)

The nullspace of A1 is two dimensional (from Z in (142)), so A1 represents a plane in four dimensions. Similarly define a second plane in terms of A2 ∈ R2×4 : ¯ · ¾ ¸ ½ ¯ a31 a32 a33 a34 4 ¯ (144) x = b2 A2 , x ∈ R ¯ a41 a42 a43 a44

If the two planes are independent (meaning any line in one is linearly independent ¸ of any line from the other), they will intersect at a point because · A1 is invertible; then A2 ½ A1 ∩ A2 = x ∈ R4

¯ · ¸ · ¸¾ ¯ A1 b1 ¯ ¯ A2 x = b2

(145) 2

2.5.1.2.2 Exercise. Linear program. Minimize a hyperplane over affine set A in the nonnegative orthant minimize x

cTx

subject to Ax = b xº0

(146)

where A = {x | Ax = b}. Two cases of interest are drawn in Figure 30. Graphically illustrate and explain optimal solutions indicated in the caption. Why is α⋆ negative in both cases? Is there solution on the vertical axis? H

2.5. SUBSPACE REPRESENTATIONS

87

(a) c

A = {x | Ax = b} {x | cT x = α}

A = {x | Ax = b}

(b) c {x | cT x = α} Figure 30: Minimizing hyperplane over affine set A in nonnegative orthant R2+ . (a) Optimal solution is • . (b) Optimal objective α⋆ = −∞.

88

CHAPTER 2. CONVEX GEOMETRY

2.5.2

Intersection of subspaces

The intersection of nullspaces associated with two matrices A ∈ Rm×n and B ∈ Rk×n can be expressed most simply as µ· ¸¶ · ¸ A A n N (A) ∩ N (B) = N , {x ∈ R | x = 0} (147) B B the nullspace of their rowwise concatenation. Suppose the columns of a matrix Z constitute a basis for N (A) while the columns of a matrix W constitute a basis for N (BZ ). Then [146, 12.4.2]

§

N (A) ∩ N (B) = R(ZW )

(148)

If each basis is orthonormal, then the columns of ZW constitute an orthonormal basis for the intersection. In the particular circumstance A and B are each positive semidefinite [21, 6], or in the circumstance A and B are two linearly independent dyads ( B.1.1), then   A,B ∈ SM + N (A) ∩ N (B) = N (A + B) , (149) or  A + B = u1 v1T + u2 v2T (l.i.)

§

§

2.6

Extreme, Exposed

2.6.0.0.1 Definition. Extreme point. An extreme point xε of a convex set C is a point, belonging to its closure C [39, 3.3], that is not expressible as a convex combination of points in C distinct from xε ; id est, for xε ∈ C and all x1 , x2 ∈ C \xε

§

µ x1 + (1 − µ)x2 6= xε ,

µ ∈ [0, 1]

(150) △

§

In other words, xε is an extreme point of C if and only if xε is not a point relatively interior to any line segment in C . [333, 2.10] Borwein & Lewis offer: [51, 4.1.6] An extreme point of a convex set C is a point xε in C whose relative complement C \ xε is convex. The set consisting of a single point C = {xε } is itself an extreme point.

§

2.6. EXTREME, EXPOSED

89

§

§

2.6.0.0.2 Theorem. Extreme existence. [285, 18.5.3] [26, II.3.5] A nonempty closed convex set containing no lines has at least one extreme point. ⋄

§

Definition. Face, edge. [185, A.2.3]

2.6.0.0.3

ˆ A face F of convex set C is a convex subset F ⊆ C such that every

closed line segment x1 x2 in C , having a relatively interior point (x ∈ rel int x1 x2) in F , has both endpoints in F . The zero-dimensional faces of C constitute its extreme points. The empty set and C itself are conventional faces of C . [285, 18]

§

ˆ All faces F are extreme sets by definition; id est, for F ⊆ C and all x1 , x2 ∈ C \F

µ x1 + (1 − µ)x2 ∈ / F ,

µ ∈ [0, 1]

ˆ A one-dimensional face of a convex set is called an edge.

(151) △

§

Dimension of a face is the penultimate number of affinely independent points ( 2.4.2.3) belonging to it; dim F = sup dim{x2 − x1 , x3 − x1 , . . . , xρ − x1 | xi ∈ F , i = 1 . . . ρ} (152) ρ

The point of intersection in C with a strictly supporting hyperplane identifies an extreme point, but not vice versa. The nonempty intersection of any supporting hyperplane with C identifies a face, in general, but not vice versa. To acquire a converse, the concept exposed face requires introduction:

2.6.1

Exposure

§

2.6.1.0.1 Definition. Exposed face, exposed point, vertex, facet. [185, A.2.3, A.2.4]

ˆ F is an exposed face of an n-dimensional convex set C iff there is a supporting hyperplane ∂H to C such that F = C ∩ ∂H

(153)

Only faces of dimension −1 through n − 1 can be exposed by a hyperplane.

90

CHAPTER 2. CONVEX GEOMETRY

A

B C

D

Figure 31: Closed convex set in R2 . Point A is exposed hence extreme; a classical vertex. Point B is extreme but not an exposed point. Point C is exposed and extreme; zero-dimensional exposure makes it a vertex. Point D is neither an exposed or extreme point although it belongs to a one-dimensional exposed face. [185, A.2.4] [308, 3.6] Closed face AB is exposed; a facet. The arc is not a conventional face, yet it is composed entirely of extreme points. Union of all rotations of this entire set about its vertical edge produces another convex set in three dimensions having no edges; but that convex set produced by rotation about horizontal edge containing D has edges.

§

§

2.6. EXTREME, EXPOSED

91

ˆ An

exposed point, the definition of vertex, is equivalent to a zero-dimensional exposed face; the point of intersection with a strictly supporting hyperplane.

ˆ A facet is an (n − 1)-dimensional exposed face of an n-dimensional

convex set C ; facets exist in one-to-one correspondence with the (n − 1)-dimensional faces.2.27

ˆ {exposed points} = {extreme points} {exposed faces} ⊆ {faces}

2.6.1.1

△

Density of exposed points

§

§

For any closed convex set C , its exposed points constitute a dense subset of its extreme points; [285, 18] [313] [308, 3.6, p.115] dense in the sense [348] that closure of that subset yields the set of extreme points. For the convex set illustrated in Figure 31, point B cannot be exposed because it relatively bounds both the facet AB and the closed quarter circle, each bounding the set. Since B is not relatively interior to any line segment in the set, then B is an extreme point by definition. Point B may be regarded as the limit of some sequence of exposed points beginning at vertex C . 2.6.1.2

Face transitivity and algebra

Faces of a convex set enjoy transitive relation. If F1 is a face (an extreme set) of F2 which in turn is a face of F3 , then it is always true that F1 is a face of F3 . (The parallel statement for exposed faces is false. [285, 18]) For example, any extreme point of F2 is an extreme point of F3 ; in this example, F2 could be a face exposed by a hyperplane supporting polyhedron F3 . [208, def.115/6, p.358] Yet it is erroneous to presume that a face, of dimension 1 or more, consists entirely of extreme points. Nor is a face of dimension 2 or more entirely composed of edges, and so on. For the polyhedron in R3 from Figure 20, for example, the nonempty faces exposed by a hyperplane are the vertices, edges, and facets; there are no more. The zero-, one-, and two-dimensional faces are in one-to-one correspondence with the exposed faces in that example.

§

2.27

This coincidence occurs simply because the facet’s dimension is the same as the dimension of the supporting hyperplane exposing it.

92

CHAPTER 2. CONVEX GEOMETRY

Define the smallest face F that contains some element G of a convex set C : F(C ∋ G) (154) videlicet, C ⊇ F(C ∋ G) ∋ G . An affine set has no faces except itself and the empty set. The smallest face that contains G of the intersection of convex set C with an affine set A [222, 2.4] [223]

§

F((C ∩ A) ∋ G) = F(C ∋ G) ∩ A

(155)

equals the intersection of A with the smallest face that contains G of set C . 2.6.1.3

§

Boundary

(confer 2.1.7) The classical definition of boundary of a set C presumes nonempty interior: ∂ C = C \ int C

(15)

§

More suitable for the study of convex sets is the relative boundary; defined [185, A.2.1.2] rel ∂ C = C \ rel int C (156) the boundary relative to the affine hull of C , conventionally equivalent to:

§

2.6.1.3.1 Definition. Conventional boundary of convex set. [185, C.3.1] The relative boundary ∂ C of a nonempty convex set C is the union of all exposed faces of C . △ Equivalence of this definition to (156) comes about because it is conventionally presumed that any supporting hyperplane, central to the definition of exposure, does not contain C . [285, p.100] Any face F of convex set C (that is not C itself) belongs to rel ∂ C . ( 2.8.2.1) In the exception when C is a single point {x} , (11)

§

rel ∂{x} = {x}\{x} = ∅ ,

§

§

x ∈ Rn

(157)

A bounded convex polyhedron ( 2.3.2, 2.12.0.0.1) having nonempty interior, for example, in R has a boundary constructed from two points, in R2 from at least three line segments, in R3 from convex polygons, while a convex

2.7. CONES

93

polychoron (a bounded polyhedron in R4 [348]) has a boundary constructed from three-dimensional convex polyhedra. A halfspace is partially bounded by a hyperplane; its interior therefore excludes that hyperplane. By Definition 2.6.1.3.1, an affine set has no relative boundary.

2.7

Cones

In optimization, convex cones achieve prominence because they generalize subspaces. Most compelling is the projection analogy: Projection on a subspace can be ascertained from projection on its orthogonal complement ( E), whereas projection on a closed convex cone can be determined from projection instead on its algebraic complement ( 2.13, E.9.2.1); called the polar cone.

§

§

2.7.0.0.1 Definition. The one-dimensional set

§

Ray.

{ζ Γ + B | ζ ≥ 0 , Γ 6= 0} ⊂ Rn

(158)

defines a halfline called a ray in nonzero direction Γ ∈ Rn having base B ∈ Rn . When B = 0, a ray is the conic hull of direction Γ ; hence a convex cone. △ The relative boundary of a single ray, base 0, in any dimension is the origin because that is the union of all exposed faces not containing the entire set. Its relative interior is the ray itself excluding the origin.

2.7.1

Cone defined

A set X is called, simply, cone if and only if Γ ∈ X ⇒ ζ Γ ∈ X for all ζ ≥ 0

(159)

where X denotes closure of cone X . An example of such a cone is the union of two opposing quadrants; e.g., X = {x ∈ R2 | x1 x2 ≥ 0} which is not convex. [346, 2.5] Similar examples are shown in Figure 32 and Figure 36. All cones can be defined by an aggregate of rays emanating exclusively from the origin (but not all cones are convex). Hence all closed cones contain

§

94

CHAPTER 2. CONVEX GEOMETRY

X 0

0

(a)

(b)

X Figure 32: (a) Two-dimensional nonconvex cone drawn truncated. Boundary of this cone is itself a cone. Each polar half is itself a convex cone. (b) This convex cone (drawn truncated) is a line through the origin in any dimension. It has no relative boundary, while its relative interior comprises entire line.

0

Figure 33: This nonconvex cone in R2 is a pair of lines through the origin. [232, 2.4]

§

2.7. CONES

95

0 Figure 34: Boundary of a convex cone in R2 is a nonconvex cone; a pair of rays emanating from the origin.

X 0

Figure 35: Nonconvex cone X drawn truncated in R2 . Boundary is also a cone. [232, 2.4] Cone exterior is convex cone.

§

96

CHAPTER 2. CONVEX GEOMETRY

X

X

Figure 36: Truncated nonconvex cone X = {x ∈ R2 | x1 ≥ x2 , x1 x2 ≥ 0}. Boundary is also a cone. [232, 2.4] Cartesian axes drawn for reference. Each half (about the origin) is itself a convex cone.

§

the origin 0 and are unbounded, excepting the simplest cone {0}. empty set ∅ is not a cone, but its conic hull is; cone ∅ = {0}

2.7.2

The

(98)

Convex cone

We call the set K ⊆ RM a convex cone iff Γ1 , Γ2 ∈ K ⇒ ζ Γ1 + ξ Γ2 ∈ K for all ζ , ξ ≥ 0

(160)

Apparent from this definition, ζ Γ1 ∈ K and ξ Γ2 ∈ K ∀ ζ , ξ ≥ 0 ; meaning, K is a cone. Set K is convex since, for any particular ζ , ξ ≥ 0 µ ζ Γ1 + (1 − µ) ξ Γ2 ∈ K ∀ µ ∈ [0, 1]

(161)

because µ ζ , (1 − µ) ξ ≥ 0. Obviously, {X } ⊃ {K}

(162)

the set of all convex cones is a proper subset of all cones. The set of convex cones is a narrower but more familiar class of cone, any member

2.7. CONES

97

of which can be equivalently described as the intersection of a possibly (but not necessarily) infinite number of hyperplanes (through the origin) and halfspaces whose bounding hyperplanes pass through the origin; a halfspace-description ( 2.4). Convex cones need not be full-dimensional.

§

Figure 37: Not a cone; ironically, the three-dimensional flared horn (with or without its interior) resembling mathematical symbol ≻ denoting cone membership and partial order.

§

More familiar convex cones include a Lorentz cone (confer Figure 44) (a.k.a: second-order cone, quadratic cone, circular cone ( 2.9.2.5.1), unbounded ice-cream cone united with its interior) Kℓ =

½·

§

x t

¸

¾ ∈ R × R | kxkℓ ≤ t , n

§

ℓ=2

(163)

and any polyhedral cone ( 2.12.1.0.1); e.g., any orthant generated by Cartesian half-axes ( 2.1.3). Esoteric examples of convex cones include the point at the origin, any line through the origin, any ray having the origin as base such as the nonnegative real line R+ in subspace R , any halfspace partially bounded by a hyperplane through the origin, the positive N semidefinite cone SM + (176), the cone of Euclidean distance matrices EDM (839) ( 6), any subspace, and Euclidean vector space Rn .

§

98

CHAPTER 2. CONVEX GEOMETRY

2.7.2.1

cone invariance

More Euclidean bodies are cones, it seems, than are not.2.28 This class of convex body, the convex cone, is invariant to scaling, linear and single- or many-valued inverse linear transformation, vector summation, and Cartesian product, but is not invariant to translation. [285, p.22]

§ §

2.7.2.1.1 Theorem. Cone intersection. [285, 2, 19] Intersection of an arbitrary collection of convex cones is a convex cone. Intersection of an arbitrary collection of closed convex cones is a closed convex cone. [240, 2.3] Intersection of a finite number of polyhedral cones ( 2.12.1.0.1, Figure 48 p.134) remains a polyhedral cone. ⋄

§

§

The property pointedness is associated with a convex cone.

§

2.7.2.1.2 Definition. Pointed convex cone. (confer 2.12.2.2) A convex cone K is pointed iff it contains no line. Equivalently, K is not pointed iff there exists any nonzero direction Γ ∈ K such that −Γ ∈ K . If the origin is an extreme point of K or, equivalently, if K ∩ −K = {0}

(164)

§

then K is pointed, and vice versa. [308, 2.10] A convex cone is pointed iff the origin is the smallest nonempty face of its closure. △

§

Then a pointed closed convex cone, by principle of separating hyperplane ( 2.4.2.7), has a strictly supporting hyperplane at the origin. The simplest and only bounded [346, p.75] convex cone K = {0} ⊆ Rn is pointed by convention, but has empty interior. Its relative boundary is the empty set (157) while its relative interior is the point itself (11). The pointed convex cone that is a halfline emanating from the origin in Rn has the origin as relative boundary while its relative interior is the halfline itself, excluding the origin. 2.28

confer Figures: 24 32 33 34 35 36 37 39 41 48 52 55 57 58 59 60 61 62 63 130 143 166

2.7. CONES

99

§

2.7.2.1.3 Theorem. Pointed cones. [51, 3.3.15, exer.20] n A closed convex cone K ⊂ R is pointed if and only if there exists a normal α such that the set C , {x ∈ K | hx , αi = 1} (165) is closed, bounded, and K = cone C . Equivalently, K is pointed if and only if there exists a vector β normal to a hyperplane strictly supporting K ; id est, for some positive scalar ǫ hx , βi ≥ ǫkxk ∀ x ∈ K

(166) ⋄

§

If closed convex cone K is not pointed, then it has no extreme point.2.29 Yet a pointed closed convex cone has only one extreme point [39, 3.3]: the exposed point residing at the origin; its vertex. And from the cone intersection theorem it follows that an intersection of convex cones is pointed if at least one of the cones is. 2.7.2.2

Pointed closed convex cone and partial order

Relation ¹ is a partial order on some set if the relation possesses2.30 reflexivity (x ¹ x) antisymmetry (x ¹ z , z ¹ x ⇒ x = z) transitivity (x ¹ y , y ¹ z ⇒ x ¹ z),

(x ¹ y , y ≺ z ⇒ x ≺ z)

§

A pointed closed convex cone K induces partial order on Rn or Rm×n , [21, 1] [302, p.7] essentially defined by vector or matrix inequality; x ¹ z

⇔

z−x∈ K

(167)

x ≺ z

⇔

z − x ∈ rel int K

(168)

K

K

2.29

nor does it have extreme directions (§2.8.1). A set is totally ordered if it further obeys a comparability property of the relation: for each and every x and y from the set, x ¹ y or y ¹ x ; e.g., R is the smallest unbounded totally ordered and connected set. 2.30

100

CHAPTER 2. CONVEX GEOMETRY

R2 C1 (a)

x+K

y x

C2 (b)

y−K

Figure 38: (confer Figure 66) (a) Point x is the minimum element of set C1 with respect to cone K because cone translated to x ∈ C1 contains entire set. (Cones drawn truncated.) (b) Point y is a minimal element of set C2 with respect to cone K because negative cone translated to y ∈ C2 contains only y . These concepts, minimum/minimal, become equivalent under a total order.

2.7. CONES

101

Neither x or z is necessarily a member of K for these relations to hold. Only when K is the nonnegative orthant do these inequalities reduce to ordinary entrywise comparison. ( 2.13.4.2.3) Inclusive of that special case, we ascribe nomenclature generalized inequality to comparison with respect to a pointed closed convex cone. We say two points x and y are comparable when x ¹ y or y ¹ x with respect to pointed closed convex cone K . Visceral mechanics of actually comparing points, when cone K is not an orthant, are well illustrated in the example of Figure 60 which relies on the equivalent membership-interpretation in definition (167) or (168). Comparable points and the minimum element of some vector- or matrix-valued partially ordered set are thus well defined, so nonincreasing sequences with respect to cone K can therefore converge in this sense: Point x ∈ C is the (unique) minimum element of set C with respect to cone K iff for each and every z ∈ C we have x ¹ z ; equivalently, iff C ⊆ x + K .2.31 A closely related concept, minimal element, is useful for partially ordered sets having no minimum element: Point x ∈ C is a minimal element of set C with respect to pointed closed convex cone K if and only if (x − K) ∩ C = x . (Figure 38) No uniqueness is implied here, although implicit is the assumption: dim K ≥ dim aff C . In words, a point that is a minimal element is smaller (with respect to K) than any other point in the set to which it is comparable. Further properties of partial order with respect to pointed closed convex cone K are not defining:

§

homogeneity (x ¹ y , λ ≥ 0 ⇒ λx ¹ λz), additivity (x ¹ z , u ¹ v ⇒ x + u ¹ z + v), 2.7.2.2.1

(x ≺ y , λ > 0 ⇒ λx ≺ λz) (x ≺ z , u ¹ v ⇒ x + u ≺ z + v)

Definition. Proper cone: a cone that is

ˆ pointed ˆ closed ˆ convex ˆ has nonempty interior (is full-dimensional).

2.31

△

Borwein & Lewis [51, §3.3, exer.21] ignore possibility of equality to x + K in this condition, and require a second condition: . . . and C ⊂ y + K for some y in Rn implies x ∈ y +K.

102

CHAPTER 2. CONVEX GEOMETRY

§

A proper cone remains proper under injective linear transformation. [85, 5.1] Examples of proper cones are the positive semidefinite cone SM + in the ambient space of symmetric matrices ( 2.9), the nonnegative real line R+ in vector space R , or any orthant in Rn .

2.8

§

Cone boundary

§

Every hyperplane supporting a convex cone contains the origin. [185, A.4.2] Because any supporting hyperplane to a convex cone must therefore itself be a cone, then from the cone intersection theorem it follows: 2.8.0.0.1 Lemma. Cone faces. Each nonempty exposed face of a convex cone is a convex cone.

§

[26, II.8] ⋄

2.8.0.0.2 Theorem. Proper-cone boundary. Suppose a nonzero point Γ lies on the boundary ∂K of proper cone K in Rn . Then it follows that the ray {ζ Γ | ζ ≥ 0} also belongs to ∂K . ⋄ Proof. By virtue of its propriety, a proper cone guarantees existence of a strictly supporting hyperplane at the origin. [285, cor.11.7.3]2.32 Hence the origin belongs to the boundary of K because it is the zero-dimensional exposed face. The origin belongs to the ray through Γ , and the ray belongs to K by definition (159). By the cone faces lemma, each and every nonempty exposed face must include the origin. Hence the closed line segment 0Γ must lie in an exposed face of K because both endpoints do by Definition 2.6.1.3.1. That means there exists a supporting hyperplane ∂H to K containing 0Γ . So the ray through Γ belongs both to K and to ∂H . ∂H must therefore expose a face of K that contains the ray; id est, {ζ Γ | ζ ≥ 0} ⊆ K ∩ ∂H ⊂ ∂K

(169) ¨

Proper cone {0} in R0 has no boundary (156) because (11) rel int{0} = {0} 2.32

(170)

Rockafellar’s corollary yields a supporting hyperplane at the origin to any convex cone in Rn not equal to Rn .

2.8. CONE BOUNDARY

103

The boundary of any proper cone in R is the origin. The boundary of any convex cone whose dimension exceeds 1 can be constructed entirely from an aggregate of rays emanating exclusively from the origin.

2.8.1

Extreme direction

The property extreme direction arises naturally in connection with the pointed closed convex cone K ⊂ Rn , being analogous to extreme point. [285, 18, p.162]2.33 An extreme direction Γε of pointed K is a vector corresponding to an edge that is a ray emanating from the origin.2.34 Nonzero direction Γε in pointed K is extreme if and only if

§

ζ1 Γ1 + ζ2 Γ2 6= Γε

∀ ζ1 , ζ2 ≥ 0 ,

∀ Γ1 , Γ2 ∈ K \{ζ Γε ∈ K | ζ ≥ 0}

(171)

In words, an extreme direction in a pointed closed convex cone is the direction of a ray, called an extreme ray, that cannot be expressed as a conic combination of directions of any rays in the cone distinct from it. An extreme ray is a one-dimensional face of K . By (99), extreme direction Γε is not a point relatively interior to any line segment in K \{ζ Γε ∈ K | ζ ≥ 0}. Thus, by analogy, the corresponding extreme ray {ζ Γε ∈ K | ζ ≥ 0} is not a ray relatively interior to any plane segment 2.35 in K . 2.8.1.1

extreme distinction, uniqueness

An extreme direction is unique, but its vector representation Γε is not because any positive scaling of it produces another vector in the same (extreme) direction. Hence an extreme direction is unique to within a positive scaling. When we say extreme directions are distinct, we are referring to distinctness of rays containing them. Nonzero vectors of various length in the same extreme direction are therefore interpreted to be identical extreme directions.2.36 2.33

We diverge from Rockafellar’s extreme direction: “extreme point at infinity”. An edge (§2.6.0.0.3) of a convex cone is not necessarily a ray. A convex cone may ∗ contain an edge that is a line; e.g., a wedge-shaped polyhedral cone (K in Figure 39). 2.35 A planar fragment; in this context, a planar cone. 2.36 Like vectors, an extreme direction can be identified by the Cartesian point at the vector’s head with respect to the origin. 2.34

104

CHAPTER 2. CONVEX GEOMETRY

∂K

∗

K

Figure 39: K is a pointed polyhedral cone having empty interior in R3 (drawn truncated in a plane parallel to the floor upon which you stand). Dual cone ∗ K is a wedge whose truncated boundary is illustrated (drawn perpendicular ∗ to the floor). In this particular instance, K ⊂ int K (excepting the origin). Cartesian coordinate axes drawn for reference.

2.8. CONE BOUNDARY

105

The extreme directions of the polyhedral cone in Figure 24 (p.70), for example, correspond to its three edges. For any pointed polyhedral cone, there is a one-to-one correspondence of one-dimensional faces with extreme directions. The extreme directions of the positive semidefinite cone ( 2.9) comprise the infinite set of all symmetric rank-one matrices. [21, 6] [182, III] It is sometimes prudent to instead consider the less infinite but complete normalized set, for M > 0 (confer (209))

§

§

§

{zz T ∈ SM | kzk = 1}

(172)

The positive semidefinite cone in one dimension M = 1, S+ the nonnegative real line, has one extreme direction belonging to its relative interior; an idiosyncrasy of dimension 1. Pointed closed convex cone K = {0} has no extreme direction because extreme directions are nonzero by definition.

ˆ If closed convex cone K is not pointed, then it has no extreme directions and no vertex. [21, §1] Conversely, pointed closed convex cone K is equivalent to the convex hull of its vertex and all its extreme directions. [285, §18, p.167] That is the

practical utility of extreme direction; to facilitate construction of polyhedral sets, apparent from the extremes theorem:

§

§

§

§

2.8.1.1.1 Theorem. (Klee) Extremes. [308, 3.6] [285, 18, p.166] (confer 2.3.2, 2.12.2.0.1) Any closed convex set containing no lines can be expressed as the convex hull of its extreme points and extreme rays. ⋄ It follows that any element of a convex set containing no lines may be expressed as a linear combination of its extreme elements; e.g., Example 2.9.2.4.1. 2.8.1.2

Generators

In the narrowest sense, generators for a convex set comprise any collection of points and directions whose convex hull constructs the set. When the extremes theorem applies, the extreme points and directions are called generators of a convex set. An arbitrary collection of generators

106

CHAPTER 2. CONVEX GEOMETRY

for a convex set includes its extreme elements as a subset; the set of extreme elements of a convex set is a minimal set of generators for that convex set. Any polyhedral set has a minimal set of generators whose cardinality is finite. When the convex set under scrutiny is a closed convex cone, conic combination of generators during construction is implicit as shown in Example 2.8.1.2.1 and Example 2.10.2.0.1. So, a vertex at the origin (if it exists) becomes benign. We can, of course, generate affine sets by taking the affine hull of any collection of points and directions. We broaden, thereby, the meaning of generator to be inclusive of all kinds of hulls. Any hull of generators is loosely called a vertex-description. ( 2.3.4) Hulls encompass subspaces, so any basis constitutes generators for a vertex-description; span basis R(A).

§

2.8.1.2.1 Example. Application of extremes theorem. Given an extreme point at the origin and N extreme rays, denoting the i th extreme direction by Γi ∈ Rn , then the convex hull is (81) © ª P = [0© Γ1 Γ2 · · · ΓN ] a ζ | aT 1 = 1, a º 0, ζ ≥ 0 ª T = (173) ©[Γ1 Γ2 · · · ΓN ] a ζ | a 1 ≤ª1, a ºn 0, ζ ≥ 0 = [Γ1 Γ2 · · · ΓN ] b | b º 0 ⊂ R a closed convex set that is simply a conic hull like (97).

2.8.2

2

Exposed direction

§

§

2.8.2.0.1 Definition. Exposed point & direction of pointed convex cone. [285, 18] (confer 2.6.1.0.1)

ˆ When a convex cone has a vertex, an exposed point, it resides at the origin; there can be only one.

ˆ In the closure of a pointed convex cone, an exposed direction is the

direction of a one-dimensional exposed face that is a ray emanating from the origin.

ˆ {exposed directions} ⊆ {extreme directions}

△

2.8. CONE BOUNDARY

107

For a proper cone in vector space Rn with n ≥ 2, we can say more: {exposed directions} = {extreme directions}

(174)

It follows from Lemma 2.8.0.0.1 for any pointed closed convex cone, there is one-to-one correspondence of one-dimensional exposed faces with exposed directions; id est, there is no one-dimensional exposed face that is not a ray base 0. The pointed closed convex cone EDM2 , for example, is a ray in isomorphic subspace R whose relative boundary ( 2.6.1.3.1) is the origin. The conventionally exposed directions of EDM2 constitute the empty set ∅ ⊂ {extreme direction}. This cone has one extreme direction belonging to its relative interior; an idiosyncrasy of dimension 1.

§

2.8.2.1

Connection between boundary and extremes

§

§

2.8.2.1.1 Theorem. Exposed. [285, 18.7] (confer 2.8.1.1.1) Any closed convex set C containing no lines (and whose dimension is at least 2) can be expressed as closure of the convex hull of its exposed points and exposed rays. ⋄ From Theorem 2.8.1.1.1,  (156)    = conv{exposed points and exposed rays} \ rel int C    = conv{extreme points and extreme rays} \ rel int C

rel ∂ C = C \ rel int C

(175)

Thus each and every extreme point of a convex set (that is not a point) resides on its relative boundary, while each and every extreme direction of a convex set (that is not a halfline and contains no line) resides on its relative boundary because extreme points and directions of such respective sets do not belong to relative interior by definition. The relationship between extreme sets and the relative boundary actually goes deeper: Any face F of convex set C (that is not C itself) belongs to rel ∂ C , so dim F < dim C . [285, 18.1.3]

§

2.8.2.2

Converse caveat

It is inconsequent to presume that each and every extreme point and direction is necessarily exposed, as might be erroneously inferred from the conventional

108

CHAPTER 2. CONVEX GEOMETRY

C B A

D

0

§

Figure 40: Properties of extreme points carry over to extreme directions. [285, 18] Four rays (drawn truncated) on boundary of conic hull of two-dimensional closed convex set from Figure 31 lifted to R3 . Ray through point A is exposed hence extreme. Extreme direction B on cone boundary is not an exposed direction, although it belongs to the exposed face cone{A , B}. Extreme ray through C is exposed. Point D is neither an exposed or extreme direction although it belongs to a two-dimensional exposed face of the conic hull.

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

109

§

boundary definition ( 2.6.1.3.1); although it can correctly be inferred: each and every extreme point and direction belongs to some exposed face. Arbitrary points residing on the relative boundary of a convex set are not necessarily exposed or extreme points. Similarly, the direction of an arbitrary ray, base 0, on the boundary of a convex cone is not necessarily an exposed or extreme direction. For the polyhedral cone illustrated in Figure 24, for example, there are three two-dimensional exposed faces constituting the entire boundary, each composed of an infinity of rays. Yet there are only three exposed directions. Neither is an extreme direction on the boundary of a pointed convex cone necessarily an exposed direction. Lift the two-dimensional set in Figure 31, for example, into three dimensions such that no two points in the set are collinear with the origin. Then its conic hull can have an extreme direction B on the boundary that is not an exposed direction, illustrated in Figure 40.

2.9

Positive semidefinite (PSD) cone The cone of positive semidefinite matrices studied in this section is arguably the most important of all non-polyhedral cones whose facial structure we completely understand. −Alexander Barvinok [26, p.78]

2.9.0.0.1 Definition. Positive semidefinite cone. The set of all symmetric positive semidefinite matrices of particular dimension M is called the positive semidefinite cone: © ª SM A ∈ SM | A º 0 + , © ª = A ∈ SM | y TAy ≥ 0 ∀ kyk = 1 ª T© (176) A ∈ SM | hyy T , Ai ≥ 0 = kyk=1

= {A ∈ SM + | rank A ≤ M }

§

formed by the intersection of an infinite number of halfspaces ( 2.4.1.1) in vectorized variable2.37 A , each halfspace having partial boundary containing infinite in number when M > 1. Because y TA y = y TAT y , matrix A is almost always assumed symmetric. (§A.2.1) 2.37

110

CHAPTER 2. CONVEX GEOMETRY

the origin in isomorphic RM (M +1)/2 . It is a unique immutable proper cone in the ambient space of symmetric matrices SM . The positive definite (full-rank) matrices comprise the cone interior © ª int SM A ∈ SM | A ≻ 0 + = © ª = A ∈ SM | y TAy > 0 ∀ kyk = 1 ª T© (177) = A ∈ SM | hyy T , Ai > 0 kyk=1

= {A ∈ SM + | rank A = M }

§

while all singular positive semidefinite matrices (having at least one 0 eigenvalue) reside on the cone boundary (Figure 41); ( A.7.5) © ª M ∂ SM | A º 0, A ⊁ 0 + = A∈ S © ª = A ∈ SM | min{λ(A)i , i = 1 . . . M } = 0 (178) © ª T = A ∈ SM + | hyy , Ai = 0 for some kyk = 1 © ª = A ∈ SM + | rank A < M where λ(A) ∈ RM holds the eigenvalues of A .

△

The only symmetric positive semidefinite matrix in SM + having M 0-eigenvalues resides at the origin. ( A.7.3.0.1)

§

2.9.0.1

Membership

§

Observe the notation A º 0 ; meaning (confer 2.3.1.1), matrix A belongs to the positive semidefinite cone in the subspace of symmetric matrices, whereas A ≻ 0 denotes membership to that cone’s interior. ( 2.13.2) Notation A ≻ 0 can be read: symmetric matrix A is greater than the origin with respect to the positive semidefinite cone. This notation further implies that coordinates [sic] for orthogonal expansion of a positive (semi)definite matrix must be its (nonnegative) positive eigenvalues ( 2.13.7.1.1, E.6.4.1.1) when expanded in its eigenmatrices ( A.5.0.3). Generalizing comparison on the real line, the notation A º B denotes comparison with respect to the positive semidefinite cone; ( A.3.1) id est, A º B ⇔ A − B ∈ SM + but neither matrix A or B necessarily belongs to the positive semidefinite cone. Yet, (1427) A º B , B º 0 ⇒ A º 0 ; id est, A ∈ SM + . (confer Figure 60)

§

§

§

§

§

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

111

svec ∂ S2+ ·

γ

√

α

α β β γ

¸

2β

Minimal set of generators are the extreme directions: svec{yy T | y ∈ RM }

Figure 41: (d’Aspremont) Truncated boundary of PSD cone in S2 plotted in isometrically isomorphic R3 via svec (51); 0-contour of smallest eigenvalue (178). Lightest shading is closest, darkest shading is farthest and inside shell. Entire boundary can be constructed from© an aggregate of rays ( 2.7.0.0.1) √ ª 2 2 emanating exclusively from the origin: κ [ z1 2z1 z2 z22 ]T | κ ∈ R . A circular cone in this dimension ( 2.9.2.5), each and every ray on boundary corresponds to an extreme direction but such is not the case in any higher dimension (confer Figure 24). PSD cone geometry is not as simple in higher dimensions [26, II.12] although PSD cone is self-dual (349) in ambient real space of symmetric matrices. [182, II] PSD cone has no two-dimensional face in any dimension, its only extreme point residing at 0.

§

§

§

§

112

CHAPTER 2. CONVEX GEOMETRY

§

2.9.0.1.1 Example. Equality constraints in semidefinite program (613). Employing properties of partial order ( 2.7.2.2) for the pointed closed convex positive semidefinite cone, it is easy to show, given A + S = C Sº0 ⇔ A¹C

(179) 2

2.9.1

Positive semidefinite cone is convex

The set of all positive semidefinite matrices forms a convex cone in the ambient space of symmetric matrices because any pair satisfies definition (160); [188, 7.1] videlicet, for all ζ1 , ζ2 ≥ 0 and each and every A1 , A2 ∈ SM

§

ζ1 A1 + ζ2 A2 º 0 ⇐ A1 º 0 , A2 º 0

(180)

§

a fact easily verified by the definitive test for positive semidefiniteness of a symmetric matrix ( A): A º 0 ⇔ xTA x ≥ 0 for each and every kxk = 1

(181)

id est, for A1 , A2 º 0 and each and every ζ1 , ζ2 ≥ 0 ζ1 xTA1 x + ζ2 xTA2 x ≥ 0 for each and every normalized x ∈ RM

(182)

The convex cone SM + is more easily visualized in the isomorphic vector M (M +1)/2 whose dimension is the number of free variables in a space R symmetric M ×M matrix. When M = 2 the PSD cone is semiinfinite in expanse in R3 , having boundary illustrated in Figure 41. When M = 3 the PSD cone is six-dimensional, and so on. 2.9.1.0.1 The set

Example. Sets from maps of positive semidefinite cone. C = {X ∈ S n × x ∈ Rn | X º xxT }

(183)

§

is convex because it has Schur-form; ( A.4) T

X − xx º 0 ⇔ f (X , x) ,

·

X x xT 1

¸

º0

(184)

2.9. POSITIVE SEMIDEFINITE (PSD) CONE C

113

X

x Figure 42: Convex set C = {X ∈ S × x ∈ R | X º xxT } drawn truncated.

§

n+1 under affine e.g., Figure 42. Set C is the inverse image ( 2.1.9.0.1) of S+ n n T mapping f . The set {X ∈ S × x ∈ R | X ¹ xx } is not convex, in contrast, having no Schur-form. Yet for fixed x = xp , the set

{X ∈ S n | X ¹ xp xT p} is simply the negative semidefinite cone shifted to xp xT p .

(185) 2

2.9.1.0.2 Example. Inverse image of positive semidefinite cone. Now consider finding the set of all matrices X ∈ SN satisfying AX + B º 0

(186)

given A , B ∈ SN . Define the set X , {X | AX + B º 0} ⊆ SN

(187)

which is the inverse image of the positive semidefinite cone under affine transformation g(X) , AX + B . Set X must therefore be convex by Theorem 2.1.9.0.1. Yet we would like a less amorphous characterization of this set, so instead we consider its vectorization (32) which is easier to visualize: vec g(X) = vec(AX) + vec B = (I ⊗ A) vec X + vec B

(188)

114

CHAPTER 2. CONVEX GEOMETRY

where I ⊗ A , QΛQT ∈ SN

2

(189)

§

is block-diagonal formed by Kronecker product ( A.1.1 no.31, Assign 2 x , vec X ∈ RN 2 b , vec B ∈ RN

§D.1.2.1).

(190)

then make the equivalent problem: Find 2

vec X = {x ∈ RN | (I ⊗ A)x + b ∈ K}

(191)

K , vec SN +

(192)

where is a proper cone isometrically isomorphic with the positive semidefinite cone in the subspace of symmetric matrices; the vectorization of every element of SN + . Utilizing the diagonalization (189), vec X = {x | ΛQTx ∈ QT(K − b)} 2 = {x | ΦQTx ∈ Λ† QT(K − b)} ⊆ RN where

†

(193)

§

denotes matrix pseudoinverse ( E) and Φ , Λ† Λ

§

(194)

is a diagonal projection matrix whose entries are either 1 or 0 ( E.3). We have the complementary sum ΦQTx + (I − Φ)QTx = QTx

(195)

So, adding (I − Φ)QTx to both sides of the membership within (193) admits 2

vec X = {x ∈ RN | QTx ∈ Λ† QT(K − b) + (I − Φ)QTx} ¡ ¢ 2 = {x | QTx ∈ Φ Λ† QT(K − b) ⊕ (I − Φ)RN } 2 = {x ∈ QΛ† QT(K − b) ⊕ Q(I − Φ)RN } = (I ⊗ A)† (K − b) ⊕ N (I ⊗ A) 2

(196)

where we used the facts: linear function QTx in x on RN is a bijection, and ΦΛ† = Λ† . vec X = (I ⊗ A)† vec(SN (197) + − B) ⊕ N (I ⊗ A)

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

115

§

In words, set vec X is the vector sum of the translated PSD cone (linearly mapped onto the rowspace of I ⊗ A ( E)) and the nullspace of I ⊗ A (synthesis of fact from A.6.3 and A.7.3.0.1). Should I ⊗ A have no nullspace, then vec X = (I ⊗ A)−1 vec(SN + − B) which is the expected result. 2

§

2.9.2

§

Positive semidefinite cone boundary

For any symmetric positive semidefinite matrix A of rank ρ , there must exist a rank ρ matrix Y such that A be expressible as an outer product in Y ; [309, 6.3]

§

A = Y Y T ∈ SM + ,

rank A = ρ ,

Y ∈ RM ×ρ

(198)

Then the boundary of the positive semidefinite cone may be expressed © ª © ª M T ∂ SM | Y ∈ RM ×M −1 (199) + = A ∈ S+ | rank A < M = Y Y

§

§

Because the boundary of any convex body is obtained with closure of its relative interior ( 2.1.7, 2.6.1.3), from (177) we must also have © ª ª © M = Y Y T | Y ∈ RM ×M , rank Y = M SM + = A ∈ S+ | rank A = M (200) © ª = Y Y T | Y ∈ RM ×M

2.9.2.1

rank ρ subset of the positive semidefinite cone

For the same reason (closure), this applies more generally; for 0 ≤ ρ ≤ M © ª © ª M A ∈ SM (201) + | rank A = ρ = A ∈ S+ | rank A ≤ ρ For easy reference, we give such generally nonconvex sets a name: rank ρ subset of a positive semidefinite cone. For ρ < M this subset, nonconvex for M > 1, resides on the positive semidefinite cone boundary. 2.9.2.1.1 Exercise. Closure and rank ρ subset. Prove equality in (201).

H

For example, © ª © ª M M ∂ SM (202) + = A ∈ S+ | rank A = M − 1 = A ∈ S+ | rank A ≤ M − 1

116

√

CHAPTER 2. CONVEX GEOMETRY

svec S2+

2β

(a)

(b)

α ·

α β β γ

¸

Figure 43: (a) Projection of truncated PSD cone S2+ , truncated above γ = 1, on αβ-plane in isometrically isomorphic R3 . View is from above with respect above γ = ª 2. From these plots we might infer, to Figure 41. (b) √ © Truncated T for example, line [ 0 1/ 2 γ ] | γ ∈ R intercepts PSD cone at some large value of γ ; in fact, γ = ∞.

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

117

In S2 , each and every ray on the boundary of the positive semidefinite cone in isomorphic R3 corresponds to a symmetric rank-1 matrix (Figure 41), but that does not hold in any higher dimension. 2.9.2.2

Subspace tangent to open rank-ρ subset

When the positive semidefinite cone subset in (201) is left unclosed as in © ª M(ρ) , A ∈ SM (203) + | rank A = ρ

then we can specify a subspace tangent to the positive semidefinite cone at a particular member of manifold M(ρ). Specifically, the subspace RM tangent to manifold M(ρ) at B ∈ M(ρ) [173, 5, prop.1.1]

§

RM (B) , {XB + BX T | X ∈ RM ×M } ⊆ SM

(204)

has dimension ¶ µ ρ(ρ + 1) ρ−1 = ρ(M − ρ) + dim svec RM (B) = ρ M − 2 2

(205)

Tangent subspace RM contains no member of the positive semidefinite cone SM + whose rank exceeds ρ . Subspace RM (B) is a hyperplane supporting SM + when B ∈ M(M − 1). Another good example of tangent subspace is given in E.7.2.0.2 by (1923); RM (11T ) = ScM ⊥ , orthogonal complement to the geometric center subspace. (Figure 140, p.510)

§

2.9.2.3

Faces of PSD cone, their dimension versus rank

§

§

Each and every face of the positive semidefinite cone, having dimension less than that of the cone, is exposed. [229, 6] [200, 2.3.4] Because each and every face of the positive semidefinite cone contains the origin ( 2.8.0.0.1), each face belongs to a subspace of the same dimension. ¡ ¢ M Given positive semidefinite matrix A ∈ SM + , define F S+ ∋ A (154) as the smallest face that contains A of the positive semidefinite cone SM + . Then matrix A , having ordered diagonalization QΛQT ( A.5.1), is relatively interior to [26, II.12] [103, 31.5.3] [222, 2.4] [223] ( A.7.4.0.1) ¡ ¢ M F SM + ∋ A = {X ∈ S+ | N (X) ⊇ N (A)}

§

§

§

§

§

§

† T = {X ∈ SM + | hQ(I − ΛΛ )Q , X i = 0} A ≃ Srank +

(206)

118

CHAPTER 2. CONVEX GEOMETRY

A which is isomorphic with the convex cone Srank . Thus dimension of the + smallest face containing given matrix A is ¡ ¢ dim F SM (207) + ∋ A = rank(A)(rank(A) + 1)/2

in isomorphic RM (M +1)/2 , and each and every face of SM + is isomorphic with a positive semidefinite cone having dimension the same as the face. Observe: not all dimensions are represented, and the only zero-dimensional face is the origin. The positive semidefinite cone has no facets, for example. 2.9.2.3.1

Table: Rank k versus dimension of S3+ faces k 0 boundary 1 2 interior 3

dim F(S3+ ∋ rank-k matrix) 0 1 3 6

For the positive semidefinite cone S2+ in isometrically isomorphic R3 depicted in Figure 41, for example, rank-2 matrices belong to the interior of that face having dimension 3 (the entire closed cone), rank-1 matrices belong to relative interior of a face having dimension 1 ,2.38 and the only rank-0 matrix is the point at the origin (the zero-dimensional face). Any simultaneously diagonalizable positive semidefinite rank-k matrices belong to the same face (206). That observation leads to the following hyperplane characterization of PSD cone faces: Any rank-k < M positive semidefinite matrix A belongs to a face, of the positive semidefinite cone, described by an intersection with a hyperplane: for A = QΛQT and 0 ≤ k < M ¡ ¢ M † T F SM + ∋ A Ä rank(A) = k = {X ∈ S+ | hQ(I − ΛΛ )Q , X i = 0} ¯¿ µ · ¸¶ À ¾ ½ ¯ I ∈ Sk 0 M ¯ T Q ,X =0 = X ∈ S+ ¯ Q I − 0T 0 = SM + ∩ ∂H+

(208)

Faces are doubly indexed: continuously indexed by orthogonal matrix Q , and discretely indexed by rank k . Each and every orthogonal matrix Q The boundary constitutes all the one-dimensional faces, in R3 , which are rays emanating from the origin. 2.38

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

119

makes projectors Q(:, k+1 : M )Q(:, k+1 :¡M )T indexed by k , in other words, ¢ each projector describing a normal svec Q(:, k+1 : M )Q(:, k+1 : M )T to a supporting hyperplane ∂H+ (containing the origin) exposing a face ( 2.11) of the positive semidefinite cone containing only rank-k matrices.

§

2.9.2.4

Extreme directions of positive semidefinite cone

§

Because the positive semidefinite cone is pointed ( 2.7.2.1.2), there is a one-to-one correspondence of one-dimensional faces with extreme directions in any dimension M ; id est, because of the cone faces lemma ( 2.8.0.0.1) and the direct correspondence of exposed faces to faces of SM + , it follows there is no one-dimensional face of the positive semidefinite cone that is not a ray emanating from the origin. Symmetric dyads constitute the set of all extreme directions: For M > 1

§

{yy T ∈ SM | y ∈ RM } ⊂ ∂ SM +

(209)

this superset of extreme directions (infinite in number, confer (172)) for the positive semidefinite cone is, generally, a subset of the boundary. By the extremes theorem ( 2.8.1.1.1), the convex hull of extreme rays and the origin is the PSD cone: conv{yy T ∈ SM | y ∈ RM } = SM (210) +

§

For two-dimensional matrices (M = 2, Figure 41) {yy T ∈ S2 | y ∈ R2 } = ∂ S2+

§

(211)

while for one-dimensional matrices, in exception, (M = 1, 2.7) {yy ∈ S | y 6= 0} = int S+

(212)

Each and every extreme direction yy T makes the same angle with the identity matrix in isomorphic RM (M +1)/2 , dependent only on dimension; videlicet,2.39 ¶ µ hyy T , I i 1 T ∀ y ∈ RM (213) Á(yy , I ) = arccos = arccos √ T kyy kF kIkF M 2.39

Analogy with respect to the EDM cone is considered in [171, p.162] where it is found: angle is not constant. Extreme directions of the EDM cone can be found in §6.5.3.2. The cone’s axis is −E = 11T − I (1041).

120

CHAPTER 2. CONVEX GEOMETRY

§

2.9.2.4.1 Example. Positive semidefinite matrix from extreme directions. Diagonalizability ( A.5) of symmetric matrices yields the following results: Any symmetric positive semidefinite matrix (1394) can be written in the form X X A = λ i zi ziT = AˆAˆT = a ˆi a ˆT λº0 (214) i º 0, i

i

T a conic combination of linearly independent extreme directions (ˆ ai a ˆT i or zi zi where kzi k = 1), where λ is a vector of eigenvalues. If we limit consideration to all symmetric positive semidefinite matrices bounded via unity trace

C , {A º 0 | tr A = 1}

(86)

then any matrix A from that set may be expressed as a convex combination of linearly independent extreme directions; X 1T λ = 1 , λ º 0 (215) A= λ i zi ziT ∈ C , i

Implications are:

1. set C is convex (an intersection of PSD cone with hyperplane),

§

2. because the set of eigenvalues corresponding to a given square matrix A is unique ( A.5.0.1), no single eigenvalue can exceed 1 ; id est, I º A 3. and the converse holds: set C is an instance of Fantope (86). 2.9.2.5

2

Positive semidefinite cone is generally not circular

Extreme angle equation (213) suggests that the positive semidefinite cone might be invariant to rotation about its axis of revolution; id est, a circular cone. We investigate this now: 2.9.2.5.1 Definition. Circular cone:2.40 a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation. △ 2.40

A circular cone is assumed convex throughout, although not so by other authors. We also assume a right circular cone.

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

121

R

0

Figure 44: This solid circular cone in R3 continues upward infinitely. Axis of revolution is illustrated as vertical line through origin. R represents radius: distance measured from an extreme direction to axis of revolution. Were this a Lorentz cone, any plane slice containing axis of revolution would make a right angle.

122

CHAPTER 2. CONVEX GEOMETRY

A conic section is the intersection of a cone with any hyperplane. In three dimensions, an intersecting plane perpendicular to a circular cone’s axis of revolution produces a section bounded by a circle. (Figure 44) A prominent example of a circular cone in convex analysis is Lorentz cone (163). We also find that the positive semidefinite cone and cone of Euclidean distance matrices are circular cones, but only in low dimension. The positive semidefinite cone has axis of revolution that is the ray (base 0) through the identity matrix I . Consider a set of normalized extreme directions of the positive semidefinite cone: for some arbitrary positive constant a ∈ R+ √ {yy T ∈ SM | kyk = a} ⊂ ∂ SM (216) + The distance from each extreme direction to the axis of revolution is radius r

R , inf kyy T − cIkF = a 1 − c

1 M

(217)

which is the distance from yy T to Ma I ; the length of vector yy T − Ma I . Because distance R (in a particular dimension) from the axis of revolution to each and every normalized extreme direction is identical, the extreme directions lie on the boundary of a hypersphere in isometrically isomorphic RM (M +1)/2 . From Example 2.9.2.4.1, the convex hull (excluding vertex at the origin) of the normalized extreme directions is a conic section M C , conv{yy T | y ∈ RM , y Ty = a} = SM | hI , Ai = a} + ∩ {A ∈ S

(218)

orthogonal to identity matrix I ; hC −

a a I , I i = tr(C − I ) = 0 M M

(219)

Proof. Although the positive semidefinite cone possesses some characteristics of a circular cone, we can show it is not by demonstrating shortage of extreme directions; id est, some extreme directions corresponding to each and every angle of rotation about the axis of revolution are nonexistent: Referring to Figure 45, [355, 1-7]

§

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

123

yy T

a I M

θ

a 11T M

R

Figure 45: Illustrated is a section, perpendicular to axis of revolution, of circular cone from Figure 44. Radius R is distance from any extreme direction to axis at Ma I . Vector Ma 11T is an arbitrary reference by which to measure angle θ .

11T − Ma I , yy T − cos θ = M a2 (1 − M1 ) ­a

a I M

®

(220)

Solving for vector y we get a(1 + (M − 1) cos θ) = (1T y)2

(221)

which does not have real solution ∀ 0 ≤ θ ≤ 2π in every matrix dimension M . ¨ From the foregoing proof we can conclude that the positive semidefinite cone might be circular but only in matrix dimensions 1 and 2. Because of a shortage of extreme directions, conic section (218) cannot be hyperspherical by the extremes theorem ( 2.8.1.1.1, Figure 40).

§

2.9.2.5.2 Exercise. Circular semidefinite cone. Prove the PSD cone to be circular in matrix dimensions 1 and 2 while it is

124

CHAPTER 2. CONVEX GEOMETRY H

a rotation of Lorentz cone (163) in matrix dimension 2 .2.41 2.9.2.5.3

Example. PSD cone inscription in three dimensions.

§

Theorem. Gerˇsgorin discs. [188, 6.1] [339] [237, p.140] m m For p ∈ R+ given A = [Aij ] ∈ S , then all eigenvalues of A belong to the union of m closed intervals on the real line;        m  m m S S 1 P λ(A) ∈ ξ ∈ R |ξ − Aii | ≤ ̺i , pj |Aij | = [Aii −̺i , Aii +̺i ]  pi j=1 i=1  i=1     j 6= i (222) Furthermore, if a union of k of these m [intervals] forms a connected region that is disjoint from all the remaining n − k [intervals], then there are precisely k eigenvalues of A in this region. ⋄ To apply the theorem to determine positive semidefiniteness of symmetric matrix A , we observe that for each i we must have Aii ≥ ̺i

(223)

m=2

(224)

Suppose so A ∈ S2 . Vectorizing A as in (51), svec A belongs to isometrically isomorphic R3 . Then we have m2 m−1 = 4 inequalities, in the matrix entries Aij with Gerˇsgorin parameters p = [pi ] ∈ R2+ , p1 A11 ≥ ±p2 A12 p2 A22 ≥ ±p1 A12

(225)

which describe an intersection of four halfspaces in Rm(m+1)/2 . That intersection creates the proper polyhedral cone K ( 2.12.1) whose construction is illustrated in Figure 46. Drawn truncated is the boundary of the positive semidefinite cone svec S2+ and the bounding hyperplanes supporting K .

§

√ ¸ ¾ · p α√ β/ 2 γ+α 1 2 2 √ | α + β ≤ γ , show 2 Hint: Given cone β γ β/ 2 a vector rotation that is positive semidefinite under the same inequality. 2.41

½·

β γ−α

¸

is

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

125

0 -1 -0.5

p=

1/2 1

¸

p1 A11 ≥ ±p2 A12 p2 A22 ≥ ±p1 A12

0

0.5

·

0.5 1

1

1 0.75 0.5 0.25 0

0 -1 -0.5 0

0.5

0.5

p=

·

1 1

¸

1

1

1 0.75 0.5 0.25 0

0 -1 -0.5

A11 p=

·

2 1

¸

1

0.5

0

√

2A12 0.5 1 1 0.75 0.5 0.25 0

A22

svec ∂ S2+ Figure 46: Proper polyhedral cone K , created by intersection of halfspaces, inscribes PSD cone in isometrically isomorphic R3 as predicted by Gerˇsgorin discs theorem for A = [Aij ] ∈ S2 . Hyperplanes supporting K intersect along boundary of PSD cone. Four extreme directions of K coincide with extreme directions of PSD cone.

126

CHAPTER 2. CONVEX GEOMETRY

Created by means of Gerˇsgorin discs, K always belongs to the positive semidefinite cone for any nonnegative value of p ∈ Rm + . Hence any point in K corresponds to some positive semidefinite matrix A . Only the extreme directions of K intersect the positive semidefinite cone boundary in this dimension; the four extreme directions of K are extreme directions of the positive semidefinite cone. As p1 /p2 increases in value from 0, two extreme directions of K sweep the entire boundary of this positive semidefinite cone. Because the entire positive semidefinite cone can be swept by K , the system of linear inequalities √ ¸ · p1 ±p2 /√2 0 T svec A º 0 (226) Y svec A , 0 ±p1 / 2 p2 when made dynamic can replace a semidefinite constraint A º 0 ; id est, for K = {z | Y Tz º 0} ⊂ svec Sm + given p where Y ∈ Rm(m+1)/2×m2

(227)

m−1

svec A ∈ K ⇒ A ∈ Sm +

(228)

∃ p Ä Y T svec A º 0 ⇔ A º 0

(229)

but

§

In other words, diagonal dominance [188, p.349, 7.2.3] Aii ≥

m X j=1 j 6= i

|Aij | ,

∀i = 1...m

(230)

is only a sufficient condition for membership to the PSD cone; but by dynamic weighting p in this dimension, it was made necessary and sufficient. 2 In higher dimension (m > 2), boundary of the positive semidefinite cone is no longer constituted completely by its extreme directions (symmetric rank-one matrices); its geometry becomes intricate. How all the extreme directions can be swept by an inscribed polyhedral cone,2.42 similarly to the foregoing example, remains an open question. 2.42

It is not necessary to sweep the entire boundary in higher dimension.

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

127

2.9.2.5.4 Exercise. Dual inscription. ∗ Find dual proper polyhedral cone K from Figure 46. 2.9.2.6

H

Boundary constituents of the positive semidefinite cone

2.9.2.6.1 Lemma. Sum of positive semidefinite matrices. For A,B ∈ SM + rank(A + B) = rank(µA + (1 − µ)B)

(231) ⋄

over the open interval (0, 1) of µ .

Proof. Any positive semidefinite matrix belonging to the PSD cone has an eigen decomposition that is a positively scaled sum of linearly independent symmetric dyads. By the linearly independent dyads definition in B.1.1.0.1, rank of the sum A + B is equivalent to the number of linearly independent dyads constituting it. Linear independence is insensitive to further positive scaling by µ . The assumption of positive semidefiniteness prevents annihilation of any dyad from the sum A + B . ¨

§

§

§

2.9.2.6.2 Example. Rank function quasiconcavity. For A,B ∈ Rm×n [188, 0.4]

(confer 3.9)

rank A + rank B ≥ rank(A + B)

(232)

§

that follows from the fact [309, 3.6] dim R(A) + dim R(B) = dim R(A + B) + dim(R(A) ∩ R(B))

(233)

For A,B ∈ SM + rank A + rank B ≥ rank(A + B) ≥ min{rank A , rank B}

(1410)

that follows from the fact N (A + B) = N (A) ∩ N (B) ,

A,B ∈ SM +

(149)

Rank is a quasiconcave function on SM + because the right-hand inequality in (1410) has the concave form (607); videlicet, Lemma 2.9.2.6.1. 2

§

From this example we see, unlike convex functions, quasiconvex functions are not necessarily continuous. ( 3.9) We also glean:

128

CHAPTER 2. CONVEX GEOMETRY

2.9.2.6.3 Theorem. Convex subsets of positive semidefinite cone. Subsets of the positive semidefinite cone SM + , for 0 ≤ ρ ≤ M M SM + (ρ) , {X ∈ S+ | rank X ≥ ρ}

(234)

M are pointed convex cones, but not closed unless ρ = 0 ; id est, SM + (0) = S+ . ⋄

Proof. Given ρ , a subset SM + (ρ) is convex if and only if convex combination of any two members has rank at least ρ . That is confirmed applying identity (231) from Lemma 2.9.2.6.1 to (1410); id est, for A,B ∈ SM + (ρ) on the closed interval µ ∈ [0, 1] rank(µA + (1 − µ)B) ≥ min{rank A , rank B}

(235)

It can similarly be shown, almost identically to proof of the lemma, any conic combination of A,B in subset SM + (ρ) remains a member; id est, ∀ ζ , ξ ≥ 0 rank(ζ A + ξB) ≥ min{rank(ζ A) , rank(ξB)} Therefore, SM + (ρ) is a convex cone.

(236) ¨

Another proof of convexity can be made by projection arguments: 2.9.2.7

Projection on SM + (ρ)

Because these cones SM + (ρ) indexed by ρ (234) are convex, projection on them is straightforward. Given a symmetric matrix H having diagonalization H , QΛQT ∈ SM ( A.5.1) with eigenvalues Λ arranged in nonincreasing order, then its Euclidean projection (minimum-distance projection) on SM + (ρ)

§

PSM H = QΥ⋆ QT + (ρ) corresponds to a map of its eigenvalues: ½ max {ǫ , Λii } , i = 1 . . . ρ ⋆ Υii = max {0 , Λii } , i = ρ+1 . . . M where ǫ is positive but arbitrarily close to 0.

(237)

(238)

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

129

2.9.2.7.1 Exercise. Projection on open convex cones. Prove (238) using Theorem E.9.2.0.1.

H

Because each H ∈ SM has unique projection on SM + (ρ) (despite possibility of repeated eigenvalues in Λ), we may conclude it is a convex set by the Bunt-Motzkin theorem ( E.9.0.0.1). Compare (238) to the well-known result regarding Euclidean projection on a rank ρ subset of the positive semidefinite cone ( 2.9.2.1)

§

§

M M SM + \ S+ (ρ + 1) = {X ∈ S+ | rank X ≤ ρ} M PSM H = QΥ⋆ QT + \S+ (ρ+1)

§

(239) (240)

As proved in 7.1.4, this projection of H corresponds to the eigenvalue map ½ max {0 , Λii } , i = 1 . . . ρ ⋆ Υii = (1287) 0, i = ρ+1 . . . M Together these two results (238) and (1287) mean: A higher-rank solution to projection on the positive semidefinite cone lies arbitrarily close to any given lower-rank projection, but not vice versa. Were the number of nonnegative eigenvalues in Λ known a priori not to exceed ρ , then these two different projections would produce identical results in the limit ǫ → 0. 2.9.2.8

Uniting constituents

Interior of the PSD cone int SM + is convex by Theorem 2.9.2.6.3, for example, because all positive semidefinite matrices having rank M constitute the cone interior. All positive semidefinite matrices of rank less than M constitute the cone boundary; an amalgam of positive semidefinite matrices of different rank. Thus each nonconvex subset of positive semidefinite matrices, for 0 < ρ < M {Y ∈ SM + | rank Y = ρ}

(241)

having rank ρ successively 1 lower than M , appends a nonconvex constituent to the cone boundary; but only in their union is the boundary complete: (confer 2.9.2)

§

∂ SM + =

M −1 [ ρ=0

{Y ∈ SM + | rank Y = ρ}

(242)

130

CHAPTER 2. CONVEX GEOMETRY

The composite sequence, the cone interior in union with each successive constituent, remains convex at each step; id est, for 0 ≤ k ≤ M M [

ρ=k

{Y ∈ SM + | rank Y = ρ}

(243)

is convex for each k by Theorem 2.9.2.6.3. 2.9.2.9

Peeling constituents

Proceeding the other way: To peel constituents off the complete positive semidefinite cone boundary, one starts by removing the origin; the only rank-0 positive semidefinite matrix. What remains is convex. Next, the extreme directions are removed because they constitute all the rank-1 positive semidefinite matrices. What remains is again convex, and so on. Proceeding in this manner eventually removes the entire boundary leaving, at last, the convex interior of the PSD cone; all the positive definite matrices. 2.9.2.9.1 Exercise. Difference A − B . What about the difference of matrices A,B belonging to the positive semidefinite cone? Show:

ˆ The difference of any two points on the boundary belongs to the boundary or exterior.

ˆ The difference A − B , where A belongs to the boundary while B is H

interior, belongs to the exterior.

2.9.3

Barvinok’s proposition

Barvinok posits existence and quantifies an upper bound on rank of a positive semidefinite matrix belonging to the intersection of the PSD cone with an affine subset:

§

§

2.9.3.0.1 Proposition. (Barvinok) Affine intersection with PSD cone. [26, II.13] [24, 2.2] Consider finding a matrix X ∈ SN satisfying Xº0,

hAj , X i = bj ,

j =1 . . . m

(244)

2.9. POSITIVE SEMIDEFINITE (PSD) CONE

131

given nonzero linearly independent Aj ∈ SN and real bj . Define the affine subset A , {X | hAj , X i = bj , j = 1 . . . m} ⊆ SN (245)

If the intersection A ∩ SN + is nonempty given a number m of equalities, then there exists a matrix X ∈ A ∩ SN + such that rank X (rank X + 1)/2 ≤ m

(246)

whence the upper bound2.43 rank X ≤

¹√

8m + 1 − 1 2

º

(247)

Given desired rank instead, equivalently, m < (rank X + 1)(rank X + 2)/2

§

(248)

An extreme point of A ∩ SN + satisfies (247) and (248). (confer 4.1.1.3) T A matrix X , R R is an extreme point if and only if the smallest face that contains X of A ∩ SN + has dimension 0 ; [222, 2.4] [223] id est, iff (154) ¡ ¢ dim F (A ∩ SN (249) +)∋ X £ ¤ T T = rank(X )(rank(X ) + 1)/2 − rank svec RA1 R svec RA2 R · · · svec RAm RT

§

equals 0 in isomorphic RN (N +1)/2 . Now the intersection A ∩ SN + is assumed bounded: Assume a given nonzero upper bound ρ on rank, a number of equalities m = (ρ + 1)(ρ + 2)/2

(250)

and matrix dimension N ≥ ρ + 2 ≥ 3. If the intersection is nonempty and bounded, then there exists a matrix X ∈ A ∩ SN + such that rank X ≤ ρ

(251)

This represents a tightening of the upper bound; a reduction by exactly 1 of the bound provided by (247) given the same specified number m (250) of equalities; id est, √ 8m + 1 − 1 −1 (252) rank X ≤ 2 ⋄ 2.43

§4.1.1.3 contains an intuitive explanation. This bound is itself limited above, of course, by N ; a tight limit corresponding to an interior point of SN + .

132

CHAPTER 2. CONVEX GEOMETRY

When the intersection A ∩ SN + is known a priori to consist only of a single point, then Barvinok’s proposition provides the greatest upper bound on its rank not exceeding N . The intersection can be a single nonzero point only if the number of linearly independent hyperplanes m constituting A satisfies2.44 N (N + 1)/2 − 1 ≤ m ≤ N (N + 1)/2 (253)

2.10

Conic independence (c.i.)

In contrast to extreme direction, the property conically independent direction is more generally applicable; inclusive of all closed convex cones (not only pointed closed convex cones). Arbitrary given directions {Γi ∈ Rn , i = 1 . . . N } are conically independent if and only if, for all ζ ∈ RN + (confer 2.1.2, 2.4.2.3)

§

§

Γi ζi + · · · + Γj ζj − Γℓ = 0 ,

i 6= · · · = 6 j 6= ℓ = 1 . . . N

(254)

has no solution ζ ; in words, iff no direction from the given set can be expressed as a conic combination of those remaining; e.g., Figure 47. [Matlab implementation of test (254) on Wıκımization] It is evident that linear independence (l.i.) of N directions implies their conic independence;

ˆ l.i. ⇒ c.i.

Arranging any set of generators for a particular convex cone in a matrix columnar, X , [ Γ1 Γ2 · · · ΓN ] ∈ Rn×N (255)

then the relationship l.i. ⇒ c.i. suggests: the number of l.i. generators in the columns of X cannot exceed the number of c.i. generators. Denoting by k the number of conically independent generators contained in X , we have the most fundamental rank inequality for convex cones dim aff K = dim aff[ 0 X ] = rank X ≤ k ≤ N

(256)

Whereas N directions in n dimensions can no longer be linearly independent once N exceeds n , conic independence remains possible: For N > 1, N (N + 1)/2 − 1 independent hyperplanes in RN (N +1)/2 can make a line N tangent to svec ∂ SN + at a point because all one-dimensional faces of S+ are exposed. Because a pointed convex cone has only one vertex, the origin, there can be no intersection of svec ∂ SN + with any higher-dimensional affine subset A that will make a nonzero point. 2.44

2.10. CONIC INDEPENDENCE (C.I.)

133

0

0

0

(a)

(b)

(c)

Figure 47: Vectors in R2 : (a) affinely and conically independent, (b) affinely independent but not conically independent, (c) conically independent but not affinely independent. None of the examples exhibits linear independence. (In general, a.i. < c.i.)

2.10.0.0.1

Table: Maximum number of c.i. directions dimension n 0 1 2 3 .. .

sup k (pointed) sup k (not pointed) 0 1 2 ∞ .. .

0 2 4 ∞ .. .

Assuming veracity of this table, there is an apparent vastness between two and three dimensions. The finite numbers of conically independent directions indicate:

ˆ Convex cones in dimensions 0, 1, and 2 must be polyhedral. (§2.12.1) Conic independence is certainly one convex idea that cannot be completely explained by a two-dimensional picture as Barvinok suggests [26, p.vii]. From this table it is also evident that dimension of Euclidean space cannot exceed the number of conically independent directions possible;

ˆ n ≤ sup k

134

CHAPTER 2. CONVEX GEOMETRY

K (a)

K ∂K

∗

(b)

Figure 48: (a) A pointed polyhedral cone (drawn truncated) in R3 having six facets. The extreme directions, corresponding to six edges emanating from the origin, are generators for this cone; not linearly independent but they ∗ must be conically independent. (b) The boundary of dual cone K (drawn ∗ truncated) is now added to the drawing of same K . K is polyhedral, proper, and has the same number of extreme directions as K has facets.

2.10. CONIC INDEPENDENCE (C.I.)

2.10.1

135

Preservation of conic independence

§

§

Independence in the linear ( 2.1.2.1), affine ( 2.4.2.4), and conic senses can be preserved under linear transformation. Suppose a matrix X ∈ Rn×N (255) holds a conically independent set columnar. Consider a transformation on the domain of such matrices T (X) : Rn×N → Rn×N , XY

(257)

where fixed matrix Y , [ y1 y2 · · · yN ] ∈ RN ×N represents linear operator T . Conic independence of {Xyi ∈ Rn , i = 1 . . . N } demands, by definition (254), Xyi ζi + · · · + Xyj ζj − Xyℓ = 0 ,

i 6= · · · = 6 j 6= ℓ = 1 . . . N

(258)

N have no solution ζ ∈ RN + . That is ensured by conic independence of {yi ∈ R } and by R(Y )∩ N (X) = 0 ; seen by factoring out X .

2.10.1.1

§

linear maps of cones

[21, 7] If K is a convex cone in Euclidean space R and T is any linear mapping from R to Euclidean space M , then T (K) is a convex cone in M and x ¹ y with respect to K implies T (x) ¹ T (y) with respect to T (K). If K has nonempty interior in R , then so does T (K) in M . If T is a linear bijection, then x ¹ y ⇔ T (x) ¹ T (y). And if K is closed, then so is T (K). If F is a face of K , then T (F ) is a face of T (K). Linear bijection is only a sufficient condition for closedness; convex polyhedra ( 2.12) are invariant to any linear or inverse linear transformation [26, I.9] [285, p.44, thm.19.3].

§

2.10.2

§

Pointed closed convex K & conic independence

§

The following bullets can be derived from definitions (171) and (254) in conjunction with the extremes theorem ( 2.8.1.1.1): The set of all extreme directions from a pointed closed convex cone K ⊂ Rn is not necessarily a linearly independent set, yet it must be a conically independent set; (compare Figure 24 on page 70 with Figure 48a)

ˆ {extreme directions} ⇒ {c.i.}

When a conically independent set of directions from pointed closed convex cone K is known to comprise generators, conversely, then all directions from that set must be extreme directions of the cone;

136

CHAPTER 2. CONVEX GEOMETRY

ˆ {extreme directions} ⇔ {c.i. generators of pointed closed convex K} Barker & Carlson [21, §1] call the extreme directions a minimal generating

set for a pointed closed convex cone. A minimal set of generators is therefore a conically independent set of generators, and vice versa,2.45 for a pointed closed convex cone. An arbitrary collection of n or fewer distinct extreme directions from pointed closed convex cone K ⊂ Rn is not necessarily a linearly independent set; e.g., dual extreme directions (453) from Example 2.13.11.0.3.

ˆ {≤ n extreme directions in R } ; {l.i.} n

Linear dependence of few extreme directions is another convex idea that cannot be explained by a two-dimensional picture as Barvinok suggests [26, p.vii]; indeed, it only first comes to light in four dimensions! But there is a converse: [308, 2.10.9]

§

ˆ {extreme directions} ⇐ {l.i. generators of closed convex K}

2.10.2.0.1 Example. Vertex-description of halfspace H about origin. From n + 1 points in Rn we can make a vertex-description of a convex cone that is a halfspace H , where {xℓ ∈ Rn , ℓ = 1 . . . n} constitutes a minimal set of generators for a hyperplane ∂H through the origin. An example is illustrated in Figure 49. By demanding the augmented set {xℓ ∈ Rn , ℓ = 1 . . . n + 1} be affinely independent (we want vector xn+1 not parallel to ∂H ), then H = =

S

(ζ xn+1 + ∂H)

ζ ≥0

{ζ xn+1 + cone{xℓ ∈ Rn , ℓ = 1 . . . n} | ζ ≥ 0}

(259)

= cone{xℓ ∈ Rn , ℓ = 1 . . . n + 1} a union of parallel hyperplanes. Cardinality is one step beyond dimension of the ambient space, but {xℓ ∀ ℓ} is a minimal set of generators for this convex cone H which has no extreme elements. 2 2.45

This converse does not hold for nonpointed closed convex cones as Table 2.10.0.0.1 implies; e.g., ponder four conically independent generators for a plane (n = 2, Figure 47).

2.10. CONIC INDEPENDENCE (C.I.)

137

H

x2

x3 0 ∂H x1 Figure 49: Minimal set of generators X = [ x1 x2 x3 ] ∈ R2×3 (not extreme directions) for halfspace about origin; affinely and conically independent.

2.10.2.0.2 Exercise. Enumerating conically independent directions. Describe a nonpointed polyhedral cone in three dimensions having more than 8 conically independent generators. (confer Table 2.10.0.0.1) H

2.10.3

Utility of conic independence

Perhaps the most useful application of conic independence is determination of the intersection of closed convex cones from their halfspace-descriptions, or representation of the sum of closed convex cones from their vertex-descriptions. T

Ki

A halfspace-description for the intersection of any number of closed convex cones Ki can be acquired by pruning normals; specifically, only the conically independent normals from the aggregate of all the halfspace-descriptions need be retained.

P

Ki

Generators for the sum of any number of closed convex cones Ki can be determined by retaining only the conically independent generators from the aggregate of all the vertex-descriptions.

138

CHAPTER 2. CONVEX GEOMETRY

Such conically independent sets are not necessarily unique or minimal.

2.11

When extreme means exposed

For any convex polyhedral set in Rn having nonempty interior, distinction between the terms extreme and exposed vanishes [308, 2.4] [103, 2.2] for faces of all dimensions except n ; their meanings become equivalent as we saw in Figure 20 (discussed in 2.6.1.2). In other words, each and every face of any polyhedral set (except the set itself) can be exposed by a hyperplane, and vice versa; e.g., Figure 24. Lewis [229, 6] [200, 2.3.4] claims nonempty extreme proper subsets and n the exposed subsets coincide for S+ ; id est, each and every face of the positive semidefinite cone, whose dimension is less than dimension of the cone, is exposed. A more general discussion of cones having this property can be found in [318]; e.g., Lorentz cone (163) [20, II.A].

§

§

§

§

§

§

2.12

Convex polyhedra

Every polyhedron, such as the convex hull (81) of a bounded list X , can be expressed as the solution set of a finite system of linear equalities and inequalities, and vice versa. [103, 2.2]

§

2.12.0.0.1 Definition. Convex polyhedra, halfspace-description. A convex polyhedron is the intersection of a finite number of halfspaces and hyperplanes; P = {y | Ay º b , C y = d} ⊆ Rn (260) where coefficients A and C generally denote matrices. Each row of C is a vector normal to a hyperplane, while each row of A is a vector inward-normal to a hyperplane partially bounding a halfspace. △

§

By the halfspaces theorem in 2.4.1.1.1, a polyhedron thus described is a closed convex set having possibly empty interior; e.g., Figure 20. Convex polyhedra2.46 are finite-dimensional comprising all affine sets ( 2.3.1), 2.46

§

We consider only convex polyhedra throughout, but acknowledge the existence of concave polyhedra. [348, Kepler-Poinsot Solid ]

2.12. CONVEX POLYHEDRA

139

polyhedral cones, line segments, rays, halfspaces, convex polygons, solids [208, def.104/6, p.343], polychora, polytopes,2.47 etcetera. It follows from definition (260) by exposure that each face of a convex polyhedron is a convex polyhedron. The projection of any polyhedron on a subspace remains a polyhedron. More generally, image and inverse image of a convex polyhedron under any linear transformation remains a convex polyhedron; [26, I.9] [285, thm.19.3] the foremost consequence being, invariance of polyhedral set closedness. When b and d in (260) are 0, the resultant is a polyhedral cone. The set of all polyhedral cones is a subset of convex cones:

§

2.12.1

Polyhedral cone

From our study of cones, we see: the number of intersecting hyperplanes and halfspaces constituting a convex cone is possibly but not necessarily infinite. When the number is finite, the convex cone is termed polyhedral. That is the primary distinguishing feature between the set of all convex cones and polyhedra; all polyhedra, including polyhedral cones, are finitely generated [285, 19]. (Figure 50) We distinguish polyhedral cones in the set of all convex cones for this reason, although all convex cones of dimension 2 or less are polyhedral.

§

2.12.1.0.1 Definition. Polyhedral cone, halfspace-description.2.48 (confer (97)) A polyhedral cone is the intersection of a finite number of halfspaces and hyperplanes about the origin; K = {y | Ay º 0 , Cy = 0} ⊆ Rn

= {y | Ay º 0 , Cy º 0 , Cy ¹ 0}     A     C yº0 = y|   −C

(a) (b) (261) (c)

where coefficients A and C generally denote matrices of finite dimension. Each row of C is a vector normal to a hyperplane containing the origin, 2.47

Some authors distinguish bounded polyhedra via the designation polytope. [103, §2.2] Rockafellar [285, §19] proposes affine sets be handled via complementary pairs of affine inequalities; e.g., Cy º d and Cy ¹ d which, when taken together, can present severe difficulties to some interior-point methods of numerical solution. 2.48

140

CHAPTER 2. CONVEX GEOMETRY polyhedra convex cones polyhedral cones

bounded polyhedra

Figure 50: Polyhedral cones are finitely generated, unbounded, and convex.

while each row of A is a vector inward-normal to a hyperplane containing the origin and partially bounding a halfspace. △

§

§

A polyhedral cone thus defined is closed, convex ( 2.4.1.1), has only a finite number of generators ( 2.8.1.2), and can have empty interior. (Minkowski) Conversely, any finitely generated convex cone is polyhedral. (Weyl) [308, 2.8] [285, thm.19.1] From the definition it follows that any single hyperplane through the origin, or any halfspace partially bounded by a hyperplane through the origin is a polyhedral cone. The most familiar example of polyhedral cone is any quadrant (or orthant, 2.1.3) generated by Cartesian half-axes. Esoteric examples of polyhedral cone include the point at the origin, any line through the origin, any ray having the origin as base such as the nonnegative real line R+ in subspace R , polyhedral flavor (proper) Lorentz cone (290), any subspace, and Rn . More polyhedral cones are illustrated in Figure 48 and Figure 24.

§

§

2.12.2

Vertices of convex polyhedra

§

By definition, a vertex ( 2.6.1.0.1) always lies on the relative boundary of a convex polyhedron. [208, def.115/6, p.358] In Figure 20, each vertex of the polyhedron is located at the intersection of three or more facets, and every edge belongs to precisely two facets [26, VI.1, p.252]. In Figure 24, the only vertex of that polyhedral cone lies at the origin.

§

2.12. CONVEX POLYHEDRA

141

The set of all polyhedral cones is clearly a subset of convex polyhedra and a subset of convex cones (Figure 50). Not all convex polyhedra are bounded; evidently, neither can they all be described by the convex hull of a bounded set of points as defined in (81). Hence a universal vertex-description of polyhedra in terms of that same finite-length list X (71):

§

2.12.2.0.1 Definition. Convex polyhedra, vertex-description. (confer 2.8.1.1.1) Denote the truncated a-vector ai:ℓ =

"

ai .. . aℓ

#

(262)

By discriminating a suitable finite-length generating list (or set) arranged columnar in X ∈ Rn×N , then any particular polyhedron may be described © ª P = Xa | aT (263) 1:k 1 = 1 , am:N º 0 , {1 . . . k} ∪ {m . . . N } = {1 . . . N }

where 0 ≤ k ≤ N and 1 ≤ m ≤ N +1. Setting k = 0 removes the affine equality condition. Setting m = N +1 removes the inequality. △ Coefficient indices in (263) may or may not be overlapping, but all the coefficients are assumed constrained. From (73), (81), and (97), we summarize how the coefficient conditions may be applied; affine sets −→ aT 1:k 1 = 1 polyhedral cones −→ am:N º 0

¾

←− convex hull (m ≤ k)

(264)

It is always possible to describe a convex hull in the region of overlapping indices because, for 1 ≤ m ≤ k ≤ N T {am:k | aT m:k 1 = 1, am:k º 0} ⊆ {am:k | a1:k 1 = 1, am:N º 0}

(265)

Members of a generating list are not necessarily vertices of the corresponding polyhedron; certainly true for (81) and (263), some subset of list members reside in the polyhedron’s relative interior. Conversely, when boundedness (81) applies, convex hull of the vertices is a polyhedron identical to convex hull of the generating list.

142

CHAPTER 2. CONVEX GEOMETRY

2.12.2.1

Vertex-description of polyhedral cone

Given closed convex cone K in a subspace of Rn having any set of generators for it arranged in a matrix X ∈ Rn×N as in (255), then that cone is described setting m = 1 and k = 0 in vertex-description (263): K = cone X = {Xa | a º 0} ⊆ Rn

(97)

a conic hull of N generators.

2.12.2.2

§

Pointedness

[308, 2.10] Assuming all generators constituting the columns of X ∈ Rn×N are nonzero, polyhedral cone K is pointed ( 2.7.2.1.2) if and only if there is no nonzero a º 0 that solves Xa = 0 ; id est, iff

§

find a subject to Xa = 0 1T a = 1 aº0

(266)

2.49 is infeasible or iff N (X) ∩ RN +=0 . A proper polyhedral cone in Rn must have at least n linearly independent generators, or be the intersection of at least n halfspaces whose partial boundaries have normals that are linearly independent. Otherwise, the cone will contain at least one line and there can be no vertex; id est, the cone cannot otherwise be pointed. Euclidean vector space Rn or any halfspace is a good example of full-dimensional nonpointed polyhedral cone; hence, has no vertex. Examples of pointed closed convex cones K are not limited to polyhedral cones: the origin, any 0-based ray in a subspace, any two-dimensional V-shaped cone in a subspace, any Lorentz cone, the cone of Euclidean M M distance matrices EDMN in SN h , the proper cones: S+ in ambient S , any orthant in Rn or Rm×n ; e.g., the nonnegative real line R+ in vector space R . 2.49

∗

If rank X = n , then the dual cone K (§2.13.1) is pointed. (283)

2.12. CONVEX POLYHEDRA

143

S = {s | s º 0 , 1Ts ≤ 1}

1

Figure 51: Unit simplex S in R3 is a unique solid tetrahedron but not regular.

144

CHAPTER 2. CONVEX GEOMETRY

2.12.3

Unit simplex

A peculiar subset of the nonnegative orthant with halfspace-description S , {s | s º 0 , 1Ts ≤ 1} ⊆ Rn+

(267)

is a unique bounded convex polyhedron called unit simplex (Figure 51) having nonempty interior, n + 1 facets, n + 1 vertices, and dimension dim S = n

(268)

The origin supplies one vertex while heads of the standard basis [188] [309] {ei , i = 1 . . . n} in Rn constitute those remaining;2.50 thus its vertex-description:

2.12.3.1

S = conv {0, {ei , i = 1 . . . n}} © ª = [ 0 e1 e2 · · · en ] a | aT 1 = 1 , a º 0

(269)

Simplex

The unit simplex comes from a class of general polyhedra called simplex, having vertex-description: [89] [285] [346] [103] conv{xℓ ∈ Rn } | ℓ = 1 . . . k+1 , dim aff{xℓ } = k , n ≥ k

(270)

So defined, a simplex is a closed bounded convex set having possibly empty interior. Examples of simplices, by increasing affine dimension, are: a point, any line segment, any triangle and its relative interior, a general tetrahedron, any five-vertex polychoron, and so on.

§

2.12.3.1.1 Definition. Simplicial cone. A proper polyhedral ( 2.7.2.2.1) cone K in Rn is called simplicial iff K has exactly n extreme directions; [20, II.A] equivalently, iff proper K has exactly n linearly independent generators contained in any given set of generators. △

§

ˆ simplicial cone ⇒ proper polyhedral cone

There are an infinite variety of simplicial cones in Rn ; e.g., Figure 24, Figure 52, Figure 61. Any orthant is simplicial, as is any rotation thereof. In R0 the unit simplex is the point at the origin, in R the unit simplex is the line segment [0, 1], in R2 it is a triangle and its relative interior, in R3 it is the convex hull of a tetrahedron (Figure 51), in R4 it is the convex hull of a pentatope [348], and so on. 2.50

2.12. CONVEX POLYHEDRA

145

Figure 52: Two views of a simplicial cone and its dual in R3 (second view on next page). Semiinfinite boundary of each cone is truncated for illustration. Each cone has three facets (confer 2.13.11.0.3). Cartesian axes are drawn for reference.

§

146

CHAPTER 2. CONVEX GEOMETRY

2.13. DUAL CONE & GENERALIZED INEQUALITY

2.12.4

147

Converting between descriptions

§

Conversion between halfspace- (260) (261) and vertex-description (81) (263) is nontrivial, in general, [15] [103, 2.2] but the conversion is easy for simplices. [57, 2.2.4] Nonetheless, we tacitly assume the two descriptions to be equivalent. [285, 19, thm.19.1] We explore conversions in 2.13.4, 2.13.9, and 2.13.11:

§

§

2.13

§

§

§

Dual cone & generalized inequality & biorthogonal expansion

These three concepts, dual cone, generalized inequality, and biorthogonal expansion, are inextricably melded; meaning, it is difficult to completely discuss one without mentioning the others. The dual cone is critical in tests for convergence by contemporary primal/dual methods for numerical solution of conic problems. [366] [257, 4.5] For unique minimum-distance projection ∗ on a closed convex cone K , the negative dual cone −K plays the role that orthogonal complement plays for subspace projection.2.51 ( E.9.2.1) Indeed, ∗ −K is the algebraic complement in Rn ;

§

§

∗

K ⊞ −K = Rn

(1946)

where ⊞ denotes unique orthogonal vector sum. One way to think of a pointed closed convex cone is as a new kind of coordinate system whose basis is generally nonorthogonal; a conic system, very much like the familiar Cartesian system whose analogous cone is the first quadrant (the nonnegative orthant). Generalized inequality ºK is a formalized means to determine membership to any pointed closed convex cone K ( 2.7.2.2) whereas biorthogonal expansion is, fundamentally, an expression of coordinates in a pointed conic system whose axes are linearly independent but not necessarily orthogonal. When cone K is the nonnegative orthant, then these three concepts come into alignment with the Cartesian prototype: biorthogonal expansion becomes orthogonal expansion, the dual cone becomes identical to the orthant, and generalized inequality obeys a total order entrywise.

§

2.51

Namely, projection on a subspace is ascertainable from its projection on the orthogonal complement.

148

2.13.1

CHAPTER 2. CONVEX GEOMETRY

Dual cone

§

For any set K (convex or not), the dual cone [99, 4.2] © ª ∗ K , y ∈ Rn | hy , xi ≥ 0 for all x ∈ K

(271)

§

is a unique cone2.52 that is always closed and convex because it is an intersection of halfspaces ( 2.4.1.1.1). Each halfspace has inward-normal x , belonging to K , and boundary containing the origin; e.g., Figure 53a. When cone K is convex, there is a second and equivalent construction: ∗ Dual cone K is the union of each and every vector y inward-normal to a hyperplane supporting K ( 2.4.2.6.1); e.g., Figure 53b. When K is represented by a halfspace-description such as (261), for example, where

§

 aT 1 . A ,  ..  ∈ Rm×n , aT m 

 cT 1 . C ,  ..  ∈ Rp×n cT p 

(272)

then the dual cone can be represented as the conic hull ∗

K = cone{a1 , . . . , am , ±c1 , . . . , ±cp }

(273)

a vertex-description, because each and every conic combination of normals from the halfspace-description of K yields another inward-normal to a hyperplane supporting K . ∗ K can also be constructed pointwise using projection theory from E.9.2: for PK x the Euclidean projection of point x on closed convex cone K

§

∗

−K = {x − PK x | x ∈ Rn } = {x ∈ Rn | PK x = 0}

(1947)

2.13.1.0.1 Exercise. Manual dual cone construction. Perhaps the most instructive graphical method of dual cone construction is cut-and-try. Find the dual of each polyhedral cone from Figure 54 by using dual cone equation (271). H 2.52

∗

◦

The dual cone is the negative polar cone defined by many authors; K = −K . [185, §A.3.2] [285, §14] [38] [26] [308, §2.7]

2.13. DUAL CONE & GENERALIZED INEQUALITY

K (a)

149

∗

K

0 K

∗

y

(b)

K 0

∗

Figure 53: Two equivalent constructions of dual cone K in R2 : (a) Showing construction by intersection of halfspaces about 0 (drawn truncated). Only those two halfspaces whose bounding hyperplanes have inward-normal corresponding to an extreme direction of this pointed closed convex cone K ⊂ R2 need be drawn; by (342). (b) Suggesting construction by union of inward-normals y to each and every hyperplane ∂H+ supporting K . This interpretation is valid when K is convex because existence of a supporting hyperplane is then guaranteed ( 2.4.2.6).

§

150

CHAPTER 2. CONVEX GEOMETRY

R2

R3

1

0.8

0.6

∂K

0.4

(a)

∗

K

0.2

∂K

K

∗

q

0

−0.2

−0.4

K

−0.6

∗

−0.8

−1

−0.5

0

0.5

1

1.5

∗

x ∈ K ⇔ hy , xi ≥ 0 for all y ∈ G(K )

(339)

Figure 54: Dual cone construction by right angle. Each extreme direction of a proper polyhedral cone is orthogonal to a facet of its dual cone, and vice versa, in any dimension. ( 2.13.6.1) (a) This characteristic guides graphical construction of dual cone in two dimensions: It suggests finding dual-cone boundary ∂ by making right angles with extreme directions of polyhedral cone. The construction is then pruned so that each dual boundary vector does not exceed π/2 radians in angle with each and every vector from polyhedral cone. Were dual cone in R2 to narrow, Figure 55 would be reached in limit. (b) Same polyhedral cone and its dual continued into three dimensions. (confer Figure 61)

§

(b)

2.13. DUAL CONE & GENERALIZED INEQUALITY

151

K K

∗

0

∗

Figure 55: Polyhedral cone K is a halfspace about origin in R2 . Dual cone K is a ray base 0, hence has empty interior in R2 ; so K cannot be pointed, hence has no extreme directions. (Both convex cones appear truncated.)

2.13.1.0.2 Exercise. Dual cone definitions. What is {x ∈ Rn | xTz ≥ 0 ∀ z ∈ Rn } ? What is {x ∈ Rn | xTz ≥ 1 ∀ z ∈ Rn } ? What is {x ∈ Rn | xTz ≥ 1 ∀ z ∈ Rn+ } ?

H

∗

As defined, dual cone K exists even when the affine hull of the original cone is a proper subspace; id est, even when the original cone has empty ∗ interior. Rockafellar formulates dimension of K and K . [285, 14]2.53 To further motivate our understanding of the dual cone, consider the ease with which convergence can be ascertained in the following optimization problem (p):

§

§

2.13.1.0.3 Example. Dual problem. (confer 4.1) Duality is a powerful and widely employed tool in applied mathematics for a number of reasons. First, the dual program is always convex even if the primal is not. Second, the number of variables in the dual is equal to the number of constraints in the primal which is often less than the number of 2.53

His monumental work Convex Analysis has not one figure or illustration. [26, §II.16] for a good illustration of Rockafellar’s recession cone [39].

See

152

CHAPTER 2. CONVEX GEOMETRY

f (x , zp ) or f (x)

f (xp , z) or g(z) x

z

Figure 56: (Drawing by Lucas V. Barbosa) This serves as mnemonic icon for primal and dual problems, although objective functions from conic problems (276p) (276d) are linear. When problems are strong duals, duality gap is 0 ; meaning, functions f (x) , g(z) (dotted) kiss at saddle value as depicted at center. Otherwise, dual functions never meet (f (x) > g(z)) by (274).

2.13. DUAL CONE & GENERALIZED INEQUALITY

153

§

variables in the primal program. Third, the maximum value achieved by the dual problem is often equal to the minimum of the primal. [277, 2.1.3] When not equal, the dual always provides a bound on the primal optimal objective. For convex problems, the dual variables provide necessary and sufficient optimality conditions: Essentially, Lagrange duality theory concerns representation of a given optimization problem as half a minimax problem. [285, 36] [57, 5.4] Given any real function f (x, z)

§

§

minimize maximize f (x, z) ≥ maximize minimize f (x, z) x

z

z

(274)

x

always holds. When minimize maximize f (x, z) = maximize minimize f (x, z) x

z

z

(275)

x

we have strong duality and then a saddle value [140] exists. (Figure 56) [282, p.3] Consider primal conic problem (p) (over cone K) and its corresponding dual problem (d): [272, 3.3.1] [222, 2.1] [223] given vectors α , β and matrix constant C

§

minimize (p)

x

αTx

maximize y,z

subject to x ∈ K Cx = β

§

β Tz ∗

subject to y ∈ K C Tz + y = α

(d)

(276)

Observe: the dual problem is also conic, and its objective function value never exceeds that of the primal; αTx ≥ β Tz xT(C Tz + y) ≥ (C x)T z

(277)

T

x y≥0

which holds by definition (271). Under the sufficient condition: (276p) is a convex problem and satisfies Slater’s condition,2.54 then equality xTy = 0 is achieved; each problem (p) and (d) attains the same optimal value of its objective and each problem is called a strong dual to the other because the 2.54

A convex problem, essentially, has convex objective function optimized over a convex set. (§4) In this context, (p) is convex if K is a convex cone. Slater’s condition is satisfied whenever any primal strictly feasible point exists. (p.272)

154

CHAPTER 2. CONVEX GEOMETRY

duality gap (primal−dual optimal objective difference) becomes 0. Then (p) and (d) are together equivalent to the minimax problem αTx − β Tz

minimize x, y, z

∗

(p)−(d)

subject to x ∈ K , y ∈ K C x = β , C Tz + y = α

(278)

§

whose optimal objective always has the saddle value 0 (regardless of the particular convex cone K and other problem parameters). [334, 3.2] Thus determination of convergence for either primal or dual problem is facilitated. Were convex cone K polyhedral ( 2.12.1), then problems (p) and (d) would be linear programs.2.55 Were K a positive semidefinite cone, then problem (p) has the form of prototypical semidefinite program 2.56 (613) with (d) its dual. It is sometimes possible to solve a primal problem by way of its dual; advantageous when the dual problem is easier to solve than the primal problem, for example, because it can be solved analytically, or has some special structure that can be exploited. [57, 5.5.5] ( 4.2.3.1) 2

§

§

2.13.1.1

§

Key properties of dual cone

ˆ For any cone, (−K) = −K ˆ For any cones K and K , K ⊆ K ⇒ K ⊇ K ˆ (Cartesian product) For closed convex cones K ∗

1

∗

2

1

∗ 1

2

∗ 2

§

[308, 2.7]

and K2 , their Cartesian product K = K1 × K2 is a closed convex cone, and 1

∗

∗

∗

K = K1 × K2

(279)

where each dual is determined with respect to a cone’s ambient space.

ˆ (conjugation) [285, §14] [99, §4.5] [308, p.52] When K is any convex cone, dual of the dual cone equals closure of the original cone; ∗∗

K = K 2.55

(280)

The self-dual nonnegative orthant yields the primal prototypical linear program and its dual. 2.56 A semidefinite program is any convex program constraining a variable to any subset of a positive semidefinite cone. The qualifier prototypical conventionally means: a program having linear objective, affine equality constraints, but no inequality constraints except for cone membership.

2.13. DUAL CONE & GENERALIZED INEQUALITY

155

is the intersection of all halfspaces about the origin that contain K . ∗∗∗ ∗ Because K = K always holds, ∗

∗

K = (K)

(281)

When convex cone K is closed, then dual of the dual cone is the original ∗∗ cone; K = K ⇔ K is a closed convex cone: [308, p.53, p.95] ∗

K = {x ∈ Rn | hy , xi ≥ 0 ∀ y ∈ K }

(282)

ˆ If any cone K has nonempty interior, then K is pointed; ∗

∗

K nonempty interior ⇒ K pointed

(283) ∗

If the closure of any convex cone K is pointed, conversely, then K has nonempty interior; ∗

K pointed ⇒ K nonempty interior

(284)

Given that a cone K ⊂ Rn is closed and convex, K is pointed if and only ∗ ∗ ∗ if K − K = Rn ; id est, iff K has nonempty interior. [51, 3.3 exer.20]

§

ˆ (vector sum) [285, thm.3.8] For convex cones K

1

and K2

K1 + K2 = conv(K1 ∪ K2 )

(285)

is a convex cone.

ˆ (dual vector-sum) [285, §16.4.2] [99, §4.6] For convex cones K

1

∗

∗

∗

∗

K1 ∩ K2 = (K1 + K2 ) = (K1 ∪ K2 )

ˆ (closure of vector sum of duals)

2.57

∗

∗

and K2 (286)

For closed convex cones K1 and K2 ∗

∗

∗

(K1 ∩ K2 ) = K1 + K2 = conv(K1 ∪ K2 )

§

§

(287)

[308, p.96] where operation closure becomes superfluous under the sufficient condition K1 ∩ int K2 6= ∅ [51, 3.3 exer.16, 4.1 exer.7]. 2.57

These parallel analogous results for subspaces R1 , R2 ⊆ Rn ; [99, §4.6] ⊥ (R1 + R2 )⊥ = R⊥ 1 ∩ R2 ⊥ (R1 ∩ R2 )⊥ = R⊥ 1 + R2

R⊥⊥ = R for any subspace R .

156

CHAPTER 2. CONVEX GEOMETRY

ˆ (Krein-Rutman) Given closed convex cones K ⊆ R and K ⊆ R and any linear map A : R → R , then provided int K ∩ AK 6= ∅ [51, §3.3.13, confer §4.1 exer.9] n

1

m

∗

m

1

∗

∗

(A−1 K1 ∩ K2 ) = AT K1 + K2

n

2

2

(288)

where dual of cone K1 is with respect to its ambient space Rm and dual of cone K2 is with respect to Rn , where A−1 K1 denotes inverse image ( 2.1.9.0.1) of K1 under mapping A , and where AT denotes adjoint operation. The particularly important case K2 = Rn is easy to show: for ATA = I

§

∗

(AT K1 ) , = = =

{y ∈ Rn | xTy ≥ 0 ∀ x ∈ AT K1 } {y ∈ Rn | (ATz)T y ≥ 0 ∀ z ∈ K1 } {ATw | z Tw ≥ 0 ∀ z ∈ K1 } ∗ AT K1

(289)

ˆ K is proper if and only if K is proper. ˆ K is polyhedral if and only if K is polyhedral. [308, §2.8] ˆ K is simplicial if and only if K is simplicial. (§2.13.9.2) A simplicial ∗

∗

∗

cone and its dual are proper polyhedral cones (Figure 61, Figure 52), but not the converse.

ˆ K ⊞ −K = R ⇔ K is closed and convex. (1946) ˆ Any direction in a proper cone K is normal to a hyperplane separating ∗

n

∗

K from −K .

2.13.1.2

Examples of dual cone ∗

When K is Rn , K is the point at the origin, and vice versa. ∗ When K is a subspace, K is its orthogonal complement, and vice versa. ( E.9.2.1, Figure 57) When cone K is a halfspace in Rn with n > 0 (Figure 55 for example), ∗ the dual cone K is a ray (base 0) belonging to that halfspace but orthogonal to its bounding hyperplane (that contains the origin), and vice versa.

§

2.13. DUAL CONE & GENERALIZED INEQUALITY

157 K

∗

K

0

R3

K

∗

R2 K

0

Figure 57: When convex cone K is any one Cartesian axis, its dual cone ∗ is the convex hull of all axes remaining. In R3 , dual cone K (drawn tiled and truncated) is a hyperplane through origin; its normal belongs to line K . ∗ In R2 , dual cone K is a line through origin while convex cone K is that line orthogonal.

158

CHAPTER 2. CONVEX GEOMETRY

When convex cone K is a closed halfplane in R3 (Figure 58), it is neither ∗ pointed or of nonempty interior; hence, the dual cone K can be neither of nonempty interior or pointed. When K is any particular orthant in Rn , the dual cone is identical; id est, ∗ K=K . ∗ When K is any quadrant in subspace R2 , K is a·wedge-shaped polyhedral ¸ 2 R+ ∗ cone in R3 ; e.g., for K equal to quadrant I , K = . R When K is a polyhedral flavor Lorentz cone (confer (163)) ½· ¸ ¾ x n Kℓ = ∈ R × R | kxkℓ ≤ t , ℓ ∈ {1, ∞} (290) t its dual is the proper cone [57, exmp.2.25] ½· ¸ ¾ ∗ x n Kq = Kℓ = ∈ R × R | kxkq ≤ t , t

ℓ ∈ {1, 2, ∞}

(291)

where kxkq is that norm dual to kxkℓ determined via solution to ∗ ∗ 1/ℓ + 1/q = 1. Figure 59 illustrates K = K1 and K = K1 = K∞ in R2 × R .

2.13.2

Abstractions of Farkas’ lemma

§

2.13.2.0.1 Corollary. Generalized inequality and membership relation. ∗ [185, A.4.2] Let K be any closed convex cone and K its dual, and let x and y belong to a vector space Rn . Then ∗

y ∈ K ⇔ hy , xi ≥ 0 for all x ∈ K

(292)

which is, merely, a statement of fact by definition of dual cone (271). By closure we have conjugation: [285, thm.14.1] x ∈ K ⇔ hy , xi ≥ 0 for all y ∈ K

∗

(293)

§

which may be regarded as a simple translation of the Farkas lemma [123] as in [285, 22] to the language of convex cones, and a generalization of the well-known Cartesian fact x º 0 ⇔ hy , xi ≥ 0 for all y º 0 ∗

for which implicitly K = K = Rn+ the nonnegative orthant.

(294)

2.13. DUAL CONE & GENERALIZED INEQUALITY

159

K

K

∗

∗

Figure 58: K and K are halfplanes in R3 ; blades. Both semiinfinite convex cones appear truncated. Each cone is like K from Figure 55 but embedded in a two-dimensional subspace of R3 . Cartesian coordinate axes drawn for reference.

160

CHAPTER 2. CONVEX GEOMETRY

Membership relation (293) is often written instead as dual generalized ∗ inequalities, when K and K are pointed closed convex cones, x º 0 ⇔ hy , xi ≥ 0 for all y º 0 K

K

∗

(295)

§

§

meaning, coordinates for biorthogonal expansion of x ( 2.13.7.1.2, 2.13.8) [340] must be nonnegative when x belongs to K . Conjugating, y º 0 ⇔ hy , xi ≥ 0 for all x º 0 K

∗

(296)

K

⋄

When pointed closed convex cone K is not polyhedral, coordinate axes for biorthogonal expansion asserted by the corollary are taken from extreme directions of K ; expansion is assured by Carath´eodory’s theorem ( E.6.4.1.1). We presume, throughout, the obvious:

§

∗

x ∈ K ⇔ hy , xi ≥ 0 for all y ∈ K (293) ⇔ ∗ x ∈ K ⇔ hy , xi ≥ 0 for all y ∈ K , kyk = 1

(297)

2.13.2.0.2 Exercise. Test of dual generalized inequalities. Test Corollary 2.13.2.0.1 and (297) graphically on the two-dimensional H polyhedral cone and its dual in Figure 54.

§

(confer 2.7.2.2) When pointed closed convex cone K is implicit from context: xº0 ⇔ x∈K

(298)

x ≻ 0 ⇔ x ∈ rel int K

Strict inequality x ≻ 0 means coordinates for biorthogonal expansion of x must be positive when x belongs to rel int K . Strict membership relations ∗ are useful; e.g., for any proper cone2.58 K and its dual K ∗

x ∈ int K ⇔ hy , xi > 0 for all y ∈ K , y 6= 0 x ∈ K , x 6= 0 ⇔ hy , xi > 0 for all y ∈ int K 2.58

An open cone K is admitted to (299) and (302) by (17).

∗

(299) (300)

2.13. DUAL CONE & GENERALIZED INEQUALITY

161

Conjugating, we get the dual relations: ∗

y ∈ int K ⇔ hy , xi > 0 for all x ∈ K , x 6= 0

(301)

∗

y ∈ K , y 6= 0 ⇔ hy , xi > 0 for all x ∈ int K

(302)

Boundary-membership relations for proper cones are also useful: ∗

x ∈ ∂K ⇔ ∃ y 6= 0 Ä hy , xi = 0 , y ∈ K , x ∈ K ∗

y ∈ ∂K ⇔ ∃ x 6= 0 Ä hy , xi = 0 , x ∈ K , y ∈ K which are consistent; e.g., x ∈ ∂K ⇔ x ∈ / int K and x ∈ K .

§

(303) ∗

(304)

§

2.13.2.0.3 Example. Linear inequality. [314, 4] (confer 2.13.5.1.1) Consider a given matrix A and closed convex cone K . By membership relation we have ∗

Ay ∈ K ⇔ xTA y ≥ 0 ∀ x ∈ K ⇔ y Tz ≥ 0 ∀ z ∈ {ATx | x ∈ K} ∗ ⇔ y ∈ {ATx | x ∈ K} This implies

∗

∗

{y | Ay ∈ K } = {ATx | x ∈ K}

§

(305)

(306)

When K is the self-dual nonnegative orthant ( 2.13.5.1), for example, then ∗

{y | Ay º 0} = {ATx | x º 0}

(307)

and the dual relation ∗

{y | Ay º 0} = {ATx | x º 0}

(308)

comes by a theorem of Weyl (p.140) that yields closedness for any vertex description of a polyhedral cone. 2 2.13.2.1

Null certificate, Theorem of the alternative

If in particular xp ∈ / K a closed convex cone, then construction in Figure 53b suggests there exists a supporting hyperplane (having inward-normal ∗ belonging to dual cone K ) separating xp from K ; indeed, (293)

162

CHAPTER 2. CONVEX GEOMETRY ∗

/ K ⇔ ∃ y ∈ K Ä hy , xp i < 0 xp ∈

(309)

Existence of any one such y is a certificate of null membership. From a different perspective, xp ∈ K

or in the alternative

(310)

∗

∃ y ∈ K Ä hy , xp i < 0 By alternative is meant: these two systems are incompatible; one system is feasible while the other is not. 2.13.2.1.1 Example. Theorem of the alternative for linear inequality. Myriad alternative systems of linear inequality can be explained in terms of pointed closed convex cones and their duals. Beginning from the simplest Cartesian dual generalized inequalities (294) (with respect to nonnegative orthant Rm + ), y º 0 ⇔ xTy ≥ 0 for all x º 0

(311)

Given A ∈ Rn×m , we make vector substitution y ← ATy ATy º 0 ⇔ xTAT y ≥ 0 for all x º 0

(312)

Introducing a new vector by calculating b , Ax we get ATy º 0 ⇔ bTy ≥ 0 , b = Ax for all x º 0

(313)

By complementing sense of the scalar inequality: ATy º 0

or in the alternative

(314)

bTy < 0 , ∃ b = Ax , x º 0 If one system has a solution, then the other does not; define a convex cone K = {y | ATy º 0} , then y ∈ K or in the alternative y ∈ / K. T Scalar inequality b y < 0 can be moved to the other side of the alternative,

2.13. DUAL CONE & GENERALIZED INEQUALITY

163

but that requires some explanation: From the results of Example 2.13.2.0.3, ∗ the dual cone is K = {Ax | x º 0}. From (293) we have ∗

y ∈ K ⇔ bTy ≥ 0 for all b ∈ K ATy º 0 ⇔ bTy ≥ 0 for all b ∈ {Ax | x º 0}

(315) ∗

Given some b vector and y ∈ K , then bTy < 0 can only mean b ∈ / K . An ∗ alternative system is therefore simply b ∈ K : [185, p.59] (Farkas/Tucker) ATy º 0 , bTy < 0

or in the alternative

(316)

b = Ax , x º 0

∗

Geometrically this means, either vector b belongs to convex cone K or it ∗ does not. When b ∈ / K , then there is a vector y ∈ K normal to a hyperplane ∗ separating point b from cone K . For another example, from membership relation (292) with affine transformation of dual variable we may write: Given A ∈ Rn×m and b ∈ Rn ATx = 0 ,

b − Ay ∈ K

b − Ay ∈ K

∗

∗

⇔ xT(b − Ay) ≥ 0

⇒ xT b ≥ 0

∀x ∈ K

∀x ∈ K

(317)

(318)

From membership relation (317), conversely, suppose we allow any y ∈ Rm . Then because −xTA y is unbounded below, xT(b − Ay) ≥ 0 implies ATx = 0 : for y ∈ Rm ATx = 0 ,

In toto,

∗

b − Ay ∈ K ⇐ xT(b − Ay) ≥ 0

b − Ay ∈ K

∗

∀x ∈ K

⇔ xT b ≥ 0 , ATx = 0 ∀ x ∈ K

(319) (320)

Vector x belongs to cone K but is also constrained to lie in a subspace of Rn specified by an intersection of hyperplanes through the origin {x ∈ Rn | ATx = 0}. From this, alternative systems of generalized inequality ∗ with respect to pointed closed convex cones K and K Ay ¹ b K

∗

or in the alternative xT b < 0 , ATx = 0 , x º 0 K

(321)

164

CHAPTER 2. CONVEX GEOMETRY

derived from (320) simply by taking the complementary sense of the inequality in xT b . These two systems are alternatives; if one system has a solution, then the other does not.2.59 [285, p.201] By invoking a strict membership relation between proper cones (299), we can construct a more exotic alternative strengthened by demand for an interior point; b − Ay ≻ 0 ⇔ xT b > 0 , ATx = 0 ∀ x º 0 , x 6= 0 K

∗

(322)

K

From this, alternative systems of generalized inequality [57, pages:50,54,262] Ay ≺ b K

∗

(323)

or in the alternative xT b ≤ 0 , ATx = 0 , x º 0 , x 6= 0 K

derived from (322) by taking the complementary sense of the inequality in xT b . And from this, alternative systems with respect to the nonnegative orthant attributed to Gordan in 1873: [147] [51, 2.2] substituting A ← AT and setting b = 0 ATy ≺ 0

§

or in the alternative

(324)

Ax = 0 , x º 0 , kxk1 = 1

Ben-Israel collects related results from Farkas, Motzkin, Gordan, and Stiemke in Motzkin transposition theorem. [172] 2 2.59

If solutions at ±∞ are disallowed, then the alternative systems become instead mutually exclusive with respect to nonpolyhedral cones. Simultaneous infeasibility of the two systems is not precluded by mutual exclusivity; called a weak alternative. Ye provides an example illustrating simultaneous to the positive · ¸ infeasibility· with respect ¸ 1 0 0 1 2 semidefinite cone: x ∈ S , y ∈ R , A = , and b = where xT b means 0 0 1 0 hx , bi . A better strategy than disallowing solutions at ±∞ is to demand an interior point as in (323) or Lemma 4.2.1.1.2. Then question of simultaneous infeasibility is moot.

2.13. DUAL CONE & GENERALIZED INEQUALITY

2.13.3

§

165

Optimality condition

(confer 2.13.10.1) The general first-order necessary and sufficient condition for optimality of solution x⋆ to a minimization problem ((276p) for example) with real differentiable convex objective function f (x) : Rn → R is [284, 3]

§

∇f (x⋆ )T (x − x⋆ ) ≥ 0 ∀ x ∈ C , x⋆ ∈ C

§

(325)

where C is a convex feasible set,2.60 and where ∇f (x⋆ ) is the gradient ( 3.7) of f with respect to x evaluated at x⋆ . In words, negative gradient is normal to a hyperplane supporting the feasible set at a point of optimality. (Figure 64) Direct solution to variation inequality (325), instead of the corresponding minimization, spawned from calculus of variations. [232, p.178] [122, p.37] One solution method solves an equivalent fixed point-of-projection problem x = PC (x − ∇f (x))

(326)

that follows from a necessary and sufficient condition for projection on convex set C (Theorem E.9.1.0.2) P (x⋆−∇f (x⋆ )) ∈ C ,

hx⋆−∇f (x⋆ )−x⋆ , x−x⋆ i ≤ 0 ∀ x ∈ C

(1933)

Proof of equivalence is provided by S´andor Zolt´an N´emeth on Wıκımization. Given minimum-distance projection problem minimize x

1 kx 2

− yk2

subject to x ∈ C

(327)

on convex feasible set C for example, the equivalent fixed point problem x = PC (x − ∇f (x)) = PC (y)

(328)

is solved in one step. In the unconstrained case (C = Rn ), optimality condition (325) reduces to the classical rule (p.240) ∇f (x⋆ ) = 0 ,

x⋆ ∈ dom f

which can be inferred from the following application: 2.60

a set of all variable values satisfying the problem constraints.

(329)

166

CHAPTER 2. CONVEX GEOMETRY

2.13.3.0.1 Example. Equality constrained problem. Given a real differentiable convex function f (x) : Rn → R defined on domain Rn , a fat full-rank matrix C ∈ Rp×n , and vector d ∈ Rp , the convex optimization problem minimize f (x) x (330) subject to C x = d

§

is characterized by the well-known necessary and sufficient optimality condition [57, 4.2.3] ∇f (x⋆ ) + C T ν = 0 (331)

where ν ∈ Rp is the eminent Lagrange multiplier. [283] [232, p.188] [212] In other words, feasible solution x⋆ is optimal if and only if ∇f (x⋆ ) belongs to R(C T ). Via membership relation, we now derive condition (331) from the general first-order condition for optimality (325): For problem (330) C , {x ∈ Rn | C x = d} = {Z ξ + xp | ξ ∈ Rn−rank C }

(332)

is the feasible set where Z ∈ Rn×n−rank C holds basis N (C ) columnar, and xp is any particular solution to C x = d . Since x⋆ ∈ C , we arbitrarily choose xp = x⋆ which yields an equivalent optimality condition ∇f (x⋆ )T Z ξ ≥ 0 ∀ ξ ∈ Rn−rank C

(333)

when substituted into (325). But this is simply half of a membership relation where the cone dual to Rn−rank C is the origin in Rn−rank C . We must therefore have Z T ∇f (x⋆ ) = 0 ⇔ ∇f (x⋆ )T Z ξ ≥ 0 ∀ ξ ∈ Rn−rank C (334) meaning, ∇f (x⋆ ) must be orthogonal to N (C ). These conditions Z T ∇f (x⋆ ) = 0 ,

C x⋆ = d

(335)

are necessary and sufficient for optimality.

2

2.13.4

Discretization of membership relation

2.13.4.1

Dual halfspace-description ∗

Halfspace-description of dual cone K is equally simple as vertex-description K = cone(X) = {Xa | a º 0} ⊆ Rn

(97)

2.13. DUAL CONE & GENERALIZED INEQUALITY

167

for corresponding closed convex cone K : By definition (271), for X ∈ Rn×N as in (255), (confer (261)) © ª ∗ K = ©y ∈ Rn | z Ty ≥ 0 for all z ∈ K ª = ©y ∈ Rn | z Ty ≥ 0 for all z =ªX a , a º 0 (336) = ©y ∈ Rn | aTX Ty ≥ª0 , a º 0 = y ∈ Rn | X Ty º 0 that follows from the generalized inequality and membership corollary (294). The semi-infinity of tests specified by all z ∈ K has been reduced to a set of generators for K constituting the columns of X ; id est, the test has been discretized. Whenever cone K is known to be closed and convex, the conjugate statement must also hold; id est, given any set of generators for dual cone ∗ K arranged columnar in Y , then the consequent vertex-description of dual cone connotes a halfspace-description for K : [308, 2.8] © ª ∗ ∗∗ K = {Y a | a º 0} ⇔ K = K = z | Y Tz º 0 (337)

§

2.13.4.2

First dual-cone formula

From these two results (336) and (337) we deduce a general principle:

ˆ From any [sic] given vertex-description (97) of closed convex cone K , ∗

a halfspace-description of the dual cone K is immediate by matrix transposition (336); conversely, from any given halfspace-description (261) of K , a dual vertex-description is immediate (337). [285, p.122]

§

§

Various other converses are just a little trickier. ( 2.13.9, 2.13.11) ∗ We deduce further: For any polyhedral cone K , the dual cone K is also ∗∗ polyhedral and K = K . [308, p.56] The generalized inequality and membership corollary is discretized in the following theorem inspired by (336) and (337):

§

§

2.13.4.2.1 Theorem. Discretized membership. (confer 2.13.2.0.1)2.61 Given any set of generators ( 2.8.1.2) denoted by G(K) for closed convex ∗ cone K ⊆ Rn and any set of generators denoted G(K ) for its dual such that K = cone G(K) , 2.61

∗

∗

K = cone G(K )

Stated in [21, §1] without proof for pointed closed convex case.

(338)

168

CHAPTER 2. CONVEX GEOMETRY

then discretization of the generalized inequality and membership corollary (p.158) is necessary and sufficient for certifying cone membership: for x and y in vector space Rn ∗

x ∈ K ⇔ hγ ∗ , xi ≥ 0 for all γ ∗ ∈ G(K ) ∗

y ∈ K ⇔ hγ , yi ≥ 0 for all γ ∈ G(K) ∗

∗

∗

(339) (340) ⋄

∗

Proof. K = {G(K )a | a º 0}. y ∈ K ⇔ y = G(K )a for somePa º 0. ∗ ∗ x ∈ K ⇔ hy , xi ≥ 0 ∀ y ∈ K ⇔ hG(K )a , xi ≥ 0 ∀ a º 0 (293). a , i αi ei where ei is the i th member of a standard basis of possibly infinite P ∗ ∗ cardinality. hG(K )a , xi ≥ 0 ∀ a º 0 ⇔ αi hG(K )ei , xi ≥ 0 ∀ αi ≥ 0 ⇔ i ∗ hG(K )ei , xi ≥ 0 ∀ i . Conjugate relation (340) is similarly derived. ¨ 2.13.4.2.2 Exercise. Test of discretized dual generalized inequalities. Test Theorem 2.13.4.2.1 on Figure 54a using extreme directions there as generators. H From the discretized membership theorem we may further deduce a more surgical description of closed convex cone that prescribes only a finite number of halfspaces for its construction when polyhedral: (Figure 53(a)) ∗

K = {x ∈ Rn | hγ ∗ , xi ≥ 0 for all γ ∗ ∈ G(K )} ∗

K = {y ∈ Rn | hγ , yi ≥ 0 for all γ ∈ G(K)}

(341) (342)

2.13.4.2.3 Exercise. Partial order induced by orthant. When comparison is with respect to the nonnegative orthant K = Rn+ , then from the discretized membership theorem it directly follows: x ¹ z ⇔ xi ≤ zi ∀ i

(343)

Generate simple counterexamples demonstrating that this equivalence with entrywise inequality holds only when the underlying cone inducing partial order is the nonnegative orthant. H

2.13. DUAL CONE & GENERALIZED INEQUALITY

169

2.13.4.2.4 Example. Boundary membership to proper polyhedral cone. For a polyhedral cone, test (303) of boundary membership can be formulated as a linear program. Say proper polyhedral cone K is specified completely by generators that are arranged columnar in X = [ Γ1 · · · ΓN ] ∈ Rn×N

(255)

id est, K = {X a | a º 0}. Then membership relation ∗

c ∈ ∂K ⇔ ∃ y 6= 0 Ä hy , ci = 0 , y ∈ K , c ∈ K

(303)

may be expressed find a, y

y 6= 0

subject to cT y = 0 X Ty º 0 Xa = c aº0

(344)

This linear feasibility problem has a solution iff c ∈ ∂K . 2.13.4.3

2

smallest face of proper cone

Given nonempty convex subset C of a convex set K , the smallest face of K containing C is equivalent to the intersection of all faces of K containing C . [285, p.164] By (282), membership relation (303) means that each and every point on boundary ∂K of proper cone K belongs to a hyperplane ∗ supporting K whose normal y belongs to dual cone K . It follows that the smallest face F , containing C ⊂ ∂K ⊂ Rn on boundary of proper cone K , ∗ is the intersection of all hyperplanes containing C whose normals are in K ; ∗

F(K ⊃ C) = {x ∈ K | x ⊥ K ∩ C ⊥ }

(345)

C ⊥ , {y ∈ Rn | hz , yi = 0 ∀ z ∈ C}

(346)

where

2.13.4.3.1 Example. Finding smallest face. Suppose proper polyhedral cone K is completely specified by generators arranged columnar in X = [ Γ1 · · · ΓN ] ∈ Rn×N

(255)

170

CHAPTER 2. CONVEX GEOMETRY

To find its smallest face F(K ∋ c) containing a given point c ∈ ∂K , by the discretized membership theorem 2.13.4.2.1, it is necessary and sufficient to find generators for the smallest face. We may do so one generator at a time: ∗ Consider generator Γi . If there exists a vector z ∈ K orthogonal to c but not to Γi , then Γi cannot belong to the smallest face of K containing c . Such a vector z would solve a linear feasibility problem: find z subject to cTz = 0 X Tz º 0 ΓT iz = 1

(347)

If this problem is infeasible for generator Γi ∈ K , then Γi ∈ F(K ∋ c) by (345) ∗ and (336) because Γi ⊥ K ∩ c ⊥ = {z ∈ Rn | X Tz º 0 , cTz = 0} ; in that case, Γi is a generator of F(K ∋ c). If there exists a feasible vector z for which ∗ ⊥ ΓT so Γi ∈ / F(K ∋ c) ; feasible vector z i z = 1, conversely, then Γi 6⊥ K ∩ c is a certificate of null membership. Since the constant in constraint ΓT i z = 1 is arbitrary positively, then by theorem of the alternative there is correspondence between (347) and (321) admitting the alternative linear problem: for a given point c ∈ ∂K find

a, t

a∈RN , t∈R

subject to tc − Γi = X a aº0

(348)

Now if this problem is feasible (bounded) for generator Γi ∈ K , then (347) is infeasible and Γi ∈ F(K ∋ c) is a generator of the smallest face containing c . 2 2.13.4.3.2 Exercise. Smallest face of positive semidefinite cone. Derive (206) from (345). H 2.13.4.3.3 Exercise. Smallest face of pointed closed convex cone. Show that formula (345) and algorithms (347) and (348) apply more broadly; id est, a full-dimensional cone K is an unnecessary condition.2.62 H 2.62

∗

Hint: A hyperplane, with normal in K , containing cone K is admissible.

2.13. DUAL CONE & GENERALIZED INEQUALITY

2.13.5

171

Dual PSD cone and generalized inequality ∗

The dual positive semidefinite cone K is confined to SM by convention; ∗

M M | hY , X i ≥ 0 for all X ∈ SM SM + , {Y ∈ S + } = S+

(349)

§

The positive semidefinite cone is self-dual in the ambient space of symmetric ∗ matrices [57, exmp.2.24] [37] [182, II]; K = K . Dual generalized inequalities with respect to the positive semidefinite cone in the ambient space of symmetric matrices can therefore be simply stated: (Fej´er) X º 0 ⇔ tr(Y TX) ≥ 0 for all Y º 0 (350)

§

§

Membership to this cone can be determined in the isometrically isomorphic 2 Euclidean space RM via (33). ( 2.2.1) By the two interpretations in 2.13.1, positive semidefinite matrix Y can be interpreted as inward-normal to a hyperplane supporting the positive semidefinite cone. The fundamental statement of positive semidefiniteness, y TXy ≥ 0 ∀ y ( A.3.0.0.1), evokes a particular instance of these dual generalized inequalities (350):

§

X º 0 ⇔ hyy T , X i ≥ 0 ∀ yy T(º 0)

§

(1386)

Discretization ( 2.13.4.2.1) allows replacement of positive semidefinite matrices Y with this minimal set of generators comprising the extreme directions of the positive semidefinite cone ( 2.9.2.4). 2.13.5.1

§

self-dual cones

§

§

From (125) (a consequence of the halfspaces theorem, 2.4.1.1.1), where the only finite value of the support function for a convex cone is 0 [185, C.2.3.1], or from discretized definition (342) of the dual cone we get a rather self-evident characterization of self-dual cones: \© ª ∗ K=K ⇔ K = y | γ Ty ≥ 0 (351) γ∈G(K)

In words: Cone K is self-dual iff its own extreme directions are inward-normals to a (minimal) set of hyperplanes bounding halfspaces whose intersection constructs it. This means each extreme direction of K is normal

172

(a)

CHAPTER 2. CONVEX GEOMETRY

0

K ∂K

∂K

∗

∗

K (b)

Figure 59: Two (truncated) views of a polyhedral cone K and its dual in R3 . ∗ Each of four extreme directions from K belongs to a face of dual cone K . Cone K , shrouded by its dual, is symmetrical about its axis of revolution. Each pair of diametrically opposed extreme directions from K makes a right angle. An orthant (or any rotation thereof; a simplicial cone) is not the only self-dual polyhedral cone in three or more dimensions; [20, 2.A.21] e.g., consider an equilateral having five extreme directions. In fact, every self-dual polyhedral cone in R3 has an odd number of extreme directions. [22, thm.3]

§

2.13. DUAL CONE & GENERALIZED INEQUALITY

173

to a hyperplane exposing one of its own faces; a necessary but insufficient condition for self-dualness (Figure 59, for example). Self-dual cones are of necessarily nonempty interior. [30, I] Their most prominent representatives are the orthants, the positive semidefinite cone SM + in the ambient space of symmetric matrices (349), and Lorentz cone (163) [20, II.A] [57, exmp.2.25]. In three dimensions, a plane containing the axis of revolution of a self-dual cone (and the origin) will produce a slice whose boundary makes a right angle.

§

§

§

2.13.5.1.1 Example. Linear matrix inequality. (confer 2.13.2.0.3) Consider a peculiar vertex-description for a convex cone K defined over a positive semidefinite cone (instead of a nonnegative orthant as in n definition (97)): for X ∈ S+ given Aj ∈ S n , j = 1 . . . m      hA1 , X i ..   | X º 0 ⊆ Rm K = .   hAm , X i    T (352)  svec(A1 )  ..  =  svec X | X º 0 .   svec(Am )T , {A svec X | X º 0}

where A ∈ Rm×n(n+1)/2 , and where symmetric vectorization svec is defined in (51). Cone K is indeed convex because, by (160) A svec Xp1 , A svec Xp2 ∈ K ⇒ A(ζ svec Xp1+ξ svec Xp2 ) ∈ K for all ζ , ξ ≥ 0 (353) since a nonnegatively weighted sum of positive semidefinite matrices must be positive semidefinite. ( A.3.1.0.2) Although matrix A is finite-dimensional, K n is generally not a polyhedral cone (unless m = 1 or 2) simply because X ∈ S+ .

§

2.13.5.1.2 Theorem. Inverse image closedness. [185, prop.A.2.1.12] m p [285, thm.6.7] Given affine operator g : R → R , convex set D ⊆ Rm , and convex set C ⊆ Rp Ä g −1(rel int C) 6= ∅ , then rel int g(D) = g(rel int D) , rel int g −1 C = g −1(rel int C) , g −1 C = g −1 C

(354) ⋄

174

CHAPTER 2. CONVEX GEOMETRY

By this theorem, relative interior of K may always be expressed rel int K = {A svec X | X ≻ 0}

(355)

Because dim(aff K) = dim(A svec S n ) (121) then, provided the Aj matrices are linearly independent, rel int K = int K

(13) ∗

meaning, cone K interior is nonempty ⇒ dual cone K is pointed by (283). Convex cone K can be closed, by this corollary:

§

§

2.13.5.1.3 Corollary. Cone closedness invariance. [52, 3] [47, 3] p m Given linear operator A :³R → R and closed convex cone X ⊆ Rp , convex ´ cone K = A(X ) is closed A(X ) = A(X ) if and only if N (A) ∩ X = {0}

N (A) ∩ X * rel ∂X

or

(356)

Otherwise, K = A(X ) ⊇ A(X ) ⊇ A(X ). [285, thm.6.6]

⋄

§

§

If matrix A has no nontrivial nullspace, then A svec X is an isomorphism n and range R(A) of matrix A ; ( 2.2.1.0.1, 2.10.1.1) in X between cone S+ sufficient for convex cone K to be closed and have relative boundary rel ∂K = {A svec X | X º 0 , X ⊁ 0}

(357)

Now consider the (closed convex) dual cone: ∗

K = {y = {y =© {y = ©y = y

| | | | |

hz , yi ≥ 0 for all z ∈ K} ⊆ Rm hz , yi ≥ 0 for all z = A svec X , X º 0} hA svec X , yi ≥ 0 for all X º 0} ª hsvec X , ATyi ≥ ª0 for all X º 0 svec−1 (ATy) º 0

(358)

that follows from (350) and leads to an equally peculiar halfspace-description ∗

K = {y ∈ Rm |

m X j=1

yj Aj º 0}

(359)

2.13. DUAL CONE & GENERALIZED INEQUALITY

175

n The summation inequality with respect to positive semidefinite cone S+ is known as linear matrix inequality. [55] [138] [244] [337] Although we already know that the dual cone is convex ( 2.13.1), inverse image theorem 2.1.9.0.1 ∗ certifies convexity of K which is the inverse image of positive semidefinite P n cone S+ under linear transformation g(y) , yj Aj . And although we already know that the dual cone is closed, it is certified by (354). By Theorem 2.13.5.1.2, relative interior of the dual cone may always be expressed

§

∗

rel int K = {y ∈ Rm |

m X j=1

yj Aj ≻ 0}

(360)

Function g(y) on Rm is an isomorphism when the Aj matrices are linearly ∗ independent; hence, uniquely¡ invertible. Inverse image K must therefore ¢ n have dimension equal to dim R(AT )∩ svec S+ (44) and relative boundary ∗

m

rel ∂K = {y ∈ R |

m X j=1

yj Aj º 0 ,

m X

yj Aj ⊁ 0}

(361)

j=1 ∗

When this dimension equals m , then dual cone K interior is nonempty ∗

rel int K = int K

∗

(13)

which implies: closure of convex cone K is pointed (283).

2.13.6

2

Dual of pointed polyhedral cone

In a subspace of Rn , now we consider a pointed polyhedral cone K given in terms of its extreme directions Γi arranged columnar in X = [ Γ1 Γ2 · · · ΓN ] ∈ Rn×N

§

(255)

The extremes theorem ( 2.8.1.1.1) provides the vertex-description of a pointed polyhedral cone in terms of its finite number of extreme directions and its lone vertex at the origin:

176

CHAPTER 2. CONVEX GEOMETRY

2.13.6.0.1 Definition. Pointed polyhedral cone, vertex-description. Given pointed polyhedral cone K in a subspace of Rn , denoting its i th extreme direction by Γi ∈ Rn arranged in a matrix X as in (255), then that cone may be described: (81) (confer (173) (267)) K = = =

©

ª [ 0 ©X ] a ζ | aT 1 = 1, a º 0, ζ ≥ 0ª T ©X a ζ | a 1 ≤ª1, a ºn 0, ζ ≥ 0 Xb | b º 0 ⊆ R

(362)

that is simply a conic hull (like (97)) of a finite number N of directions. Relative interior may always be expressed rel int K = {X b | b ≻ 0} ⊂ Rn

(363)

but identifying the cone’s relative boundary in this manner rel ∂K = {X b | b º 0 , b ⊁ 0}

(364) △

holds only when matrix X represents a bijection onto its range.

Whenever cone K is pointed closed and convex (not only polyhedral), then ∗ dual cone K has a halfspace-description in terms of the extreme directions Γi of K : © ª ∗ K = y | γ Ty ≥ 0 for all γ ∈ {Γi , i = 1 . . . N } ⊆ rel ∂K

(365)

§

because when {Γi } constitutes any set of generators for K , the discretization result in 2.13.4.1 allows relaxation of the requirement ∀ x ∈ K in (271) to ∀ γ ∈ {Γi } directly.2.63 That dual cone so defined is unique, identical to (271), polyhedral whenever the number of generators N is finite © ª ∗ K = y | X Ty º 0 ⊆ Rn

(336)

§

∗

and has nonempty interior because K is assumed pointed. But K is not necessarily pointed unless K has nonempty interior ( 2.13.1.1). 2.63

The extreme directions of K constitute a minimal set of generators. Formulae and conversions to vertex-description of the dual cone are in §2.13.9 and §2.13.11.

2.13. DUAL CONE & GENERALIZED INEQUALITY 2.13.6.1

177

Facet normal & extreme direction

We see from (336) that the conically independent generators of cone K (namely, the extreme directions of pointed closed convex cone K constituting the N columns of X ) each define an inward-normal to a hyperplane ∗ supporting dual cone K ( 2.4.2.6.1) and exposing a dual facet when N is ∗ finite. Were K pointed and finitely generated, then by closure the conjugate ∗ statement would also hold; id est, the extreme directions of pointed K each define an inward-normal to a hyperplane supporting K and exposing a facet when N is finite. Examine Figure 54 or Figure 59, for example. We may conclude, the extreme directions of proper polyhedral K are ∗ respectively orthogonal to the facets of K ; likewise, the extreme directions ∗ of proper polyhedral K are respectively orthogonal to the facets of K .

§

2.13.7

Biorthogonal expansion by example

2.13.7.0.1 Example. Relationship to dual polyhedral cone. Simplicial cone K illustrated in Figure 60 induces a partial order on R2 . All points greater than x with respect to K , for example, are contained in the translated cone x + K . The extreme directions Γ1 and Γ2 of K do not make ∗ an orthogonal set; neither do extreme directions Γ3 and Γ4 of dual cone K ; rather, we have the biorthogonality condition, [340] T ΓT 4 Γ1 = Γ3 Γ2 = 0 ΓT ΓT 3 Γ1 6= 0 , 4 Γ2 6= 0

(366)

Biorthogonal expansion of x ∈ K is then x = Γ1

ΓT ΓT 3x 4x + Γ 2 T T Γ3 Γ1 Γ 4 Γ2

(367)

§

T where ΓT 3 x/(Γ3 Γ1 ) is the nonnegative coefficient of nonorthogonal projection ( E.6.1) of x on Γ1 in the direction orthogonal to Γ3 (y in Figure 155, p.689), T and where ΓT 4 x/(Γ4 Γ2 ) is the nonnegative coefficient of nonorthogonal projection of x on Γ2 in the direction orthogonal to Γ4 (z in Figure 155); they are coordinates in this nonorthogonal system. Those coefficients must be nonnegative x ºK 0 because x ∈ K (298) and K is simplicial. If we ascribe the extreme directions of K to the columns of a matrix

X , [ Γ1 Γ 2 ]

(368)

178

CHAPTER 2. CONVEX GEOMETRY

Γ4

Γ2 K

∗

z x y

K Γ1 ⊥ Γ 4 Γ2 ⊥ Γ 3

w

Γ1

0

u K

∗

w−K

Γ3

∗

Figure 60: (confer Figure 155) Simplicial cone K in R2 and its dual K drawn truncated. Conically independent generators Γ1 and Γ2 constitute extreme ∗ directions of K while Γ3 and Γ4 constitute extreme directions of K . Dotted ray-pairs bound translated cones K . Point x is comparable to point z (and vice versa) but not to y ; z ºK x ⇔ z − x ∈ K ⇔ z − x ºK 0 iff ∃ nonnegative coordinates for biorthogonal expansion of z − x . Point y is not comparable to z because z does not belong to y ± K . Translating a negated cone is quite helpful for visualization: u ¹K w ⇔ u ∈ w − K ⇔ u − w ¹K 0. Points need not belong to K to be comparable; e.g., all points less than w (with respect to K) belong to w − K .

2.13. DUAL CONE & GENERALIZED INEQUALITY

179

then we find that the pseudoinverse transpose matrix X

†T

·

1 Γ4 T Γ4 Γ2

1 = Γ3 T Γ3 Γ 1

¸

(369)

holds the extreme directions of the dual cone. Therefore, x = XX † x

(375)

§

§

is the biorthogonal expansion (367) ( E.0.1), and the biorthogonality condition (366) can be expressed succinctly ( E.1.1)2.64 X †X = I

(376)

Expansion w = XX † w for any w ∈ R2 is unique if and only if the extreme directions of K are linearly independent; id est, iff X has no nullspace. 2 2.13.7.1

Pointed cones and biorthogonality

Biorthogonality condition X † X = I from Example 2.13.7.0.1 means Γ1 and Γ2 are linearly independent generators of K ( B.1.1.1); generators because every x ∈ K is their conic combination. From 2.10.2 we know that means Γ1 and Γ2 must be extreme directions of K . A biorthogonal expansion is necessarily associated with a pointed closed convex cone; pointed, otherwise there can be no extreme directions ( 2.8.1). We will address biorthogonal expansion with respect to a pointed polyhedral cone having empty interior in 2.13.8.

§ §

§

§

§

§

2.13.7.1.1 Example. Expansions implied by diagonalization. (confer 6.5.3.2.1) When matrix X ∈ RM ×M is diagonalizable ( A.5), X = SΛS −1

 w1T M ..  X  = = [ s1 · · · sM ] Λ λ i si wiT . T i=1 wM 

(1480)

Possibly confusing is the fact that formula XX † x is simultaneously the orthogonal projection of x on R(X) (1834), and a sum of nonorthogonal projections of x ∈ R(X) on the range of each and every column of full-rank X skinny-or-square (§E.5.0.0.2). 2.64

180

CHAPTER 2. CONVEX GEOMETRY

coordinates for biorthogonal expansion are its eigenvalues λ i (contained in diagonal matrix Λ) when expanded in S ;  w1T X M ..  X −1  X = SS X = [ s1 · · · sM ] λ i si wiT = . T i=1 wM X 

§

(370)

§

Coordinate value depend upon the geometric relationship of X to its linearly independent eigenmatrices si wiT . ( A.5.0.3, B.1.1)

ˆ Eigenmatrices s w

T i

are linearly independent dyads constituted by right and left eigenvectors of diagonalizable X and are generators of some pointed polyhedral cone K in a subspace of RM ×M . i

When S is real and X belongs to that polyhedral cone K , for example, then coordinates of expansion (the eigenvalues λ i ) must be nonnegative. When X = QΛQT is symmetric, coordinates for biorthogonal expansion are its eigenvalues when expanded in Q ; id est, for X ∈ SM T

X = QQ X =

M X i=1

qi qiTX

=

M X i=1

λ i qi qiT ∈ SM

(371)

becomes an orthogonal expansion with orthonormality condition QTQ = I where λ i is the i th eigenvalue of X , qi is the corresponding i th eigenvector arranged columnar in orthogonal matrix Q = [ q1 q2 · · · qM ] ∈ RM ×M

(372)

and where eigenmatrix qi qiT is an extreme direction of some pointed polyhedral cone K ⊂ SM and an extreme direction of the positive semidefinite cone SM + .

ˆ Orthogonal expansion is a special case of biorthogonal expansion of

X ∈ aff K occurring when polyhedral cone K is any rotation about the origin of an orthant belonging to a subspace.

Similarly, when X = QΛQT belongs to the positive semidefinite cone in the subspace of symmetric matrices, coordinates for orthogonal expansion

2.13. DUAL CONE & GENERALIZED INEQUALITY

181

must be its nonnegative eigenvalues (1394) when expanded in Q ; id est, for X ∈ SM + M M X X T T (373) λ i qi qiT ∈ SM qi qi X = X = QQ X = + i=1

i=1

where λ i ≥ 0 is the i th eigenvalue of X . This means matrix X simultaneously belongs to the positive semidefinite cone and to the pointed polyhedral cone K formed by the conic hull of its eigenmatrices. 2

2.13.7.1.2 Example. Expansion respecting nonpositive orthant. Suppose x ∈ K any orthant in Rn .2.65 Then coordinates for biorthogonal expansion of x must be nonnegative; in fact, absolute value of the Cartesian coordinates. Suppose, in particular, x belongs to the nonpositive orthant K = Rn− . Then the biorthogonal expansion becomes an orthogonal expansion x = XX T x =

n X i=1

−ei (−eT i x) =

n X i=1

n −ei |eT i x| ∈ R−

(374)

and the coordinates of expansion are nonnegative. For this orthant K we have orthonormality condition X TX = I where X = −I , ei ∈ Rn is a standard basis vector, and −ei is an extreme direction ( 2.8.1) of K . Of course, this expansion x = XX T x applies more broadly to domain Rn , but then the coordinates each belong to all of R . 2

§

2.13.8

Biorthogonal expansion, derivation

Biorthogonal expansion is a means for determining coordinates in a pointed conic coordinate system characterized by a nonorthogonal basis. Study of nonorthogonal bases invokes pointed polyhedral cones and their duals; extreme directions of a cone K are assumed to constitute the basis while ∗ those of the dual cone K determine coordinates. Unique biorthogonal expansion with respect to K depends upon existence of its linearly independent extreme directions: Polyhedral cone K must be pointed; then it possesses extreme directions. Those extreme directions must be linearly independent to uniquely represent any point in their span. 2.65

An orthant is simplicial and self-dual.

182

CHAPTER 2. CONVEX GEOMETRY

We consider nonempty pointed polyhedral cone K having possibly empty interior; id est, we consider a basis spanning a subspace. Then we need ∗ only observe that section of dual cone K in the affine hull of K because, by expansion of x , membership x ∈ aff K is implicit and because any breach of the ordinary dual cone into ambient space becomes irrelevant ( 2.13.9.3). Biorthogonal expansion

§

x = XX † x ∈ aff K = aff cone(X)

(375)

is expressed in the extreme directions {Γi } of K arranged columnar in X = [ Γ1 Γ2 · · · ΓN ] ∈ Rn×N

(255)

under assumption of biorthogonality X †X = I

(376)

§

where † denotes matrix pseudoinverse ( E). We therefore seek, in this ∗ section, a vertex-description for K ∩ aff K in terms of linearly independent ∗ dual generators {Γi } ⊂ aff K in the same finite quantity2.66 as the extreme directions {Γi } of K = cone(X) = {Xa | a º 0} ⊆ Rn

(97)

We assume the quantity of extreme directions N does not exceed the dimension n of ambient vector space because, otherwise, the expansion could not be unique; id est, assume N linearly independent extreme directions hence N ≤ n (X skinny 2.67 -or-square full-rank). In other words, fat full-rank matrix X is prohibited by uniqueness because of existence of an infinity of right inverses;

ˆ polyhedral cones whose extreme directions number in excess of the ambient space dimension are precluded in biorthogonal expansion. ∗

When K is contained in a proper subspace of Rn , the ordinary dual cone K will have more generators in any minimal set than K has extreme directions. 2.67 “Skinny” meaning thin; more rows than columns. 2.66

2.13. DUAL CONE & GENERALIZED INEQUALITY 2.13.8.1

183

x∈K

Suppose x belongs to K ⊆ Rn . Then x = Xa for some a º 0. Vector a is unique only when {Γi } is a linearly independent set.2.68 Vector a ∈ RN can take the form a = Bx if R(B) = RN . Then we require Xa = XBx = x and Bx = BXa = a . The pseudoinverse B = X † ∈ RN ×n ( E) is suitable when X is skinny-or-square and full-rank. In that case rank X = N , and for all c º 0 and i = 1 . . . N

§

†T a º 0 ⇔ X † Xa º 0 ⇔ aT X TX †T c ≥ 0 ⇔ ΓT i X c ≥ 0

(377)

The penultimate inequality follows from the generalized inequality and membership corollary, while the last inequality is a consequence of that corollary’s discretization ( 2.13.4.2.1).2.69 From (377) and (365) we deduce

§

∗

K ∩ aff K = cone(X †T ) = {X †T c | c º 0} ⊆ Rn

(378)

∗

is the vertex-description for that section of K in the affine hull of K because R(X †T ) = R(X) by definition of the pseudoinverse. From (283), we know ∗ K ∩ aff K must be pointed if rel int K is logically assumed nonempty with respect to aff K . Conversely, suppose full-rank skinny-or-square matrix (N ≤ n) i h ∗ ∗ ∗ (379) X †T , Γ1 Γ2 · · · ΓN ∈ Rn×N ∗

comprises the extreme directions {Γi } ⊂ aff K of the dual-cone intersection with the affine hull of K .2.70 From the discretized membership theorem and 2.68

Conic independence alone (§2.10) is insufficient to guarantee uniqueness.

2.69

a º 0 ⇔ aT X TX †T c ≥ 0 ∀ (c º 0 ⇔ ∀ (c º 0 ⇔

aT X TX †T c ≥ 0 ∀ a º 0) †T ∀i) ΓT i X c≥0

¨

Intuitively, any nonnegative vector a is a conic combination of the standard basis {ei ∈ RN }; a º 0 ⇔ ai ei º 0 for all i . The last inequality in (377) is a consequence of the fact that x = Xa may be any extreme direction of K , in which case a is a standard basis vector; a = ei º 0. Theoretically, because c º 0 defines a pointed polyhedral cone (in fact, the nonnegative orthant in RN ), we can take (377) one step further by discretizing c : ∗

† a º 0 ⇔ ΓT i Γj ≥ 0 for i, j = 1 . . . N ⇔ X X ≥ 0

In words, X † X must be a matrix whose entries are each nonnegative. ∗ 2.70 When closed convex cone K has empty interior, K has no extreme directions.

184

CHAPTER 2. CONVEX GEOMETRY

(287) we get a partial dual to (365); id est, assuming x ∈ aff cone X n ∗ o ∗ x ∈ K ⇔ γ x ≥ 0 for all γ ∈ Γi , i = 1 . . . N ⊂ ∂K ∩ aff K ∗T

∗

⇔ X †x º 0

(380) (381)

that leads to a partial halfspace-description, K =

©

ª x ∈ aff cone X | X † x º 0

(382)

† T For γ ∗ = X †T ei , any x = Xa , and for all i we have eT i X Xa = ei a ≥ 0 only when a º 0. Hence x ∈ K . When X is full-rank, then the unique biorthogonal expansion of x ∈ K becomes (375) N X ∗T † x = XX x = Γ i Γi x (383) i=1

∗T

whose coordinates Γi x must be nonnegative because K is assumed pointed closed and convex. Whenever X is full-rank, so is its pseudoinverse X † . ( E) In the present case, the columns of X †T are linearly independent ∗ and generators of the dual cone K ∩ aff K ; hence, the columns constitute its extreme directions. ( 2.10.2) That section of the dual cone is itself a polyhedral cone (by (261) or the cone intersection theorem, 2.7.2.1.1) having the same number of extreme directions as K .

§

§

2.13.8.2

x ∈ aff K

§

∗

The extreme directions of K and K ∩ aff K have a distinct relationship; ∗ because X † X = I , then for i,j = 1 . . . N , ΓT i Γi = 1, while for i 6= j , ∗ ∗ Yet neither set of extreme directions, {Γi } nor {Γi } , is ΓT i Γj = 0. necessarily orthogonal. This is a biorthogonality condition, precisely, [340, 2.2.4] [188] implying each set of extreme directions is linearly independent. ( B.1.1.1) The biorthogonal expansion therefore applies more broadly; meaning, for any x ∈ aff K , vector x can be uniquely expressed x = Xb where b ∈ RN because aff K contains the origin. Thus, for any such x ∈ R(X) (confer E.1.1), biorthogonal expansion (383) becomes x = XX † Xb = Xb .

§

§

§

2.13. DUAL CONE & GENERALIZED INEQUALITY

2.13.9

Formulae finding dual cone

2.13.9.1

Pointed K , dual, X skinny-or-square full-rank

185

We wish to derive expressions for a convex cone and its ordinary dual under the general assumptions: pointed polyhedral K denoted by its linearly independent extreme directions arranged columnar in matrix X such that rank(X ∈ Rn×N ) = N , dim aff K ≤ n

(384)

The vertex-description is given: K = {Xa | a º 0} ⊆ Rn

(385)

from which a halfspace-description for the dual cone follows directly: ∗

K = {y ∈ Rn | X T y º 0}

(386)

X ⊥ , basis N (X T )

(387)

aff cone X = aff K = {x | X ⊥T x = 0}

(388)

By defining a matrix (a columnar basis for the orthogonal complement of R(X)), we can say meaning K lies in a subspace, perhaps Rn . Thus a halfspace-description K = {x ∈ Rn | X † x º 0 , X ⊥T x = 0}

(389)

and a vertex-description2.71 from (287) ∗

K = { [ X †T X ⊥ −X ⊥ ]b | b º 0 } ⊆ Rn

(390)

These results are summarized for a pointed polyhedral cone, having linearly independent generators, and its ordinary dual: Cone Table 1

K

vertex-description X halfspace-description X † , X ⊥T 2.71

K

∗

X †T , ±X ⊥ XT

These descriptions are not unique. A vertex-description of the dual cone, for example, might use four conically independent generators for a plane (§2.10.0.0.1, Figure 47) when only three would suffice.

186 2.13.9.2

CHAPTER 2. CONVEX GEOMETRY Simplicial case

§

When a convex cone is simplicial ( 2.12.3), Cone Table 1 simplifies because then aff cone X = Rn : For square X and assuming simplicial K such that rank(X ∈ Rn×N ) = N , dim aff K = n

(391)

we have K

Cone Table S

K

∗

X †T XT

vertex-description X halfspace-description X †

For example, vertex-description (390) simplifies to ∗

K = {X †T b | b º 0} ⊂ Rn

(392)

§

∗

Now, because dim R(X) = dim R(X †T ) , ( E) the dual cone K is simplicial whenever K is. 2.13.9.3

Cone membership relations in a subspace ∗

It is obvious, by definition (271) of the ordinary dual cone K in ambient vector space R , that its determination instead in subspace M ⊆ R is identical to its intersection with M ; id est, assuming closed convex cone ∗ K ⊆ M and K ⊆ R ∗

∗

(K were ambient M) ≡ (K in ambient R) ∩ M

(393)

because © ª © ª y ∈ M | hy , xi ≥ 0 for all x ∈ K = y ∈ R | hy , xi ≥ 0 for all x ∈ K ∩ M (394) From this, a constrained membership relation for the ordinary dual cone ∗ K ⊆ R , assuming x, y ∈ M and closed convex cone K ⊆ M ∗

y ∈ K ∩ M ⇔ hy , xi ≥ 0 for all x ∈ K

§

(395)

By closure in subspace M we have conjugation ( 2.13.1.1): ∗

x ∈ K ⇔ hy , xi ≥ 0 for all y ∈ K ∩ M

(396)

2.13. DUAL CONE & GENERALIZED INEQUALITY

187

This means membership determination in subspace M requires knowledge of the dual cone only in M . For sake of completeness, for proper cone K with respect to subspace M (confer (299)) ∗

x ∈ int K ⇔ hy , xi > 0 for all y ∈ K ∩ M , y 6= 0

(397)

∗

x ∈ K , x 6= 0 ⇔ hy , xi > 0 for all y ∈ int K ∩ M

(398)

(By closure, we also have the conjugate relations.) Yet when M equals aff K for K a closed convex cone ∗

x ∈ rel int K ⇔ hy , xi > 0 for all y ∈ K ∩ aff K , y 6= 0

(399)

∗

x ∈ K , x 6= 0 ⇔ hy , xi > 0 for all y ∈ rel int(K ∩ aff K) 2.13.9.4

(400)

Subspace M = aff K

Assume now a subspace M that is the affine hull of cone K : Consider again a pointed polyhedral cone K denoted by its extreme directions arranged columnar in matrix X such that rank(X ∈ Rn×N ) = N , dim aff K ≤ n

(384)

We want expressions for the convex cone and its dual in subspace M = aff K : ∗

Cone Table A

K

K ∩ aff K

vertex-description halfspace-description

X † X , X ⊥T

X †T X T , X ⊥T

When dim aff K = n , this table reduces to Cone Table S. These descriptions facilitate work in a proper subspace. The subspace of symmetric matrices SN , for example, often serves as ambient space.2.72 2.13.9.4.1 Exercise. Conically independent columns and rows. We suspect the number of conically independent columns (rows) of X to be the same for X †T , where † denotes matrix pseudoinverse ( E). Prove whether it holds that the columns (rows) of X are c.i. ⇔ the columns (rows) of X †T are c.i. H

§

∗

N N The dual cone of positive semidefinite matrices SN + = S+ remains in S by convention, N ×N whereas the ordinary dual cone would venture into R . 2.72

188

CHAPTER 2. CONVEX GEOMETRY

1

0.8

∗

KM+

0.6

(a)

KM+

0.4

0.2

0

−0.2

∗

KM+

−0.4

−0.6

−0.8

−1

−0.5

0

0.5

1

1.5

 0 X †T(: , 3) =  0  1 



 1 1 1 X = 0 1 1  0 0 1

∗ ∂KM+

(b)

KM+ 

 1 X †T(: , 1) =  −1  0



 0 X †T(: , 2) =  1  −1

Figure 61: Simplicial cones. (a) Monotone nonnegative cone KM+ and its ∗ dual KM+ (drawn truncated) in R2 . (b) Monotone nonnegative cone and boundary of its dual (both drawn truncated) in R3 . Extreme directions of ∗ KM+ are indicated.

2.13. DUAL CONE & GENERALIZED INEQUALITY

§

189

§

2.13.9.4.2 Example. Monotone nonnegative cone. [57, exer.2.33] [328, 2] Simplicial cone ( 2.12.3.1.1) KM+ is the cone of all nonnegative vectors having their entries sorted in nonincreasing order: KM+ , {x | x1 ≥ x2 ≥ · · · ≥ xn ≥ 0} ⊆ Rn+

= {x | (ei − ei+1 )Tx ≥ 0, i = 1 . . . n−1, eT n x ≥ 0}

(401)

= {x | X † x º 0}

a halfspace-description where ei is the i th standard basis vector, and where2.73 X †T , [ e1 −e2

e2 −e3 · · · en ] ∈ Rn×n

(402)

For any vectors x and y , simple algebra demands xTy =

n X i=1

xi yi = (x1 − x2 )y1 + (x2 − x3 )(y1 + y2 ) + (x3 − x4 )(y1 + y2 + y3 ) + · · · + (xn−1 − xn )(y1 + · · · + yn−1 ) + xn (y1 + · · · + yn )

(403)

Because xi − xi+1 ≥ 0 ∀ i by assumption whenever x ∈ KM+ , we can employ dual generalized inequalities (296) with respect to the self-dual nonnegative ∗ orthant Rn+ to find the halfspace-description of the dual cone KM+ . We can say xTy ≥ 0 for all X † x º 0 [sic] if and only if y1 ≥ 0 ,

y1 + y 2 ≥ 0 ,

... ,

y1 + y 2 + · · · + y n ≥ 0

(404)

id est, xTy ≥ 0 ∀ X † x º 0 ⇔ X Ty º 0

where

X = [ e1

e1 + e2

e1 + e2 + e3 · · · 1 ] ∈ Rn×n

(405) (406)

Because X † x º 0 connotes membership of x to pointed KM+ , then by (271) the dual cone we seek comprises all y for which (405) holds; thus its halfspace-description ∗

KM+ = {y º 0} = {y | ∗

KM+

Pk

i=1

yi ≥ 0 , k = 1 . . . n} = {y | X Ty º 0} ⊂ Rn (407)

With X † in hand, we might concisely scribe the remaining vertex- and halfspace-descriptions from the tables for KM+ and its dual. Instead we use dual generalized inequalities in their derivation. 2.73

190

CHAPTER 2. CONVEX GEOMETRY x2

1

0.5

KM

0

−0.5

KM

−1

∗

KM

−1.5

−2

−1

−0.5

0

0.5

1

1.5

x1

2

∗

Figure 62: Monotone cone KM and its dual KM (drawn truncated) in R2 . The monotone nonnegative cone and its dual are simplicial, illustrated for two Euclidean spaces in Figure 61. From 2.13.6.1, the extreme directions of proper KM+ are respectively ∗ ∗ orthogonal to the facets of KM+ . Because KM+ is simplicial, the inward-normals to its facets constitute the linearly independent rows of X T by (407). Hence the vertex-description for KM+ employs the columns of X in agreement with Cone Table S because X † = X −1 . Likewise, the extreme ∗ directions of proper KM+ are respectively orthogonal to the facets of KM+ whose inward-normals are contained in the rows of X † by (401). So the ∗ vertex-description for KM+ employs the columns of X †T . 2

§

2.13.9.4.3 Example. Monotone cone. (Figure 62, Figure 63) Of nonempty interior but not pointed, the monotone cone is polyhedral and defined by the halfspace-description KM , {x ∈ Rn | x1 ≥ x2 ≥ · · · ≥ xn } = {x ∈ Rn | X ∗Tx º 0}

(408)

Its dual is therefore pointed but of empty interior, having vertex-description ∗

KM = { X ∗ b , [ e1 − e2 e2 − e3 · · · en−1 − en ] b | b º 0 } ⊂ Rn

(409)

∗

where the columns of X ∗ comprise the extreme directions of KM . Because ∗ KM is pointed and satisfies ∗

rank(X ∗ ∈ Rn×N ) = N , dim aff K ≤ n

(410)

2.13. DUAL CONE & GENERALIZED INEQUALITY

191

x3

∂KM -x2

∗

KM x3 ∂KM

x2

∗

KM

∗

Figure 63: Two views of monotone cone KM and its dual KM (drawn truncated) in R3 . Monotone cone is not pointed. Dual monotone cone has empty interior. Cartesian coordinate axes are drawn for reference.

192

CHAPTER 2. CONVEX GEOMETRY

where N = n − 1, and because KM is closed and convex, we may adapt Cone Table 1 (p.185) as follows: K

Cone Table 1*

∗

∗∗

K =K

vertex-description X∗ ∗† halfspace-description X , X ∗⊥T

X ∗†T , ±X ∗⊥ X ∗T

The vertex-description for KM is therefore KM = {[ X ∗†T X ∗⊥ −X ∗⊥ ]a | a º 0} ⊂ Rn

(411)

where X ∗⊥ = 1 and 

X ∗†

n−1

−1

−1

··· ...

−1

···

1

−1

−1

 ··· −2 −2  n − 2 n − 2 −2  . . . . .. n − 3 n − 3 . . −(n − 4) −3 1  . =  .. .. . n 3 . n − 4 . . −(n − 3) −(n − 3) .   ... 2 −(n − 2) −(n − 2) 2 ···  2 1

while

1

1

1

−(n − 1)

∗

KM = {y ∈ Rn | X ∗† y º 0 , X ∗⊥T y = 0} is the dual monotone cone halfspace-description.



      ∈ Rn−1×n     (412)

(413) 2

2.13.9.4.4 Exercise. Inside the monotone cones. Mathematically describe the respective interior of the monotone nonnegative cone and monotone cone. In three dimensions, also describe the relative interior of each face. H 2.13.9.5

More pointed cone descriptions with equality condition

Consider pointed polyhedral cone K having a linearly independent set of generators and whose subspace membership is explicit; id est, we are given the ordinary halfspace-description K = {x | Ax º 0 , C x = 0} ⊆ Rn

(261a)

2.13. DUAL CONE & GENERALIZED INEQUALITY

193

where A ∈ Rm×n and C ∈ Rp×n . This can be equivalently written in terms of nullspace of C and vector ξ : K = {Z ξ ∈ Rn | AZ ξ º 0}

(414)

where R(Z ∈ Rn×n−rank C ) , N (C ). Assuming (384) is satisfied ¡ ¢ rank X , rank (AZ )† ∈ Rn−rank C×m = m − ℓ = dim aff K ≤ n − rank C (415)

where ℓ is the number of conically dependent rows in AZ which must ˆ before the Cone Tables become applicable.2.74 be removed to make AZ ˆ , (AZ ˆ )† ∈ Rn−rank C×m−ℓ , Then results collected there admit assignment X m−ℓ×n where Aˆ ∈ R , followed with linear transformation by Z . So we get the ˆ )† skinny-or-square, vertex-description, for full-rank (AZ ˆ )† b | b º 0} K = {Z(AZ

(416)

From this and (336) we get a halfspace-description of the dual cone ∗

K = {y ∈ Rn | (Z TAˆT )† Z Ty º 0}

(417)

From this and Cone Table 1 (p.185) we get a vertex-description, (1797) ∗ ˆ )T C T −C T ]c | c º 0} K = {[ Z †T(AZ

(418)

Yet because K = {x | Ax º 0} ∩ {x | Cx = 0}

(419)

then, by (287), we get an equivalent vertex-description for the dual cone ∗

K = {x | Ax º 0}∗ + {x | Cx = 0}∗ = {[ AT C T −C T ]b | b º 0}

(420)

from which the conically dependent columns may, of course, be removed. 2.74

When the conically dependent rows (§2.10) are removed, the rows remaining must be linearly independent for the Cone Tables (p.19) to apply.

194

CHAPTER 2. CONVEX GEOMETRY

2.13.10

Dual cone-translate

First-order optimality condition (325) inspires a dual-cone variant: For any set K , the negative dual of its translation by any a ∈ Rn is © ª −(K − a)∗ , ©y ∈ Rn | hy , x − ai ≤ 0 for all x ∈ K ª (421) = y ∈ Rn | hy , xi ≤ 0 for all x ∈ K − a

§

a closed convex cone called normal cone to K at point a . (K⊥ (a) E.10.3.2.1) From this, a new membership relation like (293) for closed convex cone K : y ∈ −(K − a)∗ ⇔ hy , x − ai ≤ 0 for all x ∈ K

(422)

and by closure the conjugate x ∈ K ⇔ hy , x − ai ≤ 0 for all y ∈ −(K − a)∗ 2.13.10.1

§

(423)

first-order optimality condition - restatement

(confer 2.13.3) The general first-order necessary and sufficient condition for optimality of solution x⋆ to a minimization problem with real differentiable convex objective function f (x) : Rn → R over convex feasible set C is [284, 3]

§

−∇f (x⋆ ) ∈ −(C − x⋆ )∗ ,

§

x⋆ ∈ C

(424)

id est, the negative gradient ( 3.7) belongs to the normal cone to C at x⋆ as in Figure 64. 2.13.10.1.1 Example. Normal cone to orthant. Consider proper cone K = Rn+ , the self-dual nonnegative orthant in Rn . The normal cone to Rn+ at a ∈ K is (2014) KR⊥n+(a ∈ Rn+ ) = −(Rn+ − a)∗ = −Rn+ ∩ a⊥ , ∗

a ∈ Rn+

(425)

where −Rn+ = −K is the algebraic complement of Rn+ , and a⊥ is the orthogonal complement to range of vector a . This means: When point a is interior to Rn+ , the normal cone is the origin. If np represents number of nonzero entries in vector a ∈ ∂ Rn+ , then dim(−Rn+ ∩ a⊥ ) = n − np and there is a complementary relationship between the nonzero entries in vector a and the nonzero entries in any vector x ∈ −Rn+ ∩ a⊥ . 2

2.13. DUAL CONE & GENERALIZED INEQUALITY

195

α ≥ β ≥ γ

α β C

−∇f (x⋆ )

x⋆ γ

{z | f (z) = α} {y | ∇f (x⋆ )T (y − x⋆ ) = 0 , f (x⋆ ) = γ} Figure 64: (confer Figure 75) Shown is a plausible contour plot in R2 of some arbitrary differentiable real convex function f (x) at selected levels α , β , and γ ; id est, contours of equal level f (level sets) drawn dashed in function’s domain. From results in 3.7.2 (p.248), gradient ∇f (x⋆ ) is normal to γ-sublevel set Lγ f (525) by Definition E.9.1.0.1. From 2.13.10.1, function is minimized over convex set C at point x⋆ iff negative gradient −∇f (x⋆ ) belongs to normal cone to C there. In circumstance depicted, normal cone is a ray whose direction is coincident with negative gradient. So, gradient is normal to a hyperplane supporting both C and the γ-sublevel set.

§

§

196

CHAPTER 2. CONVEX GEOMETRY

2.13.10.1.2 Example. Optimality conditions for conic problem. Consider a convex optimization problem having real differentiable convex objective function f (x) : Rn → R defined on domain Rn minimize x

f (x)

(426)

subject to x ∈ K

Let’s first suppose that the feasible set is a pointed polyhedral cone K possessing a linearly independent set of generators and whose subspace membership is made explicit by fat full-rank matrix C ∈ Rp×n ; id est, we are given the halfspace-description, for A ∈ Rm×n K = {x | Ax º 0 , Cx = 0} ⊆ Rn

(261a)

(We’ll generalize to any convex cone K later.) Vertex-description of this cone, ˆ )† skinny-or-square full-rank, is assuming (AZ ˆ )† b | b º 0} K = {Z(AZ

(416)

§

where Aˆ ∈ Rm−ℓ×n , ℓ is the number of conically dependent rows in AZ ( 2.10) which must be removed, and Z ∈ Rn×n−rank C holds basis N (C ) columnar. From optimality condition (325), ˆ )† b − x⋆ ) ≥ 0 ∀ b º 0 ∇f (x⋆ )T (Z(AZ ˆ )† (b − b⋆ ) ≤ 0 ∀ b º 0 −∇f (x⋆ )T Z(AZ because

ˆ )† b⋆ ∈ K x⋆ , Z(AZ

(427) (428) (429)

From membership relation (422) and Example 2.13.10.1.1 h−(Z TAˆT )† Z T ∇f (x⋆ ) , b − b⋆ i ≤ 0 for all b ∈ Rm−ℓ + ⇔ −(Z TAˆT )† Z T ∇f (x⋆ ) ∈ −Rm−ℓ ∩ b⋆⊥ +

(430)

Then equivalent necessary and sufficient conditions for optimality of conic problem (426) with feasible set K are: (confer (335)) (Z TAˆT )† Z T ∇f (x⋆ ) º 0 , m−ℓ R+

b⋆ º 0 , m−ℓ R+

ˆ )† b⋆ = 0 ∇f (x⋆ )T Z(AZ

(431)

2.13. DUAL CONE & GENERALIZED INEQUALITY

197

expressible, by (417), ∗

∇f (x⋆ ) ∈ K ,

x⋆ ∈ K ,

∇f (x⋆ )T x⋆ = 0

(432)

This result (432) actually applies more generally to any convex cone K comprising the feasible set: Necessary and sufficient optimality conditions are in terms of objective gradient −∇f (x⋆ ) ∈ −(K − x⋆ )∗ ,

x⋆ ∈ K

(424)

whose membership to normal cone, assuming only cone K convexity, ∗

⊥ ⋆ −(K − x⋆ )∗ = KK (x ∈ K) = −K ∩ x⋆⊥

(2014)

equivalently expresses conditions (432). When K = Rn+ , in particular, then C = 0, A = Z = I ∈ S n ; id est, minimize x

f (x)

subject to x º 0 Rn +

(433)

§

Necessary and sufficient optimality conditions become (confer [57, 4.2.3]) ∇f (x⋆ ) º 0 , Rn +

x⋆ º 0 , Rn +

∇f (x⋆ )T x⋆ = 0

(434)

equivalent to condition (303)2.75 (under nonzero gradient) for membership to the nonnegative orthant boundary ∂Rn+ . 2 2.13.10.1.3 Example. Complementarity problem. A complementarity problem in nonlinear function f is nonconvex find z ∈ K ∗ subject to f (z) ∈ K hz , f (z)i = 0

[195]

(435)

yet bears strong resemblance to (432) and to (1950) Moreau’s decomposition ∗ theorem on page 712 for projection P on mutually polar cones K and −K . 2.75

and equivalent to well-known Karush-Kuhn-Tucker (KKT) optimality conditions [57, §5.5.3] because the dual variable becomes gradient ∇f (x).

198

CHAPTER 2. CONVEX GEOMETRY

§

Identify a sum of mutually orthogonal projections x , z −f (z) ; in Moreau’s ∗ terms, z = PK x and −f (z) = P−K∗ x . Then f (z) ∈ K ( E.9.2.2 no.4) and z is a solution to the complementarity problem iff it is a fixed point of z = PK x = PK (z − f (z))

(436)

Given that a solution exists, existence of a fixed point would be guaranteed by theory of contraction. [211, p.300] But because only nonexpansivity (Theorem E.9.3.0.1) is achievable by a projector, uniqueness cannot be assured. [189, p.155] Elegant proofs of equivalence between complementarity problem (435) and fixed point problem (436) are provided by S´andor Zolt´an N´emeth on Wıκımization. 2 2.13.10.1.4 Example. Linear complementarity problem. [82] [253] [289] Given matrix B ∈ Rn×n and vector q ∈ Rn , a prototypical complementarity problem on the nonnegative orthant K = Rn+ is linear in w = f (z) : find z º 0 subject to w º 0 wTz = 0 w = q + Bz

(437)

This problem is not convex when both vectors w and z are variable.2.76 Notwithstanding, this linear complementarity problem can be solved by identifying w ← ∇f (z) = q + Bz then substituting that gradient into (435) find z ∈ K ∗ subject to ∇f (z) ∈ K hz , ∇f (z)i = 0

(438)

2.76

But if one of them is fixed, then the problem becomes convex with a very simple geometric interpretation: Define the affine subset A , {y ∈ Rn | B y = w − q} For wTz to vanish, there must be a complementary relationship between the nonzero entries of vectors w and z ; id est, wi zi = 0 ∀ i . Given w º 0, then z belongs to the convex set of solutions: z ∈ −KR⊥n+(w ∈ Rn+ ) ∩ A = Rn+ ∩ w⊥ ∩ A where KR⊥n(w) is the normal cone to Rn+ at w (425). If this intersection is nonempty, then + the problem is solvable.

2.13. DUAL CONE & GENERALIZED INEQUALITY

199

which is simply a restatement of optimality conditions (432) for conic problem (426). Suitable f (z) is the quadratic objective from convex problem minimize z

1 T z Bz 2

+ q Tz

(439)

subject to z º 0

n which means B ∈ S+ should be (symmetric) positive semidefinite for solution 2 of (437) by this method. Then (437) has solution iff (439) does.

2.13.11

Proper nonsimplicial K , dual, X fat full-rank

Since conically dependent columns can always be removed from X to ∗ construct K or to determine K [Wıκımization], then assume we are given a set of N conically independent generators ( 2.10) of an arbitrary proper polyhedral cone K in Rn arranged columnar in X ∈ Rn×N such that N > n (fat) and rank X = n . Having found formula (392) to determine the dual of a simplicial cone, the easiest way to find a vertex-description of proper S dual ∗ cone K is to first decompose K into simplicial parts K i so that K = K i .2.77 Each component simplicial cone in K corresponds to some subset of n linearly independent columns from X . The key idea, here, is how the extreme directions of the simplicial parts must remain extreme directions of K . Finding the dual of K amounts to finding the dual of each simplicial part:

§

§

2.13.11.0.1 Theorem. Dual cone intersection. [308, 2.7] Suppose proper cone K ⊂ Rn equals the union of M simplicial cones K i whose ∗ extreme directions all coincide with those of K . Then proper dual cone K ∗ is the intersection of M dual simplicial cones K i ; id est, K=

2.77

M [

i=1

∗

Ki ⇒ K =

M \

i=1

∗

Ki

(440) ⋄

That proposition presupposes, of course, that we know how to perform simplicial decomposition efficiently; also called “triangulation”. [281] [159, §3.1] [160, §3.1] Existence of multiple simplicial parts means expansion of x ∈ K like (383) can no longer be unique because the number N of extreme directions in K exceeds dimension n of the space.

200

CHAPTER 2. CONVEX GEOMETRY

Proof. For Xi ∈ Rn×n , a complete matrix of linearly independent extreme directions (p.136) arranged columnar, corresponding simplicial K i ( 2.12.3.1.1) has vertex-description

§

K i = {Xi c | c º 0}

(441)

Now suppose, K =

M [

i=1

Ki =

M [

i=1

{Xi c | c º 0}

(442)

The union of all K i can be equivalently expressed     a        b   [ X1 X2 · · · XM ]  K = | a , b . . . c º 0  ...        c

(443)

§

Because extreme directions of the simplices K i are extreme directions of K by assumption, then by the extremes theorem ( 2.8.1.1.1), K = { [ X1 X2 · · · XM ] d | d º 0 }

(444)

Defining X , [ X1 X2 · · · XM ] (with any redundant [sic] columns optionally ∗ removed from X ), then K can be expressed, (336) (Cone Table S, p.186) ∗

T

K = {y | X y º 0} =

M \

i=1

{y |

XiTy

º 0} =

M \

i=1

∗

Ki

(445) ¨

To find the extreme directions of the dual cone, first we observe that some facets of each simplicial part K i are common to facets of K by assumption, and the union of all those common facets comprises the set of all facets of K by design. For any particular proper polyhedral cone K , the extreme ∗ directions of dual cone K are respectively orthogonal to the facets of K . ( 2.13.6.1) Then the extreme directions of the dual cone can be found among inward-normals to facets of the component simplicial cones K i ; those ∗ normals are extreme directions of the dual simplicial cones K i . From the theorem and Cone Table S (p.186),

§

∗

K =

M \

i=1

∗ Ki

=

M \

i=1

{Xi†T c | c º 0}

(446)

2.13. DUAL CONE & GENERALIZED INEQUALITY

201

∗

∗

The set of extreme directions {Γi } for proper dual cone K is therefore constituted by those conically independent generators, from the columns of all the dual simplicial matrices {Xi†T } , that do not violate discrete ∗ definition (336) of K ; o n o n ∗ ∗ ∗ Γ1 , Γ2 . . . ΓN = c.i. Xi†T(:,j) , i = 1 . . . M , j = 1 . . . n | Xi† (j,:)Γℓ ≥ 0, ℓ = 1 . . . N (447) where c.i. denotes selection of only the conically independent vectors from the argument set, argument (:,j) denotes the j th column while (j,:) denotes the j th row, and {Γℓ } constitutes the extreme directions of K . Figure 48b (p.134) shows a cone and its dual found via this algorithm.

2.13.11.0.2 Example. Dual of K nonsimplicial in subspace aff K . Given conically independent generators for pointed closed convex cone K in R4 arranged columnar in

X = [ Γ1 Γ2 Γ3



 1 1 0 0  −1 0 1 0   Γ4 ] =   0 −1 0 1  0 0 −1 −1

(448)

having dim aff K = rank X = 3, (256) then performing the most inefficient simplicial decomposition in aff K we find 

1 1 0  −1 0 1 X1 =   0 −1 0 0 0 −1  1 0 0  −1 1 0 X3 =   0 0 1 0 −1 −1



 , 



 , 



1 1 0  −1 0 0 X2 =   0 −1 1 0 0 −1  1 0 0  0 1 0 X4 =   −1 0 1 0 −1 −1

   

   

(449)

202

CHAPTER 2. CONVEX GEOMETRY

The corresponding dual simplicial cones in aff K have generators respectively columnar in 

2 1 1  −2 1 1 4X1†T =   2 −3 1 −2 1 −3  3 2 −1  −1 2 −1 4X3†T =   −1 −2 3 −1 −2 −1



 , 



 , 



1 2 1  −3 2 1 4X2†T =   1 −2 1 1 −2 −3  3 −1 2  −1 3 −2 4X4†T =   −1 −1 2 −1 −1 −2

   



(450)

  

Applying algorithm (447) we get

h

∗

∗

∗

Γ1 Γ 2 Γ 3



 1 2 3 2 i 1 1 ∗ 2 −1 −2   Γ4 =  2  4  1 −2 −1 −3 −2 −1 −2

(451)

whose rank is 3, and is the known result;2.78 a conically independent set ∗ of generators for that pointed section of the dual cone K in aff K ; id est, ∗ K ∩ aff K . 2 2.13.11.0.3 Example. Dual of proper polyhedral K in R4 . Given conically independent generators for a full-dimensional pointed closed convex cone K

X = [ Γ1 Γ2 Γ3 Γ4

2.78



1 1 0 1  −1 0 1 0 Γ5 ] =   0 −1 0 1 0 0 −1 −1

 0 1   0  0

(452)

These calculations proceed so as to be consistent with [104, §6]; as if the ambient vector space were proper subspace aff K whose dimension is 3. In that ambient space, K may be regarded as a proper cone. Yet that author (from the citation) erroneously states dimension of the ordinary dual cone to be 3 ; it is, in fact, 4.

2.13. DUAL CONE & GENERALIZED INEQUALITY

203

we count 5!/((5 − 4)! 4!) = 5 component simplices.2.79 Applying algorithm ∗ ∗ (447), we find the six extreme directions of dual cone K (with Γ2 = Γ5 ) X∗ =

h

∗

∗

∗

∗

∗

Γ 1 Γ 2 Γ 3 Γ4 Γ5

§



1 0 0 i  ∗ 1 0 0 Γ6 =   1 0 −1 1 −1 −1

 1 0   1  0

1 1 1 0 0 −1 1 0

(453)

which means, ( 2.13.6.1) this proper polyhedral K = cone(X) has six (three-dimensional) facets generated G by its {extreme directions}:    F    1          F    2       F3 G =  F4            F    5       F6

whereas dual proper polyhedral  ∗  F1         ∗       F2  G F3∗ =     F4∗           ∗ F5

Γ1 Γ2 Γ3 Γ1 Γ2 Γ1 Γ4 Γ1 Γ3 Γ4 Γ3 Γ4 Γ2 Γ3 ∗

    Γ5     Γ5    Γ5     Γ5

(454)

cone K has only five:

 Γ∗1 Γ∗2 Γ∗3 Γ∗4   ∗ ∗ ∗  Γ1 Γ2 Γ6   Γ∗1 Γ∗4 Γ∗5 Γ∗6   Γ∗3 Γ∗4 Γ∗5   ∗ ∗ ∗ ∗  Γ2 Γ3 Γ5 Γ6

(455)

∗

∗

∗

∗

Six two-dimensional cones, having generators respectively {Γ1 Γ3 } {Γ2 Γ4 } ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ {Γ1 Γ5 } {Γ4 Γ6 } {Γ2 Γ5 } {Γ3 Γ6 } , are relatively interior to dual facets; ∗ so cannot be two-dimensional faces of K (by Definition 2.6.0.0.3). We can check this result (453) by reversing the process; we find 6!/((6 − 4)! 4!) − 3 = 12 component simplices in the dual cone.2.80 Applying algorithm (447) to those simplices returns a conically independent set of generators for K equivalent to (452). 2 2.79

There are no linearly dependent combinations of three or four extreme directions in the primal cone. 2.80 Three combinations of four dual extreme directions are linearly dependent; they belong to the dual facets. But there are no linearly dependent combinations of three dual extreme directions.

204

CHAPTER 2. CONVEX GEOMETRY

2.13.11.0.4 Exercise. Reaching proper polyhedral cone interior. Name two extreme directions Γi of cone K from Example 2.13.11.0.3 whose convex hull passes through that cone’s interior. Explain why. Are there two ∗ such extreme directions of dual cone K ? H A natural question pertains to whether a theory of unique coordinates, like biorthogonal expansion, is extensible to proper cones whose extreme directions number in excess of ambient spatial dimensionality. 2.13.11.0.5 Theorem. Conic coordinates. With respect to vector v in some finite-dimensional Euclidean space Rn , define a coordinate t⋆v of point x in full-dimensional pointed closed convex cone K (456) t⋆v (x) , sup{t ∈ R | x − tv ∈ K}

Given points x and y in cone K , if t⋆v (x) = t⋆v (y) for each and every extreme direction v of K then x = y . ⋄

2.13.11.0.6 Proof. Vector x − t⋆ v must belong to the cone boundary ∂K by definition (456). So there must exist a nonzero vector λ that is inward-normal to a hyperplane supporting cone K and containing x − t⋆ v ; id est, by boundary membership relation for full-dimensional pointed closed convex cones ( 2.13.2)

§

∗

x − t⋆ v ∈ ∂K ⇔ ∃ λ 6= 0 Ä hλ , x − t⋆ vi = 0 , λ ∈ K , x − t⋆ v ∈ K (303) where ∗

K = {w ∈ Rn | hv , wi ≥ 0 for all v ∈ G(K)}

(342)

is the full-dimensional pointed closed convex dual cone. The set G(K) , of possibly infinite cardinality N , comprises generators for cone K ; e.g., its extreme directions which constitute a minimal generating set. If x − t⋆ v is nonzero, any such vector λ must belong to the dual cone boundary by conjugate boundary membership relation ∗

λ ∈ ∂K ⇔ ∃ x−t⋆ v 6= 0 Ä hλ , x−t⋆ vi = 0 , x−t⋆ v ∈ K , λ ∈ K

∗

where ∗

K = {z ∈ Rn | hλ , zi ≥ 0 for all λ ∈ G(K )}

(341)

(304)

2.13. DUAL CONE & GENERALIZED INEQUALITY

205

This description of K means: cone K is an intersection of halfspaces whose inward-normals are generators of the dual cone. Each and every face of cone K (except the cone itself) belongs to a hyperplane supporting K . Each and every vector x − t⋆ v on the cone boundary must therefore be orthogonal ∗ to an extreme direction constituting generators G(K ) of the dual cone. To the i th extreme direction v = Γi ∈ Rn of cone K , ascribe a coordinate t⋆i (x) ∈ R of x from definition (456). On domain K , the mapping   t⋆1 (x)   (457) t⋆ (x) =  ...  : Rn → RN ⋆ tN (x)

has no nontrivial nullspace. Because x − t⋆ v must belong to ∂K by definition, the mapping t⋆ (x) is equivalent to a convex problem (separable in index i) whose objective (by (303)) is tightly bounded below by 0 : t⋆ (x) ≡ arg minimize t∈RN

N P

i=1

∗T Γj(i) (x − ti Γi )

subject to x − ti Γi ∈ K ,

(458) i=1 . . . N

where index j ∈ I is dependent on i and where (by (341)) λ = Γj∗ ∈ Rn is an ∗ extreme direction of dual cone K that is normal to a hyperplane supporting K and containing x − t⋆i Γi . Because extreme-direction cardinality N for ∗ cone K is not necessarily the same as for dual cone K , index j must be judiciously selected from a set I . To prove injectivity when extreme-direction cardinality N > n exceeds spatial dimension, we need only show mapping t⋆ (x) to be invertible; [118, thm.9.2.3] id est, x is recoverable given t⋆ (x) : x = arg minimize n x ˜∈R

N P

i=1

∗T Γj(i) (˜ x − t⋆i Γi )

subject to x˜ −

t⋆i Γi

∈K,

(459) i=1 . . . N

The feasible set of this nonseparable convex problem is T an intersection of translated full-dimensional pointed closed convex cones i K + t⋆i Γi . The objective function’s linear part describes movement in normal-direction −Γj∗ for each of N hyperplanes. The optimal point of hyperplane intersection is the unique solution x when {Γj∗ } comprises n linearly independent normals that come from the dual cone and make the objective vanish. Because the

206

CHAPTER 2. CONVEX GEOMETRY ∗

dual cone K is full-dimensional, pointed, closed, and convex by assumption, ∗ there exist N extreme directions {Γj∗ } from K ⊂ Rn that span Rn . So we need simply choose N spanning dual extreme directions that make the optimal objective vanish. Because such dual extreme directions preexist by (303), t⋆ (x) is invertible. Otherwise, in the case N ≤ n , t⋆ (x) holds coordinates for biorthogonal expansion. Reconstruction of x is therefore unique. ¨ 2.13.11.1

conic coordinate computation

The foregoing proof of the conic coordinates theorem is not constructive; it establishes existence of dual extreme directions {Γj∗ } that will reconstruct a point x from its coordinates t⋆ (x) via (459), but does not prescribe the index set I . There are at least two computational methods for specifying ∗ {Γj(i) } : one is combinatorial but sure to succeed, the other is a geometric method that searches for a minimum of a nonconvex function. We describe the latter: Convex problem (P) (P)

maximize t t∈R

subject to x − tv ∈ K

minimize n λ∈ R

λT x

subject to λT v = 1 ∗ λ∈K

(D)

(460)

is equivalent to definition (456) whereas convex problem (D) is its dual;2.81 meaning, primal and dual optimal objectives are equal t⋆ = λ⋆T x assuming Slater’s condition (p.272) is satisfied. Under this assumption of strong duality, λ⋆T (x − t⋆ v) = t⋆ (1 − λ⋆T v) = 0 ; which implies, the primal problem is equivalent to minimize λ⋆T(x − tv) t∈R (P) (461) subject to x − tv ∈ K 2.81

Form a Lagrangian associated with primal problem (P): L(t , λ) = t + λT (x − t v) = λT x + t(1 − λT v) , T

sup L(t , λ) = λ x , t

λº0 T

K∗

1−λ v =0

Dual variable (Lagrange multiplier [232, p.216]) λ generally has a nonnegative sense for primal maximization with any cone membership constraint, whereas λ would have a nonpositive sense were the primal instead a minimization problem having a membership constraint.

2.13. DUAL CONE & GENERALIZED INEQUALITY

207

while the dual problem is equivalent to minimize n λ∈ R

λT (x − t⋆ v)

subject to λT v = 1 ∗ λ∈K

(D)

(462)

Instead given coordinates t⋆ (x) and a description of cone K , we propose inversion by alternating solution of primal and dual problems minimize n x∈R

N P

i=1

⋆ Γ∗T i (x − ti Γi )

subject to x − t⋆i Γi ∈ K , minimize ∗ n Γi ∈ R

N P

i=1

(463) i=1 . . . N

⋆ ⋆ Γ∗T i (x − ti Γi )

subject to Γ∗T i Γi = 1 , ∗ ∗ Γi ∈ K ,

i=1 . . . N i=1 . . . N

(464)

where dual extreme directions Γ∗i are initialized arbitrarily and ultimately ascertained by the alternation. Convex problems (463) and (464) are iterated until convergence which is guaranteed by virtue of a monotonically nonincreasing real sequence of objective values. Convergence can be fast. The mapping t⋆ (x) is uniquely inverted when the necessarily nonnegative ⋆ ⋆ objective vanishes; id est, when Γ∗T Here, a zero i (x − ti Γi ) = 0 ∀ i . objective can occur only at the true solution x . But this global optimality condition cannot be guaranteed by the alternation because the common objective function, when regarded in both primal x and dual Γ∗i variables simultaneously, is generally neither quasiconvex or monotonic. Conversely, a nonzero objective at convergence is a certificate that inversion was not performed properly. A nonzero objective indicates that the global minimum of a multimodal objective function could not be found by this alternation. That is a flaw in this particular iterative algorithm for inversion; not in theory.2.82 A numerical remedy is to reinitialize the Γ∗i to different values.

The Proof 2.13.11.0.6 that suitable dual extreme directions {Γj∗ } always exist means that a global optimization algorithm would always find the zero objective of alternation (463) (464); hence, the unique inversion x . But such an algorithm can be combinatorial. 2.82

208

CHAPTER 2. CONVEX GEOMETRY

Chapter 3 Geometry of convex functions The link between convex sets and convex functions is via the epigraph: A function is convex if and only if its epigraph is a convex set. −Werner Fenchel

We limit our treatment of multidimensional functions 3.1 to finite-dimensional Euclidean space. Then an icon for a one-dimensional (real) convex function is bowl-shaped (Figure 74), whereas the concave icon is the inverted bowl; respectively characterized by a unique global minimum and maximum whose existence is assumed. Because of this simple relationship, usage of the term convexity is often implicitly inclusive of concavity. Despite iconic imagery, the reader is reminded that the set of all convex, concave, quasiconvex, and quasiconcave functions contains the monotonic functions [189] [200, 2.3.5]; e.g., [57, 3.6, exer.3.46].

§

§

3.1

vector- or matrix-valued functions including the real functions. Appendix D, with its tables of first- and second-order gradients, is the practical adjunct to this chapter. © 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

209

210

3.1 3.1.1

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Convex function real and vector-valued function

Vector-valued function f (X) : Rp×k → RM



 f1 (X)   .. =   . fM (X)

(465)

assigns each X in its domain dom f (a subset of ambient vector space Rp×k ) to a specific element [240, p.3] of its range (a subset of RM ). Function f (X) is linear in X on its domain if and only if, for each and every Y, Z ∈ dom f and α , β ∈ R f (α Y + βZ) = αf (Y ) + βf (Z ) (466) A vector-valued function f (X) : Rp×k → RM is convex in X if and only if dom f is a convex set and, for each and every Y, Z ∈ dom f and 0 ≤ µ ≤ 1 f (µ Y + (1 − µ)Z) ¹ µf (Y ) + (1 − µ)f (Z )

(467)

RM +

As defined, continuity is implied but not differentiability (nor smoothness).3.2 Apparently some, but not all, nonlinear functions are convex. Reversing sense of the inequality flips this definition to concavity. Linear (and affine 3.5) functions attain equality in this definition. Linear functions are therefore simultaneously convex and concave. Vector-valued functions are most often compared (167) as in (467) with respect to the M -dimensional self-dual nonnegative orthant RM + , a proper 3.3 In this case, the test prescribed by (467) is simply a comparison cone. on R of each entry fi of a vector-valued function f . ( 2.13.4.2.3) The vector-valued function case is therefore a straightforward generalization of conventional convexity theory for a real function. This conclusion follows from theory of dual generalized inequalities ( 2.13.2.0.1) which asserts

§

§

§

∗

M T f convex w.r.t RM + ⇔ w f convex ∀ w ∈ G(R+ ) 3.2

(468)

Figure 65b illustrates a nondifferentiable convex function. Differentiability is certainly not a requirement for optimization of convex functions by numerical methods; e.g., [226]. 3.3 Definition of convexity can be broadened to other (not necessarily proper) cones. ∗ ∗ Referred to in the literature as K-convexity, [273] RM + (468) generalizes to K .

3.1. CONVEX FUNCTION

211

f1 (x)

f2 (x)

(a)

(b)

Figure 65: Convex real functions here have a unique minimizer x⋆ . For x∈ R , √ f1 (x) = x2 = kxk22 is strictly convex whereas nondifferentiable function f2 (x) = x2 = |x| = kxk2 is convex but not strictly. Strict convexity of a real function is only a sufficient condition for minimizer uniqueness.

§

shown by substitution of the defining inequality (467). Discretization allows ∗ relaxation ( 2.13.4.2.1) of a semiinfinite number of conditions {w ∈ RM + } to: ∗

M {w ∈ G(RM + )} ≡ {ei ∈ R , i = 1 . . . M }

(469)

§

(the standard basis for RM and a minimal set of generators ( 2.8.1.2) for RM +) from which the stated conclusion follows; id est, the test for convexity of a vector-valued function is a comparison on R of each entry.

3.1.2

strict convexity

When f (X) instead satisfies, for each and every distinct Y and Z in dom f and all 0 < µ < 1 f (µ Y + (1 − µ)Z) ≺ µf (Y ) + (1 − µ)f (Z )

(470)

RM +

then we shall say f is a strictly convex function. Similarly to (468) ∗

M T f strictly convex w.r.t RM + ⇔ w f strictly convex ∀ w ∈ G(R+ )

(471)

discretization allows relaxation of the semiinfinite number of conditions ∗ {w ∈ RM + , w 6= 0} (299) to a finite number (469). More tests for strict convexity are given in 3.7.1.0.2, 3.7.4, and 3.8.3.0.2.

§

§

§

212

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Any convex real function f (X) has unique minimum value over any convex subset of its domain. [280, p.123] Yet solution to some convex optimization problem is, in general, not unique; e.g., given minimization of a convex real function over some convex feasible solution set C minimize X

f (X)

(472)

subject to X ∈ C

any optimal solution X ⋆ comes from a convex set of optimal solutions X ⋆ ∈ {X | f (X) = inf f (Y ) } ⊆ C

(473)

Y∈C

But a strictly convex real function has a unique minimizer X ⋆ ; id est, for the optimal solution set in (473) to be a single point, it is sufficient (Figure 65) that f (X) be a strictly convex real3.4 function and set C convex. [299] Quadratic real functions xTAx + bTx + c are convex in x iff A º 0. Quadratics characterized by positive definite matrix A ≻ 0 are strictly convex. The vector 2-norm square kxk2 (Euclidean norm square) and Frobenius norm square kXk2F , for example, are strictly convex functions of their respective argument (each absolute norm is convex but not strictly convex). Figure 65a illustrates a strictly convex real function. 3.1.2.1 minimum/minimal element, dual cone characterization f (X ⋆ ) is the minimum element of its range if and only if, for each and every ∗ T w ∈ int RM + , it is the unique minimizer of w f . (Figure 66) [57, 2.6.3] ⋆ If f (X ) is a minimal element of its range, then there exists a nonzero ∗ ⋆ T ⋆ T w ∈ RM + such that f (X ) minimizes w f . If f (X ) minimizes w f for some ∗ ⋆ w ∈ int RM + , conversely, then f (X ) is a minimal element of its range.

§

3.1.2.1.1 Exercise. Cone of convex functions. Prove that relation (468) implies: the set of all vector-valued convex functions in RM is a convex cone. So, the trivial function f = 0 is convex. Indeed, any nonnegatively weighted sum of (strictly) convex functions remains (strictly) convex.3.5 Interior to the cone are the strictly convex functions. H 3.4

It is more customary to consider only a real function for the objective of a convex optimization problem because vector- or matrix-valued functions can introduce ambiguity into the optimal objective value. (§2.7.2.2, §3.1.2.1) Study of multidimensional objective functions is called multicriteria- [302] or multiobjective- or vector-optimization. 3.5 Strict case excludes cone’s point at origin. By these definitions (468) (471), positively weighted sums mixing convex and strictly convex real functions are not strictly convex

3.1. CONVEX FUNCTION

213

Rf (b) f (X ⋆ ) f (X ⋆ )

f (X)2

Rf

f (X)1

w

w (a)

f f

Figure 66: (confer Figure 38) Function range is convex for a convex problem. (a) Point f (X ⋆ ) is the unique minimum element of function range Rf . (b) Point f (X ⋆ ) is a minimal element of depicted range. (Cartesian axes drawn for reference.)

214

3.2

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Conic origins of Lagrangian

The cone of convex functions, membership relation (468), provides a foundation for what is known as a Lagrangian function. [234, p.398] [259] Consider a conic optimization problem, for proper cone K and affine subset A minimize f (x) x (474) subject to g(x) ∈ K h(x) ∈ A A Cartesian product of convex functions remains convex, so we may write  M∗     M " # " # w R+ R+ f f ∗ T T T      g convex w.r.t K ⇔ [ w λ ν ] g convex ∀ λ ∈ G K  h h ν A A⊥ (475) where A⊥ is the normal cone to A . This holds because of equality for h in convexity criterion (467) and because membership relation (423), given point a ∈ A , becomes h ∈ A ⇔ hν , h − ai = 0 for all ν ∈ G(A⊥ )

(476)

When A = 0, for example, a minimal set of generators G for A⊥ is a superset of the standard basis for RM (Example E.5.0.0.7) the ambient space of A . A real Lagrangian arises from the scalar term in (475); id est, f L , [ w T λT ν T ] g h "

3.3

#

= w T f + λT g + ν T h

(477)

Practical norm functions, absolute value

To some mathematicians, “all norms on Rn are equivalent” [146, p.53]; meaning, ratios of different norms are bounded above and below by finite predeterminable constants. But to statisticians and engineers, all norms are certainly not created equal; as evidenced by the compressed sensing revolution, begun in 2004, whose focus is predominantly 1-norm. because each summand is considered to be an entry from a vector-valued function.

3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE

215

A norm on Rn is a convex function f : Rn → R satisfying: for x, y ∈ Rn , α ∈ R [211, p.59] [146, p.52] 1. f (x) ≥ 0 (f (x) = 0 ⇔ x = 0)

(nonnegativity) (triangle inequality)3.6

2. f (x + y) ≤ f (x) + f (y) 3. f (αx) = |α|f (x)

(nonnegative homogeneity)

Convexity follows by properties 2 and 3. Most useful are 1-, 2-, and ∞-norm: kxk1 =

1T t

minimize n t∈ R

subject to −t ¹ x ¹ t

(478)

where |x| = t⋆ (entrywise absolute value equals optimal t ).3.7 kxk2 =

where kxk2 = kxk ,

minimize

t

subject to

·

t∈ R

tI x xT t

¸

º 0

(479)

n+1 S+

√ xTx = t⋆ .

kxk∞ =

minimize t∈ R

t

subject to −t1 ¹ x ¹ t1

(480)

where max{|xi | , i = 1 . . . n} = t⋆ . kxk1 =

minimize 1T (α + β) n n

α∈ R , β∈ R

subject to α , β º 0 x=α−β

(481)

where |x| = α⋆ + β ⋆ because of complementarity α⋆Tβ ⋆ = 0 at optimality. (478) (480) (481) represent linear programs, (479) is a semidefinite program. 3.6

with equality iff x = κ y where κ ≥ 0. Vector 1 may be replaced with any positive [sic] vector to get absolute value, theoretically, although 1 provides the 1-norm. 3.7

216

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Over some convex set C given vector constant y or matrix constant Y arg inf kx − yk2 = arg inf kx − yk22

(482)

arg inf kX − Y kF = arg inf kX − Y k2F

(483)

x∈C

x∈C

X∈ C

X∈ C

§

are unconstrained convex quadratic problems. Equality does not hold for a sum of norms. ( 5.4.2.3.2) Optimal solution is norm dependent: [57, p.297] minimize n x∈ R

kxk1

subject to x ∈ C

minimize 1T t n n

≡

x∈ R , t∈ R

subject to −t ¹ x ¹ t x∈C minimize n

t

subject to

·

x∈ R , t∈ R

minimize n x∈ R

kxk2

subject to x ∈ C

≡

tI x xT t

¸

º 0

(484)

(485)

n+1 S+

x∈C

minimize n x∈ R

kxk∞

subject to x ∈ C

minimize n ≡

x∈ R , t∈ R

t

subject to −t1 ¹ x ¹ t1 x∈ C

(486)

In Rn these norms represent: kxk1 is length measured along a grid (taxicab distance), kxk2 is Euclidean length, kxk∞ is maximum |coordinate|. minimize

minimize n x∈ R

kxk1

subject to x ∈ C

α∈ Rn , β∈ Rn

≡

1T (α + β)

subject to α , β º 0 x=α−β x∈ C

(487)

These foregoing problems (478)-(487) are convex whenever set C is. Their equivalence transformations make objectives smooth.

3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE

217

A = {x ∈ R3 | Ax = b}

R3

B1 = {x ∈ R3 | kxk1 ≤ 1}

Figure 67: 1-norm ball B1 is convex hull of all cardinality 1 vectors of unit norm (its vertices). Ball boundary contains all points equidistant from origin in 1-norm. Cartesian axes drawn for reference. Plane A is overhead (drawn truncated). If 1-norm ball is expanded until it kisses A (intersects ball only at boundary), then distance (in 1-norm) from origin to A is achieved. Euclidean ball would be spherical in this dimension. Only were A parallel to two axes could there be a minimum cardinality least Euclidean norm solution. Yet 1-norm ball offers infinitely many (but not all) A-orientations resulting in a minimum cardinality solution. (∞-norm ball is a cube in this dimension.)

218

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.3.0.0.1 Example. Projecting the origin on an affine subset, in 1-norm. In (1808) we interpret least norm solution to linear system Ax = b as orthogonal projection of the origin 0 on affine subset A = {x ∈ Rn | Ax = b} where A ∈ Rm×n is fat full-rank. Suppose, instead of the Euclidean metric, we use taxicab distance to do projection. Then the least 1-norm problem is stated, for b ∈ R(A) minimize kxk1 x (488) subject to Ax = b Optimal solution can be interpreted as an oblique projection on A simply because the Euclidean metric is not employed. This problem statement sometimes returns optimal x⋆ having minimum cardinality; which can be explained intuitively with reference to Figure 67: [19] Projection of the origin, in 1-norm, on affine subset A is equivalent to maximization (in this case) of the 1-norm ball until it kisses A ; rather, a kissing point in A achieves the distance in 1-norm from the origin to A . For the example illustrated (m = 1, n = 3), it appears that a vertex of the ball will be first to touch A . 1-norm ball vertices in R3 represent nontrivial points of minimum cardinality 1, whereas edges represent cardinality 2, while relative interiors of facets represent maximum cardinality 3. By reorienting affine subset A so it were parallel to an edge or facet, it becomes evident as we expand or contract the ball that a kissing point is not necessarily unique.3.8 The 1-norm ball in Rn has 2n facets and 2n vertices.3.9 For n > 0 B1 = {x ∈ Rn | kxk1 ≤ 1} = conv{±ei ∈ Rn , i = 1 . . . n}

(489)

is a vertex-description of the unit 1-norm ball. Maximization of the 1-norm ball until it kisses A is equivalent to minimization of the 1-norm ball until it no longer intersects A . Then projection of the origin on affine subset A is minimize kxk1 n x∈R

subject to Ax = b

minimize n c∈R , x∈R

c

≡ subject to x ∈ cB1 Ax = b

(490)

where cB1 = {[ I −I ]a | aT 1 = c , a º 0} 3.8

(491)

This is unlike the case for the Euclidean ball (1808) where minimum-distance projection on a convex set is unique (§E.9); all kissable faces of the Euclidean ball are single points (vertices). 3.9 The ∞-norm ball in Rn has 2n facets and 2n vertices.

3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE k/m 1

1

0.9

0.9

0.8

0.8 signed

0.7

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.2

0.4

0.6

positive

0.7

0.6

0 0

219

0.8

0 0

1

0.2

0.4

0.6

0.8

1

m/n (488)

minimize kxk1

minimize kxk1

x

x

subject to Ax = b xº0

subject to Ax = b

(493)

Figure 68: Exact recovery transition: Respectively signed [107] [109] or positive [114] [112] [113] solutions x to Ax = b with sparsity k below thick curve are recoverable. For Gaussian random matrix A ∈ Rm×n , thick curve demarcates phase transition in ability to find sparsest solution x by linear programming. These results were empirically reproduced in [36].

f2 (x) xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx

f3 (x) xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx

f4 (x) xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx

xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx

Figure 69: Under 1-norm f2 (x) , histogram (hatched) of residual amplitudes Ax − b exhibits predominant accumulation of zero-residuals. Nonnegatively constrained 1-norm f3 (x) from (493) accumulates more zero-residuals than f2 (x). Under norm f4 (x) (not discussed), histogram would exhibit predominant accumulation of (nonzero) residuals at gradient discontinuities.

220

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Then (490) is equivalent to minimize

c∈R , x∈Rn , a∈R2n

subject to

c minimize kak1 2n x = [ I −I ]a a∈R ≡ subject to [ A −A ]a = b aT 1 = c aº0 aº0 Ax = b

(492)

where x⋆ = [ I −I ]a⋆ . (confer (487)) Significance of this result:

ˆ Any

vector 1-norm minimization problem may have its variable replaced with a nonnegative variable of twice the length.

All other things being equal, nonnegative variables are easier to solve for sparse solutions. (Figure 68, Figure 69, Figure 92) The compressed sensing problem becomes easier to interpret; e.g., for A ∈ Rm×n minimize 1T x

minimize kxk1 x

subject to Ax = b xº0

≡

x

subject to Ax = b xº0

(493)

movement of a hyperplane (Figure 26, Figure 30) over a bounded polyhedron always has a vertex solution [89, p.22]. Or vector b might lie on the boundary of a proper cone K = {Ax | x º 0}. In the latter case, we find practical application of the smallest face F containing b from 2.13.4.3 to remove all columns of matrix A not belonging to F ; because those columns correspond to 0-entries in vector x . 2

§

3.3.0.0.2 Exercise. Combinatorial optimization. A device commonly employed to relax combinatorial problems is to arrange desirable solutions at vertices of bounded polyhedra; e.g., the permutation matrices of dimension n , which are factorial in number, are the extreme points of a polyhedron in the nonnegative orthant described by an intersection of 2n hyperplanes ( 2.3.2.0.4). Minimizing a linear objective function over a bounded polyhedron is a convex problem (a linear program) that always has an optimal solution residing at a vertex. [89] What about minimizing other functions? Given some nonsingular matrix A , geometrically describe three circumstances under which there are

§

3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE

221

likely to exist vertex solutions to minimize kAxk1 n

(494)

x∈R

subject to x ∈ P

H

optimized over some bounded polyhedron P .3.10

3.3.1

k smallest/largest entries

Sum of the k smallest entries of x ∈ Rn is the optimal objective value from: for 1 ≤ k ≤ n n n P P T π(x)i = maximize k t + 1T z x y π(x)i = minimize n n

i=n−k+1

y∈ R

subject to 0 ¹ y ¹ 1 1T y = k

≡

i=n−k+1

z∈ R , t∈ R

subject to x º t 1 + z z¹0 (495)

which are dual linear programs, where π(x)1 = max{xi , i = 1 . . . n} where π is a nonlinear permutation-operator sorting its vector argument into nonincreasing order. Finding k smallest entries of an n-length vector x is expressible as an infimum of n!/(k!(n − k)!) linear functions of x . The sum P π(x)i is therefore a concave function of x ; in fact, monotonic ( 3.7.1.0.1) in this instance. Sum of the k largest entries of x ∈ Rn is the optimal objective value from: [57, exer.5.19] k k P P T π(x)i = maximize x y π(x)i = minimize k t + 1T z n n y∈ R z∈ R , t∈ R i=1 i=1 ≡ subject to 0 ¹ y ¹ 1 subject to x ¹ t 1 + z zº0 1T y = k (496) which are dual linear programs. Finding k largest entries of an n-length vector x is expressible as a supremum of n!/(k!(n − k)!) linear functions of x . (Figure 71) The summation is therefore a convex function (and monotonic in this instance, 3.7.1.0.1).

§

§

3.10

Hint: Suppose, for example, P belongs to an orthant and A were orthogonal. Begin with A = I and apply level sets of the objective, as in Figure 64 and Figure 67. ·Or rewrite ¸ x the problem as a linear program like (484) and (486) but in a composite variable ← y. t

222

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Let Π x be a permutation of entries xi such that their absolute value becomes arranged in nonincreasing order: |Π x|1 ≥ |Π x|2 ≥ · · · ≥ |Π x|n . Sum of the k largest entries of |x| ∈ Rn is a norm, by properties of vector norm ( 3.3), and is the optimal objective value of a linear program:

§

kxkn , k

k P

i=1

|Π x|i =

minimize n z∈ R , t∈ R

k t + 1T z

(497) subject to −t 1 − z ¹ x ¹ t 1 + z zº0 µ ¶ n where the norm subscript derives from a binomial coefficient , and k kxknn , kxk1 kxkn , kxk∞

(498)

1

Finding k largest absolute entries of an n-length vector x is expressible as a supremum of 2k n!/(k!(n − k)!) linear functions of x ; hence, this norm is convex (nonmonotonic) in x . [57, exer.6.3e] minimize n x∈ R

kxkn k

subject to x ∈ C

3.3.2

minimize

k t + 1T z

subject to

−t 1 − z ¹ x ¹ t 1 + z zº0 (499) x∈C

z∈ Rn , t∈ R , x∈ Rn

≡

clipping

Clipping negative vector entries is accomplished: kx+ k1 =

minimize n t∈ R

1T t

subject to x ¹ t 0¹ t where, for x = [xi , i = 1 . . . n] ∈ Rn " ) # x , x ≥ 0 1 i i x+ , t⋆ = , i = 1 . . . n = (x + |x|) 2 0, xi < 0

(500)

(501)

3.4. INVERTED FUNCTIONS AND ROOTS

223

(clipping) minimize 1T t n n

minimize n x∈ R

kx+ k1

subject to x ∈ C

3.4

x∈ R , t∈ R

subject to x ¹ t 0¹ t x∈C

≡

(502)

Inverted functions and roots

A given function f is convex iff −f is concave. Both functions are loosely referred to as convex since −f is simply f inverted about the abscissa axis, and minimization of f is equivalent to maximization of −f . A given positive function f is convex iff 1/f is concave; f inverted about ordinate 1 is concave. Minimization of f is maximization of 1/f . We wish to implement objectives of the form x−1 . Suppose we have a 2×2 matrix ¸ · x z ∈ R2 (503) T , z y which is positive semidefinite by (1454) when T º0

⇔

x > 0 and xy ≥ z 2

(504)

A polynomial constraint such as this is therefore called a conic constraint.3.11 This means we may formulate convex problems, having inverted variables, as semidefinite programs in Schur-form; e.g., minimize x∈R

subject to x > 0 x∈C rather

3.11

minimize

x−1

x, y∈ R

≡

1 x > 0, y ≥ x

y ·

x 1 subject to 1 y x∈C ⇔

·

x 1 1 y

¸

º0

¸

º0

(505)

(506)

In this dimension, the convex cone formed from the set of all values {x , y , z} satisfying constraint (504) is called a rotated quadratic or circular cone or positive semidefinite cone.

224

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

(inverted) For vector x = [xi , i = 1 . . . n] ∈ Rn minimize n x∈R

n P

i=1

minimize n

x−1 i

subject to x ≻ 0 x∈C

x∈R , y∈R

≡

subject to

rather ¡ ¢ x ≻ 0 , y ≥ tr δ(x)−1

3.4.1

⇔

·

xi √ n

y ·

xi √ n x∈C √

n y

¸

√

n y

¸

º0,

º0,

i=1 . . . n (507)

i=1 . . . n

(508)

fractional power

[136] To implement an objective of the form xα for positive α , we quantize α and work instead with that approximation. Choose nonnegative integer q for adequate quantization of α like so: α,

k 2q

(509)

where k ∈ {0, 1, 2 . . . 2 q − 1}. Any k from that set may be written q P k = b i 2i−1 where b i ∈ {0, 1}. Define vector y = [yi , i = 0 . . . q] with y0 = 1 : i=1

3.4.1.1

positive

Then we have the equivalent semidefinite program for maximizing a concave function xα , for quantized 0 ≤ α < 1 maximize q+1

maximize xα

x∈R , y∈R

x∈R

subject to x > 0 x∈C

≡

subject to

yq "

yi−1 yi yi x b i

#

º0,

i=1 . . . q

x∈C

where nonnegativity of yq is enforced by maximization; id est, # " y y i−1 i º 0 , i=1 . . . q x > 0 , yq ≤ xα ⇔ yi x b i

(510)

(511)

3.4. INVERTED FUNCTIONS AND ROOTS 3.4.1.2

225

negative

It is also desirable to implement an objective of the form x−α for positive α . The technique is nearly the same as before: for quantized 0 ≤ α < 1 minimizeq+1

z "

x , z∈R , y∈R

minimize x∈R

x

−α

# yi−1 yi º0, yi x b i # " z 1 º0 1 yq

subject to ≡

subject to x > 0 x∈C

x∈C

rather

x > 0, z ≥ x

3.4.1.3

−α

⇔

# yi−1 yi º0, yi x b i # " z 1 º0 1 yq

"

i=1 . . . q

(512)

i=1 . . . q (513)

positive inverted

Now define vector t = [ti , i = 0 . . . q] with t0 = 1. To implement an objective x1/α for quantized 0 ≤ α < 1 as in (509) minimize q+1

minimize x∈R

y∈R , t∈R

x1/α

subject to x ≥ 0 x∈C

subject to

≡

y "

ti−1 ti ti y b i

#

º0,

x = tq ≥ 0 x∈C

i=1 . . . q

(514)

rather x ≥ 0 , y ≥ x1/α

⇔

"

ti−1 ti ti y b i

x = tq ≥ 0

#

º0,

i=1 . . . q

(515)

226

3.5

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Affine function

§

A function f (X) is affine when it is continuous and has the dimensionally extensible form (confer 2.9.1.0.2) f (X) = AX + B

(516)

All affine functions are simultaneously convex and concave. This means: both −AX + B and AX + B , for example, are convex functions of X . When B = 0, f (X) is a linear function. Affine multidimensional functions are recognized by existence of no multivariate terms (multiplicative in argument entries) and no polynomial terms of degree higher than 1 ; id est, entries of the function are characterized only by linear combinations of argument entries plus constants. The real affine function in Figure 70 illustrates hyperplanes, in its domain, constituting contours of equal function-value (level sets (521)).

§

3.5.0.0.1 Example. Engineering control. [363, 2.2]3.12 For X ∈ SM and matrices A , B , Q , R of any compatible dimensions, for example, the expression XAX is not affine in X whereas

g(X) =

·

R B TX XB Q + ATX + XA

is an affine multidimensional function. engineering control. [55] [138]

¸

(517)

Such a function is typical in 2

(confer Figure 16) Any single- or many-valued inverse of an affine function is affine. The interpretation from this citation of {X ∈ SM | g(X) º 0} as “an intersection between a linear subspace and the cone of positive semidefinite matrices” is incorrect. (See §2.9.1.0.2 for a similar example.) The conditions they state under which strong duality holds for semidefinite programming are conservative. (confer §4.2.3.0.1) 3.12

3.5. AFFINE FUNCTION

227

A f (z)

z2

C H+

a z1

H−

{ {z ∈ Rz ∈2|Ra 2Tz| a Tz = κ1 } {z ∈ R 2| a T z = κ3 }= κ2 }

Figure 70: Three hyperplanes intersecting convex set C ⊂ R2 from Figure 28. Cartesian axes in R3 : Plotted is affine subset A = f (R2 ) ⊂ R2 × R ; a plane with third dimension. Sequence of hyperplanes, w.r.t domain R2 of affine function f (z) = aTz + b : R2 → R , is increasing in direction of gradient a ( 3.7.0.0.3) because affine function increases in normal direction (Figure 26). Minimization of aTz + b over C is equivalent to minimization of aTz .

§

228

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.5.0.0.2 Example. Linear objective. Consider minimization of a real affine function f (z) = aTz + b over the convex feasible set C in its domain R2 illustrated in Figure 70. Since scalar b is fixed, the problem posed is the same as the convex optimization minimize aTz z

(518)

subject to z ∈ C

§

whose objective of minimization is a real linear function. Were convex set C polyhedral ( 2.12), then this problem would be called a linear program. Were convex set C an intersection with a positive semidefinite cone, then this problem would be called a semidefinite program. There are two distinct ways to visualize this problem: one in the objective function’s domain R2 ,· the ¸other including the ambient space of the objective R2 function’s range as in . Both visualizations are illustrated in Figure 70. R Visualization in the function domain is easier because of lower dimension and because

ˆ level sets (521) of any affine function are affine. (§2.1.9) In this circumstance, the level sets are parallel hyperplanes with respect to R2 . One solves optimization problem (518) graphically by finding that hyperplane intersecting feasible set C furthest right (in the direction of negative gradient −a ( 3.7)). 2

§

When a differentiable convex objective function f is nonlinear, the negative gradient −∇f is a viable search direction (replacing −a in (518)). ( 2.13.10.1, Figure 64) [140] Then the nonlinear objective function can be replaced with a dynamic linear objective; linear as in (518).

§

§

§

3.5.0.0.3 Example. Support function. [185, C.2.1- C.2.3.1] n For arbitrary set Y ⊆ R , its support function σY (a) : Rn → R is defined σY (a) , sup aTz z∈Y

(519)

3.5. AFFINE FUNCTION

229 {aTz1 + b1 | a ∈ R} sup aTpz i i

{aTz2 + b2 | a ∈ R}

+ bi

{aTz3 + b3 | a ∈ R} a {aTz4 + b4 | a ∈ R} {aTz5 + b5 | a ∈ R} Figure 71: Pointwise supremum of convex functions remains convex; by epigraph intersection. Supremum of affine functions in variable a evaluated at argument ap is illustrated. Topmost affine function per a is supremum.

a positively homogeneous function of direction a whose range contains ±∞. [232, p.135] For each z ∈ Y , aTz is a linear function of vector a . Because σY (a) is a pointwise supremum of linear functions, it is convex in a (Figure 71). Application of the support function is illustrated in Figure 29a for one particular normal a . Given nonempty closed bounded convex sets Y and Z in Rn and nonnegative scalars β and γ [346, p.234] σβ Y+γZ (a) = βσY (a) + γσZ (a)

(520) 2

3.5.0.0.4 Exercise. Level sets. Given a function f and constant κ , its level sets are defined Lκκ f , {z | f (z) = κ}

(521)

Give two distinct examples of convex function, that are not affine, having convex level sets. H

230

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS f (x)

q(x)

quasiconvex

convex

x

x

Figure 72: Quasiconvex function q epigraph is not necessarily convex, but convex function f epigraph is convex in any dimension. Sublevel sets are necessarily convex for either function, but sufficient only for quasiconvexity.

3.6

Epigraph, Sublevel set

It is well established that a continuous real function is convex if and only if its epigraph makes a convex set. [185] [285] [333] [346] [232] Thereby, piecewise-continuous convex functions are admitted. Epigraph is the connection between convex sets and convex functions. Its generalization to a vector-valued function f (X) : Rp×k → RM is straightforward: [273] epi f , {(X , t) ∈ Rp×k × RM | X ∈ dom f , f (X) ¹ t }

(522)

RM +

id est, f convex ⇔ epi f convex

(523)

Necessity is proven: [57, exer.3.60] Given any (X , u) , (Y , v) ∈ epi f , we must show for all µ ∈ [0, 1] that µ(X , u) + (1− µ)(Y , v) ∈ epi f ; id est, we must show f (µX + (1− µ)Y ) ¹ µ u + (1− µ)v

(524)

RM +

Yet this holds by definition because f (µX + (1−µ)Y ) ¹ µf (X)+ (1−µ)f (Y ). The converse also holds. ¨

3.6. EPIGRAPH, SUBLEVEL SET

231

3.6.0.0.1 Exercise. Epigraph sufficiency. Prove that converse: Given any (X , u) , (Y , v) ∈ epi f , if for all µ ∈ [0, 1] µ(X , u) + (1− µ)(Y , v) ∈ epi f holds, then f must be convex. H Sublevel sets of a real convex function are convex. Likewise, corresponding to each and every ν ∈ RM Lν f , {X ∈ dom f | f (X) ¹ ν } ⊆ Rp×k

(525)

RM +

sublevel sets of a vector-valued convex function are convex. As for real convex functions, the converse does not hold. (Figure 72) To prove necessity of convex sublevel sets: For any X , Y ∈ Lν f we must show for each and every µ ∈ [0, 1] that µ X + (1−µ)Y ∈ Lν f . By definition, f (µ X + (1−µ)Y ) ¹ µf (X) + (1−µ)f (Y ) ¹ ν RM +

(526)

RM +

¨ When an epigraph (522) is artificially bounded above, t ¹ ν , then the corresponding sublevel set can be regarded as an orthogonal projection of epigraph on the function domain. Sense of the inequality is reversed in (522), for concave functions, and we use instead the nomenclature hypograph. Sense of the inequality in (525) is reversed, similarly, with each convex set then called superlevel set. 3.6.0.0.2 Example. Matrix pseudofractional function. Consider a real function of two variables f (A , x) : S n × Rn → R = xTA† x

(527)

n on dom f = S+ × R(A). This function is convex simultaneously in both variables when variable matrix A belongs to the entire positive semidefinite n cone S+ and variable vector x is confined to range R(A) of matrix A .

232

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

§

To explain this, we need only demonstrate that the function epigraph is convex. Consider Schur-form (1451) from A.4: for t ∈ R ¸ · A z º0 G(A , z , t) = zT t ⇔ (528) z T(I − AA† ) = 0 t − z TA† z ≥ 0 Aº0 n+1 Inverse image of the positive semidefinite cone S+ under affine mapping G(A , z , t) is convex by Theorem 2.1.9.0.1. Of the equivalent conditions for positive semidefiniteness of G , the first is an equality demanding vector z belong to R(A). Function f (A , z) = z TA† z is convex on convex domain n S+ × R(A) because the Cartesian product constituting its epigraph © ª ¡ n+1 ¢ epi f (A , z) = (A , z , t) | A º 0 , z ∈ R(A) , z TA† z ≤ t = G−1 S+ (529)

is convex.

2

3.6.0.0.3 Exercise. Matrix product function. Continue Example 3.6.0.0.2 by introducing vector variable x and making the substitution z ← Ax . Because of matrix symmetry ( E), for all x ∈ Rn

§

f (A , z(x)) = z TA† z = xTATA† A x = xTA x = f (A , x)

(530)

whose epigraph is epi f (A , x) =

©

ª (A , x , t) | A º 0 , xTA x ≤ t

(531)

Provide two simple explanations why f (A , x) = xTA x is not a function convex simultaneously in positive semidefinite matrix A and vector x on n dom f = S+ × Rn . H

§

3.6.0.0.4 Example. Matrix fractional function. (confer 3.8.2.0.1) Continuing Example 3.6.0.0.2, now consider a real function of two variables n on dom f = S+ × Rn for small positive constant ǫ (confer (1794)) f (A , x) = ǫ xT(A + ǫ I )−1 x

(532)

3.6. EPIGRAPH, SUBLEVEL SET

233

where the inverse always exists by (1395). This function is convex n simultaneously in both variables over the entire positive semidefinite cone S+ n and all x ∈ R : Consider Schur-form (1454) from A.4: for t ∈ R ¸ · A + ǫI z º0 G(A , z , t) = zT ǫ−1 t ⇔ (533) T −1 t − ǫ z (A + ǫ I ) z ≥ 0 A + ǫI ≻ 0

§

n+1 Inverse image of the positive semidefinite cone S+ under affine mapping G(A , z , t) is convex by Theorem 2.1.9.0.1. Function f (A , z) is convex on n S+ × Rn because its epigraph is that inverse image: © ª ¡ n+1 ¢ epi f (A , z) = (A , z , t) | A + ǫ I ≻ 0 , ǫ z T(A + ǫ I )−1 z ≤ t = G−1 S+ (534) 2

3.6.1

matrix fractional projector function

Consider nonlinear function f having orthogonal projector W as argument: f (W , x) = ǫ xT(W + ǫ I )−1 x

(535)

Projection matrix W has property W † = W T = W º 0 (1839). Any orthogonal projector can be decomposed into an outer product of orthonormal matrices W = U U T where U T U = I as explained in E.3.2. From (1794) for any ǫ > 0 and idempotent symmetric W , ǫ(W + ǫ I )−1 = I − (1 + ǫ)−1 W from which ¡ ¢ f (W , x) = ǫ xT(W + ǫ I )−1 x = xT I − (1 + ǫ)−1 W x (536)

§

Therefore

lim f (W , x) = lim+ ǫ xT(W + ǫ I )−1 x = xT(I − W )x

ǫ→0+

ǫ→0

§

(537)

where I − W is also an orthogonal projector ( E.2). We learned from Example 3.6.0.0.4 that f (W , x) = ǫ xT(W + ǫ I )−1 x is n convex simultaneously in both variables over all x ∈ Rn when W ∈ S+ is

234

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

confined to the entire positive semidefinite cone (including its boundary). It is now our goal to incorporate f into an optimization problem such that an optimal solution returned always comprises a projection matrix W . The set of orthogonal projection matrices is a nonconvex subset of the positive semidefinite cone. So f cannot be convex on the projection matrices, and its equivalent (for idempotent W ) ¡ ¢ f (W , x) = xT I − (1 + ǫ)−1 W x (538)

cannot be convex simultaneously in both variables on either the positive semidefinite or symmetric projection matrices. Suppose we allow dom f to constitute the entire positive semidefinite cone but constrain W to a Fantope (85); e.g., for convex set C and 0 < k < n as in minimize ǫ xT(W + ǫ I )−1 x n n x∈R , W ∈ S

subject to 0 ¹ W ¹ I tr W = k x∈C

(539)

Although this is a convex problem, there is no guarantee that optimal W is a projection matrix because only extreme points of a Fantope are orthogonal projection matrices U U T . Let’s try partitioning the problem into two convex parts (one for x and one for W ), substitute equivalence (536), and then iterate solution of convex problem minimize xT(I − (1 + ǫ)−1 W )x n x∈R (540) subject to x ∈ C with convex problem minimize n W∈S

x⋆T(I − (1 + ǫ)−1 W )x⋆

(a) subject to 0 ¹ W ¹ I tr W = k

maximize x⋆T W x⋆ n ≡

W∈S

subject to 0 ¹ W ¹ I tr W = k

(541)

until convergence, where x⋆ represents an optimal solution of (540) from any particular iteration. The idea is to optimally solve for the partitioned variables which are later combined to solve the original problem (539). What makes this approach sound is that the constraints are separable, the

3.6. EPIGRAPH, SUBLEVEL SET

235

partitioned feasible sets are not interdependent, and the fact that the original problem (though nonlinear) is convex simultaneously in both variables.3.13 But partitioning alone does not guarantee a projector. To make orthogonal projector W a certainty, we must invoke a known analytical optimal solution to problem (541): Diagonalize optimal solution from problem (540) x⋆ x⋆T , QΛQT ( A.5.1) and set U ⋆ = Q(: , 1 : k) ∈ Rn×k per (1627c);

§

W = U ⋆ U ⋆T =

x⋆ x⋆T + Q(: , 2 : k)Q(: , 2 : k)T kx⋆ k2

(542)

Then optimal solution (x⋆ , U ⋆ ) to problem (539) is found, for small ǫ , by iterating solution to problem (540) with optimal (projector) solution (542) to convex problem (541). Proof. Optimal vector x⋆ is orthogonal to the last n − 1 columns of orthogonal matrix Q , so ⋆ f(540) = kx⋆ k2 (1 − (1 + ǫ)−1 )

(543)

⋆ after each iteration. Convergence of f(540) is proven with the observation that iteration (540) (541a) is a nonincreasing sequence that is bounded below by 0. Any bounded monotonic sequence in R is convergent. [240, 1.2] [39, 1.1] Expression (542) for optimal projector W holds at each iteration, therefore ⋆ kx⋆ k2 (1 − (1 + ǫ)−1 ) must also represent the optimal objective value f(540) at convergence. Because the objective f(539) from problem (539) is also bounded below ⋆ by 0 on the same domain, this convergent optimal objective value f(540) (for positive ǫ arbitrarily close to 0) is necessarily optimal for (539); id est,

§

§

⋆ ⋆ f(540) ≥ f(539) ≥0

(544)

⋆ lim f(540) =0

(545)

by (1610), and ǫ→0+

Since optimal (x⋆ , U ⋆ ) from problem (540) is feasible to problem (539), and because their objectives are equivalent for projectors by (536), then converged (x⋆ , U ⋆ ) must also be optimal to (539) in the limit. Because problem (539) is convex, this represents a globally optimal solution. ¨ 3.13

A convex problem has convex feasible set, and the objective surface has one and only one global minimum. [280, p.123]

236

3.6.2

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

semidefinite program via Schur

Schur complement (1451) can be used to convert a projection problem to an optimization problem in epigraph form. Suppose, for example, we are presented with the constrained projection problem studied by Hayden & Wells in [170] (who provide analytical solution): Given A ∈ RM ×M and some full-rank matrix S ∈ RM ×L with L < M kA − Xk2F

minimize X∈ SM

(546)

subject to S TXS º 0

Variable X is constrained to be positive semidefinite, but only on a subspace determined by S . First we write the epigraph form: minimize

X∈ SM , t∈ R

t (547)

subject to kA − Xk2F ≤ t S TXS º 0

§

§

Next we use Schur complement [257, 6.4.3] [231] and matrix vectorization ( 2.2): minimize

X∈ SM , t∈ R

subject to

t ·

tI vec(A − X) vec(A − X)T 1

S TXS º 0

¸

º0

(548)

§

This semidefinite program ( 4) is an epigraph form in disguise, equivalent to (546); it demonstrates how a quadratic objective or constraint can be converted to a semidefinite constraint. Were problem (546) instead equivalently expressed without the square minimize X∈ SM

kA − XkF

subject to S TXS º 0 then we get a subtle variation:

(549)

3.6. EPIGRAPH, SUBLEVEL SET minimize

237 t

X∈ SM , t∈ R

(550)

subject to kA − XkF ≤ t S TXS º 0 that leads to an equivalent for (549) (and for (546) by (483)) minimize

X∈ SM , t∈ R

subject to

t ·

tI vec(A − X) T vec(A − X) t

S TXS º 0

¸

º0

(551)

3.6.2.0.1 Example. Schur anomaly. Consider a problem abstract in the convex constraint, given symmetric matrix A minimize kXk2F − kA − Xk2F X∈ SM (552) subject to X ∈ C the minimization of a difference of two quadratic functions each convex in matrix X . Observe equality kXk2F − kA − Xk2F = 2 tr(XA) − kAk2F

(553)

So problem (552) is equivalent to the convex optimization minimize X∈ SM

tr(XA)

subject to X ∈ C

(554)

But this problem (552) does not have Schur-form minimize

X∈ SM , α , t

t−α

subject to X ∈ C kXk2F ≤ t kA − Xk2F ≥ α

§

because the constraint in α is nonconvex. ( 2.9.1.0.1)

(555)

2

238

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS 2

1.5

1

0.5

Y2

0

−0.5

−1

−1.5

−2 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Y1 2 Figure 73: Gradient in R evaluated on grid over some open disc in domain of convex quadratic bowl f (Y ) = Y T Y : R2 → R illustrated in Figure 74. Circular contours are level sets; each defined by a constant function-value.

3.7

Gradient

§

Gradient ∇f of any differentiable multidimensional function f (formally defined in D.1) maps each entry fi to a space having the same dimension as the ambient space of its domain. Notation ∇f is shorthand for gradient ∇x f (x) of f with respect to x . ∇f (y) can mean ∇y f (y) or gradient ∇x f (y) of f (x) with respect to x evaluated at y ; a distinction that should become clear from context. Gradient of a differentiable real function f (x) : RK → R with respect to its vector argument is defined 

  ∇f (x) =   

∂f (x) ∂x1 ∂f (x) ∂x2

.. .

∂f (x) ∂xK



   ∈ RK  

(1683)

while the second-order gradient of the twice differentiable real function with

3.7. GRADIENT

239

respect to its vector argument is traditionally called the Hessian ;3.14 

   2 ∇ f (x) =   

∂ 2f (x) ∂x21

∂ 2f (x) ∂x1 ∂x2

∂ 2f (x) ∂x2 ∂x1 ∂ 2f (x) ∂xK ∂x1

.. .

···

∂ 2f (x) ∂x1 ∂xK

∂ 2f (x) ∂x22

··· ...

∂ 2f (x) ∂x2 ∂xK

∂ 2f (x) ∂xK ∂x2

···

∂ 2f (x) 2 ∂xK

.. .

.. .

§



    ∈ SK  

(1684)

The gradient can be interpreted as a vector pointing in the direction of greatest change, [204, 15.6] or polar to direction of steepest descent.3.15 [350] The gradient can also be interpreted as that vector normal to a sublevel set; e.g., Figure 75, Figure 64. For the quadratic bowl in Figure 74, the gradient maps to R2 ; illustrated in Figure 73. For a one-dimensional function of real variable, for example, the gradient evaluated at any point in the function domain is just the slope (or derivative) of that function there. (confer D.1.4.1)

§

ˆ For any differentiable multidimensional function, zero gradient ∇f = 0 is a condition necessary for its unconstrained minimization [140, §3.2]:

3.7.0.0.1 Example. Projection on a rank-1 subset. For A ∈ SN having eigenvalues λ(A) = [λ i ] ∈ RN , consider the unconstrained nonconvex optimization that is a projection on the rank-1 subset ( 2.9.2.1) of positive semidefinite cone SN Defining λ1 , max i {λ(A) i } and + : corresponding eigenvector v1

§

minimize kxxT − Ak2F = minimize tr(xxT(xTx) − 2AxxT + ATA) x x ( kλ(A)k2 , λ1 ≤ 0 = (1622) 2 2 kλ(A)k − λ1 , λ1 > 0 arg minimize kxxT − Ak2F = x

3.14

(

0, λ1 ≤ 0 √ v1 λ1 , λ1 > 0

(1623)

Jacobian is the Hessian transpose, so commonly confused in matrix calculus. Newton’s direction −∇ 2 f (x)−1 ∇f (x) is better for optimization of nonlinear functions well approximated locally by a quadratic function. [140, p.105] 3.15

240

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

§

From (1713) and D.2.1, the gradient of kxxT − Ak2F is ¡ ¢ ∇x (xTx)2 − 2xTA x = 4(xTx)x − 4Ax

(556)

Ax = x(xTx)

(557)

Setting the gradient to 0

is necessary for optimal solution. Replace vector x with a normalized eigenvector vi of A ∈ SN , corresponding to a positive eigenvalue λ i , scaled by square root of that eigenvalue. Then (557) is satisfied x ← vi

p λ i ⇒ Avi = vi λ i

(558)

§

xxT = λ i vi viT is a rank-1 matrix on the positive semidefinite cone boundary, and the minimum is achieved ( 7.1.2) when λ i = λ1 is the largest positive eigenvalue of A . If A has no positive eigenvalue, then x = 0 yields the minimum. 2 Differentiability is a prerequisite neither to convexity or to numerical solution of a convex optimization problem. The gradient provides a necessary and sufficient condition (325) (424) for optimality in the constrained case, nevertheless, as it does in the unconstrained case:

ˆ For any differentiable multidimensional convex function, zero gradient ∇f = 0 is a necessary and sufficient condition for its unconstrained minimization [57, §5.5.3]: 3.7.0.0.2 Example. Pseudoinverse. The pseudoinverse matrix is the unique solution to an unconstrained convex optimization problem [146, 5.5.4]: given A ∈ Rm×n

§

minimize kXA − Ik2F n×m

(559)

¡ ¢ kXA − Ik2F = tr ATX TXA − XA − ATX T + I

(560)

X∈ R

where

3.7. GRADIENT

241

§

whose gradient ( D.2.3)

vanishes when

¡ ¢ ∇X kXA − Ik2F = 2 XAAT − AT = 0

(561)

XAAT = AT

(562)

When A is fat full-rank, then AAT is invertible, X ⋆ = AT (AAT )−1 is the pseudoinverse A† , and AA† = I . Otherwise, we can make AAT invertible by adding a positively scaled identity, for any A ∈ Rm×n X = AT(AAT + t I )−1

(563)

Invertibility is guaranteed for any finite positive value of t by (1395). Then matrix X becomes the pseudoinverse X → A† , X ⋆ in the limit t → 0+ . Minimizing instead kAX − Ik2F yields the second flavor in (1793). 2 3.7.0.0.3 Example. Hyperplane, line, described by affine function. Consider the real affine function of vector variable, (confer Figure 70) f (x) : Rp → R = aTx + b

(564)

whose domain is Rp and whose gradient ∇f (x) = a is a constant vector (independent of x). This function describes the real line R (its range), and it describes a nonvertical [185, B.1.2] hyperplane ∂H in the space Rp × R for any particular vector a (confer 2.4.2);

§

∂H =

½·

§

x T a x+b

¸

p

| x∈ R

¾

⊂ Rp × R

(565)

having nonzero normal η =

·

a −1

¸

∈ Rp × R

(566)

242

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.16 Epigraph This equivalence to a hyperplane holds only for real functions. · p¸ R of real affine function f (x) is therefore a halfspace in , so we have: R

The real affine function is to convex functions as the hyperplane is to convex sets. Similarly, the matrix-valued affine function of real variable x , for any particular matrix A ∈ RM ×N , h(x) : R → RM ×N = Ax + B

(567)

describes a line in RM ×N in direction A {Ax + B | x ∈ R} ⊆ RM ×N and describes a line in R×RM ×N ½· ¸ ¾ x | x ∈ R ⊂ R×RM ×N Ax + B whose slope with respect to x is A . 3.16

(568)

(569) 2

To prove that, consider a vector-valued affine function f (x) : Rp→ RM = Ax + b

having gradient ∇f (x) = AT ∈ Rp×M : The affine set ¸ ½· ¾ x | x ∈ Rp ⊂ Rp × RM Ax + b is perpendicular to η ,

·

∇f (x) −I

¸

∈ Rp×M × RM ×M

because η

T

µ·

x Ax + b

¸

−

·

0 b

¸¶

=0

Yet η is a vector (in Rp × RM ) only when M = 1.

∀ x ∈ Rp ¨

3.7. GRADIENT

3.7.1

243

monotonic function

A real function of real argument is called monotonic when it is exclusively nonincreasing or nondecreasing over the whole of its domain. A real differentiable function of real argument is monotonic when its first derivative (not necessarily continuous) maintains sign over the function domain.

3.7.1.0.1 Definition. Monotonicity. Multidimensional function f (X) is nondecreasing monotonic when nonincreasing monotonic when

Y º X ⇒ f (Y ) º f (X) Y º X ⇒ f (Y ) ¹ f (X)

(570)

∀ X, Y ∈ dom f .

△

For vector-valued functions compared with respect to the nonnegative orthant, it is necessary and sufficient for each entry fi to be monotonic in the same sense. Any affine function is monotonic. tr(Y TX) is a nondecreasing monotonic function of matrix X ∈ SM when constant matrix Y is positive semidefinite, for example; which follows from a result (350) of Fej´er. A convex function can be characterized by another kind of nondecreasing monotonicity of its gradient:

§

§

3.7.1.0.2 Theorem. Gradient monotonicity. [185, B.4.1.4] p×k [51, 3.1, exer.20] Given f (X) : R → R a real differentiable function with matrix argument on open convex domain, the condition h∇f (Y ) − ∇f (X) , Y − Xi ≥ 0 for each and every X, Y ∈ dom f

(571)

is necessary and sufficient for convexity of f . Strict inequality and caveat distinct Y , X provide necessary and sufficient conditions for strict convexity. ⋄

244

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

§

§

3.7.1.0.3 Example. Composition of functions. [57, 3.2.4] [185, B.2.1] Monotonic functions play a vital role determining convexity of functions constructed by transformation. Given functions g : Rk → R and h : Rn → Rk , their composition f = g(h) : Rn → R defined by f (x) = g(h(x)) ,

dom f = {x ∈ dom h | h(x) ∈ dom g}

(572)

is convex when

ˆ g is convex nondecreasing monotonic and h is convex ˆ g is convex nonincreasing monotonic and h is concave

and composite function f is concave when

ˆ g is concave nondecreasing monotonic and h is concave ˆ g is concave nonincreasing monotonic and h is convex

where ∞ (−∞) is assigned to convex (concave) g when evaluated outside its domain. For differentiable functions, these rules are consequent to (1714).

ˆ Convexity (concavity) of any g is preserved when h is affine.

2

A squared norm is convex having the same minimum because a squaring operation is convex nondecreasing monotonic on the nonnegative real line. 3.7.1.0.4 Exercise. Product and ratio of convex functions. [57, exer.3.32] In general the product or ratio of two convex functions is not convex. [213] However, there are some results that apply to functions on R [real domain]. Prove the following.3.17 (a) If f and g are convex, both nondecreasing (or nonincreasing), and positive functions on an interval, then f g is convex. (b) If f , g are concave, positive, with one nondecreasing and the other nonincreasing, then f g is concave. (c) If f is convex, nondecreasing, and positive, and g is concave, nonincreasing, and positive, then f /g is convex. H 3.17

Hint: Prove §3.7.1.0.4(a) by verifying Jensen’s inequality ((467) at µ = 21 ).

3.7. GRADIENT

3.7.2

245

first-order convexity condition, real function

Discretization of w º 0 in (468) invites refocus to the real-valued function:

§

§

§

§

§

§

3.7.2.0.1 Theorem. Necessary and sufficient convexity condition. [57, 3.1.3] [122, I.5.2] [366, 1.2.3] [39, 1.2] [308, 4.2] [284, 3] For real differentiable function f (X) : Rp×k → R with matrix argument on open convex domain, the condition (confer D.1.7)

§

f (Y ) ≥ f (X) + h∇f (X) , Y − Xi for each and every X, Y ∈ dom f

(573)

is necessary and sufficient for convexity of f .

⋄

When f (X) : Rp → R is a real differentiable convex function with vector argument on open convex domain, there is simplification of the first-order condition (573); for each and every X, Y ∈ dom f f (Y ) ≥ f (X) + ∇f (X)T (Y − X)

§

§

(574)

From this we can find ∂H− a unique [346, p.220-229] nonvertical [185, B.1.2] hyperplane ¸ ( 2.4), expressed in terms of function gradient, supporting epi f · X : videlicet, defining f (Y ∈ / dom f ) , ∞ [57, 3.1.7] at f (X) · ¸ µ· ¸ · ¸¶ £ ¤ Y Y X T ∈ epi f ⇔ t ≥ f (Y ) ⇒ ∇f (X) −1 − ≤0 t t f (X) (575)

§

This means, for each and every point X in the domain of a real convex function ¸f (X) , there exists a hyperplane ∂H−· in Rp¸× R having normal · ∇f (X) X supporting the function epigraph at ∈ ∂H− −1 f (X) µ· ¸ · ¸¶ ¾ ½· ¸ · p ¸ £ ¤ Y X Y R T ∂H− = − = 0 (576) ∈ ∇f (X) −1 t f (X) t R One such supporting hyperplane (confer Figure 29(a)) is illustrated in Figure 74 for a convex quadratic. From (574) we deduce, for each and every X , Y ∈ dom f ∇f (X)T (Y − X) ≥ 0 ⇒ f (Y ) ≥ f (X)

(577)

246

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

f (Y ) ·

∇f (X) −1

¸

∂H− Figure 74: When a real function f is differentiable at each point in its open domain, there is an intuitive geometric interpretation of function convexity in terms of its gradient ∇f and its epigraph: Drawn is a convex quadratic bowl in R2 × R (confer Figure 152, p.652); f (Y ) = Y T Y : R2 → R versus Y on some open disc in R2 . Unique strictly supporting hyperplane ∂H− ∈ R2 × R (only partially drawn) and its normal vector [ ∇f (X)T −1 ]T at the particular point of support [ X T f (X) ]T are illustrated. The interpretation: At each and every coordinate Y , there is a unique hyperplane containing [ Y T f (Y ) ]T and supporting the epigraph of convex differentiable f .

3.7. GRADIENT

247

α

α ≥ β ≥ γ β

Y −X ∇f (X)

γ {Z | f (Z ) = α}

{Y | ∇f (X)T (Y − X) = 0 , f (X) = α} (578) Figure 75:(confer Figure 64) Shown is a plausible contour plot in R2 of some arbitrary real differentiable convex function f (Z ) at selected levels α , β , and γ ; contours of equal level f (level sets) drawn in the function’s domain. A convex function has convex sublevel sets Lf (X) f (579). [285, 4.6] The sublevel set whose boundary is the level set at α , for instance, comprises all the shaded regions. For any particular convex function, the family comprising all its sublevel sets is nested. [185, p.75] Were sublevel sets not convex, we may certainly conclude the corresponding function is neither convex. Contour plots of real affine functions are illustrated in Figure 26 and Figure 70.

§

248

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

meaning, the gradient at X identifies a supporting hyperplane there in Rp {Y ∈ Rp | ∇f (X)T (Y − X) = 0}

(578)

to the convex sublevel sets of convex function f (confer (525)) Lf (X) f , {Z ∈ dom f | f (Z) ≤ f (X)} ⊆ Rp

(579)

illustrated for an arbitrary real convex function in Figure 75 and Figure 64. That supporting hyperplane is unique for twice differentiable f . [204, p.501]

3.7.3

first-order convexity condition, vector function

Now consider the first-order necessary and sufficient condition for convexity of a vector-valued function: Differentiable function f (X) : Rp×k → RM is convex if and only if dom f is open, convex, and for each and every X, Y ∈ dom f ¯ →Y −X d ¯¯ (580) f (Y ) º f (X) + df (X) = f (X) + ¯ f (X + t (Y − X)) dt t=0 M R+

→Y −X

where df (X) is the directional derivative 3.18 [204] [310] of f at X in direction Y − X . This, of course, follows from the real-valued function case: by dual generalized inequalities ( 2.13.2.0.1), ¿ À →Y −X →Y −X f (Y ) − f (X) − df (X) º 0 ⇔ f (Y ) − f (X) − df (X) , w ≥ 0 ∀ w º 0

§

RM +

RM +

∗

(581) where →Y −X

3.18



  df (X) =  

¡ ¢ tr ∇f1 (X)T (Y − X) ¡ ¢ tr ∇f2 (X)T (Y − X)    ∈ RM ..  . ¡ ¢ T tr ∇fM (X) (Y − X)

(582)

We extend the traditional definition of directional derivative in §D.1.4 so that direction may be indicated by a vector or a matrix, thereby broadening the scope of the Taylor series (§D.1.7). The right-hand side of the inequality (580) is the first-order Taylor series expansion of f about X .

3.7. GRADIENT

249

Necessary and sufficient discretization (468) allows relaxation of the semiinfinite number of conditions {w º 0} instead to {w ∈ {ei , i = 1 . . . M }} the extreme directions of the self-dual nonnegative orthant. Each extreme →Y −X

direction picks out a real entry fi and df (X) i from the vector-valued function and its directional derivative, then Theorem 3.7.2.0.1 applies. The vector-valued function case (580) is therefore a straightforward application of the first-order convexity condition for real functions to each entry of the vector-valued function.

3.7.4

second-order convexity condition

Again, by discretization (468), we are obliged only to consider each individual entry fi of a vector-valued function f ; id est, the real functions {fi }. For f (X) : Rp → RM , a twice differentiable vector-valued function with vector argument on open convex domain, ∇ 2fi (X) º 0 ∀ X ∈ dom f ,

i=1 . . . M

(583)

p S+

is a necessary and sufficient condition for convexity of f . Obviously, when M = 1, this convexity condition also serves for a real function. Condition (583) demands nonnegative curvature, intuitively, hence precluding points of inflection as in Figure 76 (p.257). Strict inequality in (583) provides only a sufficient condition for strict convexity, but that is nothing new; videlicet, the strictly convex real function fi (x) = x4 does not have positive second derivative at each and every x ∈ R . Quadratic forms constitute a notable exception where the strict-case converse holds reliably.

§ §

3.7.4.0.1 Exercise. Real fractional function. (confer 3.4, 3.6.0.0.4) Prove that real function f (x, y) = x/y is not convex on the first quadrant. Also exhibit this in a plot of the function. (f is quasilinear (p.258) on {y > 0} , in fact, and nonmonotonic even on the first quadrant.) H 3.7.4.0.2 Exercise. Stress function. p Define |x − y| , (x − y)2 and

X = [ x1 · · · xN ] ∈ R1×N

(71)

250

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

×N , consider function Given symmetric nonnegative data [hij ] ∈ SN ∩ RN +

f (vec X) =

N −1 X

N X

i=1 j=i+1

(|xi − xj | − hij )2 ∈ R

(1254)

Take the gradient and Hessian of f . Then explain why f is not a convex function; id est, why doesn’t second-order condition (583) apply to the constant positive semidefinite Hessian matrix you found. For N = 6 and hij data from (1325), apply line theorem 3.8.3.0.1 to plot f along some arbitrary lines through its domain. H 3.7.4.1

second-order ⇒ first-order condition

For a twice-differentiable real function fi (X) : Rp → R having open domain, a consequence of the mean value theorem from calculus allows compression of its complete Taylor series expansion about X ∈ dom fi ( D.1.7) to three terms: On some open interval of kY k2 , so that each and every line segment [X , Y ] belongs to dom fi , there exists an α ∈ [0, 1] such that [366, 1.2.3] [39, 1.1.4]

§

§

§

1 fi (Y ) = fi (X) + ∇fi (X)T (Y −X) + (Y −X)T ∇ 2fi (αX + (1 − α)Y )(Y −X) 2 (584) The first-order condition for convexity (574) follows directly from this and the second-order condition (583).

3.8

Matrix-valued convex function

We need different tools for matrix argument: We are primarily interested in continuous matrix-valued functions g(X). We choose symmetric g(X) ∈ SM because matrix-valued functions are most often compared (585) with respect to the positive semidefinite cone SM + in the ambient space of symmetric matrices.3.19 Function symmetry is not a necessary requirement for convexity; indeed, for A ∈ Rm×p and B ∈ Rm×k , g(X) = AX + B is a convex (affine) function in X on domain Rp×k with respect to the nonnegative orthant Rm×k . Symmetric convex functions share the same + benefits as symmetric matrices. Horn & Johnson [188, §7.7] liken symmetric matrices to real numbers, and (symmetric) positive definite matrices to positive real numbers. 3.19

3.8. MATRIX-VALUED CONVEX FUNCTION

251

3.8.0.0.1 Definition. Convex matrix-valued function: 1) Matrix-definition. A function g(X) : Rp×k → SM is convex in X iff dom g is a convex set and, for each and every Y, Z ∈ dom g and all 0 ≤ µ ≤ 1 [200, 2.3.7]

§

g(µ Y + (1 − µ)Z) ¹ µ g(Y ) + (1 − µ)g(Z )

(585)

SM +

Reversing sense of the inequality flips this definition to concavity. Strict convexity is defined less a stroke of the pen in (585) similarly to (470). 2) Scalar-definition. It follows that g(X) : Rp×k → SM is convex in X iff wTg(X)w : Rp×k → R is convex in X for each and every kwk = 1 ; shown by substituting the defining inequality (585). By dual generalized inequalities we have the equivalent but more broad criterion, ( 2.13.5)

§

g convex w.r.t SM + ⇔ hW , gi convex for each and every W º 0 (586) SM +

∗

Strict convexity on both sides requires caveat W 6= 0. Because the set of all extreme directions for the self-dual positive semidefinite cone ( 2.9.2.4) comprises a minimal set of generators for that cone, discretization ( 2.13.4.2.1) allows replacement of matrix W with symmetric dyad wwT as proposed. △

§ §

3.8.0.0.2 Example. Taxicab distance matrix. Consider an n-dimensional vector space Rn with metric induced by the 1-norm. Then distance between points x1 and x2 is the norm of their difference: kx1 − x2 k1 . Given a list of points arranged columnar in a matrix X = [ x1 · · · xN ] ∈ Rn×N

(71)

then we could define a taxicab distance matrix N N ×N T D1 (X) , (I ⊗ 1T n ) | vec(X)1 − 1 ⊗ X | ∈ Sh ∩ R+  0 kx1 − x2 k1 kx1 − x3 k1 · · · kx1 − xN k1  kx1 − x2 k1 0 kx2 − x3 k1 · · · kx2 − xN k1   =  kx1 − x3 k1 kx2 − x3 k1 0 kx3 − xN k1  . . .. .  .. .. .. . kx1 − xN k1 kx2 − xN k1 kx3 − xN k1 · · · 0



  (587)    

252

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

where 1n is a vector of ones having dim 1n = n and where ⊗ represents Kronecker product. This matrix-valued function is convex with respect to the nonnegative orthant since, for each and every Y, Z ∈ Rn×N and all 0 ≤ µ ≤ 1 D1 (µ Y + (1 − µ)Z)

¹

×N RN +

µ D1 (Y ) + (1 − µ)D1 (Z )

(588) 2

3.8.0.0.3 Exercise. 1-norm distance matrix. The 1-norm is called taxicab distance because to go from one point to another in a city by car, road distance is a sum of grid lengths. Prove (588). H

3.8.1

first-order convexity condition, matrix function

§

From the scalar-definition ( 3.8.0.0.1) of a convex matrix-valued function, for differentiable function g and for each and every real vector w of unit norm kwk = 1, we have →Y −X

wTg(Y )w ≥ wTg(X)w + wT dg(X) w

(589)

that follows immediately from the first-order condition (573) for convexity of a real function because

→Y −X

→Y −X ­ ® wT dg(X) w = ∇X wTg(X)w , Y − X

§

(590)

§

where dg(X) is the directional derivative ( D.1.4) of function g at X in direction Y − X . By discretized dual generalized inequalities, ( 2.13.5) À ¿ →Y −X T g(Y ) − g(X) − dg(X) º 0 ⇔ g(Y ) − g(X) − dg(X) , ww ≥ 0 ∀ wwT(º 0) →Y −X

SM +

SM +

(591) For each and every X , Y ∈ dom g (confer (580)) →Y −X

g(Y ) º g(X) + dg(X)

(592)

SM +

must therefore be necessary and sufficient for convexity of a matrix-valued function of matrix variable on open convex domain.

∗

3.8. MATRIX-VALUED CONVEX FUNCTION

3.8.2

253

epigraph of matrix-valued function, sublevel sets

We generalize epigraph to a continuous matrix-valued function [32, p.155] g(X) : Rp×k → SM : epi g , {(X , T ) ∈ Rp×k × SM | X ∈ dom g , g(X) ¹ T }

(593)

SM +

from which it follows g convex ⇔ epi g convex

(594)

LS g , {X ∈ dom g | g(X) ¹ S } ⊆ Rp×k

(595)

§

Proof of necessity is similar to that in 3.6 on page 230. Sublevel sets of a matrix-valued convex function corresponding to each and every S ∈ SM (confer (525)) SM +

are convex. There is no converse. 3.8.2.0.1 Example. Matrix fractional function redux. [32, p.155] Generalizing Example 3.6.0.0.4 consider a matrix-valued function of two n×N variables on dom g = SN for small positive constant ǫ (confer (1794)) +×R g(A , X) = ǫ X(A + ǫ I )−1 X T

(596)

where the inverse always exists by (1395). This function is convex simultaneously in both variables over the entire positive semidefinite cone SN + and all X ∈ Rn×N : Consider Schur-form (1454) from A.4: for T ∈ S n ¸ · A + ǫ I XT º0 G(A , X , T ) = X ǫ−1 T ⇔ (597) −1 T T − ǫ X(A + ǫ I ) X º 0 A + ǫI ≻ 0

§

+n By Theorem 2.1.9.0.1, inverse image of the positive semidefinite cone SN + under affine mapping G(A , X , T ) is convex. Function g(A , X ) is convex n×N on SN because its epigraph is that inverse image: +×R © ª ¡ +n ¢ epi g(A , X ) = (A , X , T ) | A + ǫ I ≻ 0 , ǫ X(A + ǫ I )−1 X T ¹ T = G−1 SN + (598) 2

254

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.8.3

second-order condition, matrix function

The following line theorem is a potent tool for establishing convexity of a multidimensional function. To understand it, what is meant by line must first be solidified. Given a function g(X) : Rp×k → SM and particular X , Y ∈ Rp×k not necessarily in that function’s domain, then we say a line {X + t Y | t ∈ R} passes through dom g when X + t Y ∈ dom g over some interval of t ∈ R .

§

3.8.3.0.1 Theorem. Line theorem. [57, 3.1.1] p×k M Matrix-valued function g(X) : R → S is convex in X if and only if it remains convex on the intersection of any line with its domain. ⋄ Now we assume a twice differentiable function. 3.8.3.0.2 Definition. Differentiable convex matrix-valued function. Matrix-valued function g(X) : Rp×k → SM is convex in X iff dom g is an open convex set, and its second derivative g ′′(X + t Y ) : R → SM is positive semidefinite on each point of intersection along every line {X + t Y | t ∈ R} that intersects dom g ; id est, iff for each and every X , Y ∈ Rp×k such that X + t Y ∈ dom g over some open interval of t ∈ R

Similarly, if

d2 g(X + t Y ) º 0 dt2 M

(599)

d2 g(X + t Y ) ≻ 0 dt2 M

(600)

S+

§

S+

then g is strictly convex; the converse is generally false. [57, 3.1.4]3.20 △

§

3.8.3.0.3 Example. Matrix inverse. (confer 3.4.1) The matrix-valued function X µ is convex on int SM for −1 ≤ µ ≤ 0 + or 1 ≤ µ ≤ 2 and concave for 0 ≤ µ ≤ 1. [57, 3.6.2] In particular, the function g(X) = X −1 is convex on int SM For each and every Y ∈ SM + . ( D.2.1, A.3.1.0.5)

§

§

§

d2 g(X + t Y ) = 2(X + t Y )−1 Y (X + t Y )−1 Y (X + t Y )−1 º 0 2 dt M S+

3.20

The strict-case converse is reliably true for quadratic forms.

(601)

3.8. MATRIX-VALUED CONVEX FUNCTION

255

on some open interval of t ∈ R such that X + t Y ≻ 0. Hence, g(X) is convex in X . This result is extensible;3.21 tr X −1 is convex on that same 2 domain. [188, 7.6, prob.2] [51, 3.1, exer.25]

§

§

3.8.3.0.4 Example. Matrix squared. Iconic real function f (x) = x2 is strictly convex on R . The matrix-valued function g(X) = X 2 is convex on the domain of symmetric matrices; for X , Y ∈ SM and any open interval of t ∈ R ( D.2.1)

§

d2 d2 g(X + t Y ) = (X + t Y )2 = 2Y 2 (602) dt2 dt2 which is positive semidefinite when Y is symmetric because then Y 2 = Y T Y (1401).3.22 A more appropriate matrix-valued counterpart for f is g(X) = X TX which is a convex function on domain {X ∈ Rm×n } , and strictly convex whenever X is skinny-or-square full-rank. This matrix-valued function can be generalized to g(X) = X TAX which is convex whenever matrix A is positive semidefinite (p.664), and strictly convex when A is positive definite and X is skinny-or-square full-rank (Corollary A.3.1.0.5). 2

The matrix 2-norm (spectral norm) coincides with largest singular value: p (603) kXk2 , sup kXak2 = σ(X)1 = λ(X TX)1 = minimize t t∈R kak=1 ¸ · tI X º0 subject to X T tI This supremum of a family of convex functions in X must be convex because it constitutes an intersection of epigraphs of convex functions. 3.8.3.0.5 Exercise. Squared maps. Give seven examples of distinct polyhedra P for which the set n {X TX | X ∈ P} ⊆ S+

(604)

were convex. Is this set convex, in general, for any polyhedron P ? (confer (1182) (1189)) Is the epigraph of function g(X) = X TX convex for any polyhedral domain? H 3.21

d/dt tr g(X + t Y ) = tr d/dt g(X + t Y ). [189, p.491] By (1419) in §A.3.1, changing the domain instead to all symmetric and nonsymmetric positive semidefinite matrices, for example, will not produce a convex function. 3.22

256

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.8.3.0.6 Example. Matrix exponential. The matrix-valued function g(X) = eX : SM → SM is convex on the subspace of circulant [154] symmetric matrices. Applying the line theorem, for all t ∈ R and circulant X , Y ∈ SM , from Table D.2.7 we have d 2 X+ t Y e = Y eX+ t Y Y º 0 , dt2 M

(XY )T = XY

(605)

S+

because all circulant matrices are commutative and, for symmetric matrices, XY = Y X ⇔ (XY )T = XY (1418). Given symmetric argument, the matrix exponential always resides interior to the cone of positive semidefinite matrices in the symmetric matrix subspace; eA ≻ 0 ∀ A ∈ SM (1791). Then for any matrix Y of compatible dimension, Y TeA Y is positive semidefinite. ( A.3.1.0.5) The subspace of circulant symmetric matrices contains all diagonal matrices. The matrix exponential of any diagonal matrix eΛ exponentiates each individual entry on the main diagonal. [233, 5.3] So, changing the function domain to the subspace of real diagonal matrices reduces the matrix exponential to a vector-valued function in an isometrically isomorphic subspace RM ; known convex ( 3.1.1) from the real-valued function case [57, 3.1.5]. 2

§

§

§

§

There are, of course, multifarious methods to determine function convexity, [57] [39] [122] each of them efficient when appropriate. 3.8.3.0.7 Exercise. log det function. Matrix determinant is neither a convex or concave function, in general, but its inverse is convex when domain is restricted to interior of a positive semidefinite cone. [32, p.149] Show by three different methods: On interior of the positive semidefinite cone, log det X = − log det X −1 is concave. H

3.9

Quasiconvex

§

§

Quasiconvex functions [57, 3.4] [185] [308] [346] [227, 2] are useful in practical problem solving because they are unimodal (by definition when nonmonotonic); a global minimum is guaranteed to exist over any convex set in the function domain; e.g., Figure 76.

3.9. QUASICONVEX

257

Figure 76: Iconic unimodal differentiable quasiconvex function of two variables graphed in R2 × R on some open disc in R2 . Note reversal of curvature in direction of gradient.

3.9.0.0.1 Definition. Quasiconvex function. f (X) : Rp×k → R is a quasiconvex function of matrix X iff dom f is a convex set and for each and every Y, Z ∈ dom f , 0 ≤ µ ≤ 1 f (µ Y + (1 − µ)Z) ≤ max{f (Y ) , f (Z )}

(606)

A quasiconcave function is determined: f (µ Y + (1 − µ)Z) ≥ min{f (Y ) , f (Z )}

(607) △

§

Unlike convex functions, quasiconvex functions are not necessarily continuous; e.g., quasiconcave rank(X) on SM + ( 2.9.2.6.2) and card(x) on RM . Although insufficient for convex functions, convexity of each and + every sublevel set serves as a definition of quasiconvexity:

258

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.9.0.0.2 Definition. Quasiconvex multidimensional function. Scalar-, vector-, or matrix-valued function g(X) : Rp×k → SM is a quasiconvex function of matrix X iff dom g is a convex set and the sublevel set corresponding to each and every S ∈ SM LS g = {X ∈ dom g | g(X) ¹ S } ⊆ Rp×k

(595)

is convex. Vectors are compared with respect to the nonnegative orthant RM + while matrices are with respect to the positive semidefinite cone SM . + Convexity of the superlevel set corresponding to each and every S ∈ SM , likewise LSg = {X ∈ dom g | g(X) º S } ⊆ Rp×k (608) is necessary and sufficient for quasiconcavity.

3.9.0.0.3 Exercise. Nonconvexity of matrix product. Consider real function f on a positive definite domain # " · ¸ rel int SN X1 + f (X) = tr(X1 X2 ) , X, ∈ dom f , X2 rel int SN +

△

(609)

with superlevel sets Ls f = {X ∈ dom f | hX1 , X2 i ≥ s }

(610)

Prove: f (X) is not quasiconcave except when N = 1, nor is it quasiconvex unless X1 = X2 . H

3.9.1

bilinear

Bilinear function3.23 xTy of vectors x and y is quasiconcave (monotonic) on the entirety of the nonnegative orthants only when vectors are of dimension 1.

3.9.2

quasilinear

When a function is simultaneously quasiconvex and quasiconcave, it is called quasilinear. Quasilinear functions are completely determined by convex level sets. One-dimensional function f (x) = x3 and vector-valued signum function sgn(x) , for example, are quasilinear. Any monotonic function is quasilinear 3.24 (but not vice versa, Exercise 3.7.4.0.1). 3.23 3.24

Convex envelope of bilinear functions is well known. [3] e.g., a monotonic concave function is quasiconvex, but dare not confuse these terms.

3.10. SALIENT PROPERTIES

3.10 1.

259

Salient properties of convex and quasiconvex functions

ˆ A convex function is assumed continuous but not necessarily differentiable on the relative interior of its domain. [285, §10] ˆ A quasiconvex function is not necessarily a continuous function.

2. convex epigraph ⇔ convexity ⇒ quasiconvexity ⇔ convex sublevel sets. convex hypograph ⇔ concavity ⇒ quasiconcavity ⇔ convex superlevel. monotonicity ⇒ quasilinearity ⇔ convex level sets. 3. 4.

log-convex ⇒ convex ⇒ quasiconvex. 3.25 concave ⇒ quasiconcave ⇐ log-concave ⇐ positive concave.

ˆ (homogeneity) Convexity, concavity, quasiconvexity, and quasiconcavity are invariant to nonnegative scaling of function. ˆ g convex ⇔ −g concave g quasiconvex ⇔ −g quasiconcave g log-convex ⇔ 1/g log-concave

§

§

5. The line theorem ( 3.8.3.0.1) translates identically to quasiconvexity (quasiconcavity). [57, 3.4.2] 6. (affine transformation of argument) Composition g(h(X)) of convex (concave) function g with any affine function h : Rm×n → Rp×k remains convex (concave) in X ∈ Rm×n , where h(Rm×n ) ∩ dom g 6= ∅ . [185, B.2.1] Likewise for the quasiconvex (quasiconcave) functions g . 7.

3.25

ˆ

§

§

– Nonnegatively weighted sum of convex (concave) functions remains convex (concave). ( 3.1.2.1.1) – Nonnegatively weighted nonzero sum of strictly convex (concave) functions remains strictly convex (concave). – Nonnegatively weighted maximum (minimum) of convex3.26 (concave) functions remains convex (concave).

Log-convex means: logarithm of function f is convex on dom f . Supremum and maximum of convex functions are proven convex by intersection of epigraphs. 3.26

260

CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

ˆ

§

– Pointwise supremum (infimum) of convex (concave) functions remains convex (concave). (Figure 71) [285, 5] – Sum of quasiconvex functions not necessarily quasiconvex. – Nonnegatively weighted maximum (minimum) of quasiconvex (quasiconcave) functions remains quasiconvex (quasiconcave). – Pointwise supremum (infimum) of quasiconvex (quasiconcave) functions remains quasiconvex (quasiconcave).

Chapter 4 Semidefinite programming Prior to 1984, linear and nonlinear programming,4.1 one a subset of the other, had evolved for the most part along unconnected paths, without even a common terminology. (The use of “programming” to mean “optimization” serves as a persistent reminder of these differences.) −Forsgren, Gill, & Wright (2002) [133] Given some application of convex analysis, it may at first seem puzzling why the search for its solution ends abruptly with a formalized statement of the problem itself as a constrained optimization. The explanation is: typically we do not seek analytical solution because there are relatively few. ( C, 3.6.2) If a problem can be expressed in convex form, rather, then there exist computer programs providing efficient numerical global solution. [153] [359] [360] [358] [366] The goal, then, becomes conversion of a given problem (perhaps a nonconvex or combinatorial problem statement) to an equivalent convex form or to an alternation of convex subproblems convergent to a solution of the original problem: By the fundamental theorem of Convex Optimization, any locally optimal point of a convex problem is globally optimal. [57, 4.2.2] [284, 1] Given convex real objective function g and convex feasible set C ⊆ dom g , which is the set of all variable values satisfying the problem constraints, we pose a

§

§

§

§

4.1

nascence of interior-point methods of solution [338] [356]. Linear programming ⊂ (convex ∩ nonlinear) programming. © 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

261

262

CHAPTER 4. SEMIDEFINITE PROGRAMMING

generic convex optimization problem minimize X

g(X)

(611)

subject to X ∈ C

where constraints are abstract here in membership of variable X to convex feasible set C . Inequality constraint functions of a convex optimization problem are convex while equality constraint functions are conventionally affine, but not necessarily so. Affine equality constraint functions, as opposed to the superset of all convex equality constraint functions having convex level sets ( 3.5.0.0.4), make convex optimization tractable. Similarly, the problem

§

maximize X

g(X)

(612)

subject to X ∈ C

is called convex were g a real concave function and feasible set C convex. As conversion to convex form is not always possible, there is much ongoing research to determine which problem classes have convex expression or relaxation. [32] [55] [138] [257] [322] [136]

4.1

Conic problem Still, we are surprised to see the relatively small number of submissions to semidefinite programming (SDP) solvers, as this is an area of significant current interest to the optimization community. We speculate that semidefinite programming is simply experiencing the fate of most new areas: Users have yet to understand how to pose their problems as semidefinite programs, and the lack of support for SDP solvers in popular modelling languages likely discourages submissions. −SIAM News, 2002. [105, p.9]

§

§

(confer p.153) Consider a conic problem (p) and its dual (d): [272, 3.3.1] [222, 2.1] [223] minimize (p)

x

c Tx

subject to x ∈ K Ax = b

maximize y,s

b Ty ∗

subject to s ∈ K A Ty + s = c

(d)

(276)

4.1. CONIC PROBLEM

263 ∗

where K is a closed convex cone, K is its dual, matrix A is fixed, and the remaining quantities are vectors. When K is a polyhedral cone ( 2.12.1), then each conic problem becomes a linear program [89]; the self-dual nonnegative orthant providing the prototypical primal linear program and its dual. More generally, each optimization problem is convex when K is a closed convex cone. Unlike the optimal objective value, a solution to each problem is not necessarily unique; in other words, the optimal solution set {x⋆ } or {y ⋆ , s⋆ } is convex and may comprise more than a single point although the corresponding optimal objective value is unique when the feasible set is nonempty.

§

4.1.1

a Semidefinite program

When K is the self-dual cone of positive semidefinite matrices in the subspace of symmetric matrices, then each conic problem is called a semidefinite program (SDP); [257, 6.4] primal problem (P) having matrix variable X ∈ S n while corresponding dual (D) has matrix slack variable S ∈ S n and vector variable y = [yi ] ∈ Rm : [10] [11, 2] [366, 1.3.8]

§

minimize n X∈ S

(P)

§

hC , X i

subject to X º 0 A svec X = b

§

maximize m n

y∈R , S∈ S

hb , yi

subject to S º 0 svec−1 (AT y) + S = C

(D) (613)

This is the prototypical semidefinite program and its dual, where matrix C ∈ S n and vector b ∈ Rm are fixed, as is  svec(A1 )T ..  ∈ Rm×n(n+1)/2 A , . svec(Am )T 

(614)

where Ai ∈ S n , i = 1 . . . m , are given. Thus

 hA1 , X i ..  A svec X =  . hAm , X i m P svec−1 (AT y) = yi Ai 

i=1

(615)

264

CHAPTER 4. SEMIDEFINITE PROGRAMMING

The vector inner-product for matrices is defined in the Euclidean/Frobenius sense in the isomorphic vector space Rn(n+1)/2 ; id est, hC , X i , tr(C TX) = svec(C )T svec X

(33)

where svec X defined by (51) denotes symmetric vectorization. Linear programming, invented by Dantzig in 1947, is now integral to modern technology. But the same cannot yet be said of semidefinite programming, whose roots can be traced back to Bellman & Fan in 1963. [93] Numerical performance of contemporary general-purpose semidefinite program solvers is limited: Computational intensity varies as O(m2 n) (m < n) based on interior-point methods (Frisch 1955, Fiacco & McCormick 1968) that produce solutions no more relatively accurate than 1E-8. There are no solvers available capable of handling in excess of n =10,000 variables without significant, sometimes crippling, loss of precision or time.4.2 [33] [256, p.258] [63, p.3] Nevertheless, semidefinite programming has emerged recently to prominence primarily because it admits a new class of problem previously unsolvable by convex optimization techniques, [55] secondarily because it theoretically subsumes other convex techniques: (Figure 77) linear programming and quadratic programming and second-order cone programming.4.3 Determination of the Riemann mapping function from complex analysis [266] [29, 8, 13], for example, can be posed as a semidefinite program.

§ §

4.1.1.1

Maximal complementarity

§

It has been shown that contemporary interior-point methods (developed circa 1990 [138]), [357] [269] [257] [11] [57, 11] [133] for numerical solution of semidefinite programs, can converge to a solution of maximal complementarity; [162, 5] [365] [235] [145] not a vertex-solution but a solution of highest cardinality or rank among all optimal solutions.4.4

§

4.2

Heuristics are not ruled out by SIOPT; indeed I would suspect that most successful methods have (appropriately described) heuristics under the hood - my codes certainly do. . . . Of course, there are still questions relating to high-accuracy and speed, but for many applications a few digits of accuracy suffices and overnight runs for non-real-time delivery is acceptable. −Nicholas I. M. Gould, Editor-in-Chief, SIOPT 4.3 SOCP came into being in the 1990s; it is not posable as a quadratic program. [231] 4.4 This characteristic might be regarded as a disadvantage to this method of numerical solution, but this behavior is not certain and depends on solver implementation.

4.1. CONIC PROBLEM

265 semidefinite program

second-order cone program

quadratic program linear program

Figure 77: Venn diagram of programming hierarchy. Semidefinite program subsumes other program classes excepting geometric program. Second-order cone program and quadratic program each subsume linear program.

§

[366, 2.5.3] 4.1.1.2

Reduced-rank solution

A simple rank reduction algorithm, for construction of a primal optimal solution X ⋆ to (613P) satisfying an upper bound on rank governed by Proposition 2.9.3.0.1, is presented in 4.3. That proposition asserts existence of feasible solutions with an upper bound on their rank; [26, II.13.1] specifically, it asserts an extreme point ( 2.6.0.0.1) of the primal feasible n set A ∩ S+ satisfies upper bound

§

rank X ≤

¹√

§

§

8m + 1 − 1 2

º

(247)

where, given A ∈ Rm×n(n+1)/2 and b ∈ Rm A , {X ∈ S n | A svec X = b} is the affine subset from primal problem (613P).

(616)

266

CHAPTER 4. SEMIDEFINITE PROGRAMMING

C

0

Γ1 P Γ2

S+3

A = ∂H Figure 78: Visualizing positive semidefinite cone in high dimension: Proper polyhedral cone S+3 ⊂ R3 representing positive semidefinite cone S3+ ⊂ S3 ; analogizing its intersection with hyperplane S3+ ∩ ∂H . Number of facets is arbitrary (analogy is not inspired by eigen decomposition). The rank-0 positive semidefinite matrix corresponds to the origin in R3 , rank-1 positive semidefinite matrices correspond to the edges of the polyhedral cone, rank-2 to the facet relative interiors, and rank-3 to the polyhedral cone interior. Vertices Γ1 and Γ2 are extreme points of polyhedron P = ∂H ∩ S+3 , and extreme directions of S+3 . A given vector C is normal to another hyperplane (not illustrated but independent w.r.t ∂H) containing line segment Γ1 Γ2 minimizing real linear function hC , X i on P . (confer Figure 26, Figure 30)

4.1. CONIC PROBLEM 4.1.1.3

267

Coexistence of low- and high-rank solutions; analogy

That low-rank and high-rank optimal solutions {X ⋆ } of (613P) coexist may be grasped with the following analogy: We compare a proper polyhedral cone S+3 in R3 (illustrated in Figure 78) to the positive semidefinite cone S3+ in isometrically isomorphic R6 , difficult to visualize. The analogy is good:

ˆ int S

3 + int S+3

is constituted by rank-3 matrices has three dimensions

ˆ boundary ∂ S boundary

3 + ∂S+3

contains rank-0, rank-1, and rank-2 matrices contains 0-, 1-, and 2-dimensional faces

ˆ the only rank-0 matrix resides in the vertex at the origin ˆ Rank-1 matrices are in one-to-one correspondence with S3+

directions of and this dimension

S+3

is not a connected set.

extreme . The set of all rank-1 symmetric matrices in

©

ª G ∈ S3+ | rank G = 1

(617)

ˆ In any SDP feasibility problem, an SDP feasible solution with the lowest

rank must be an extreme point of the feasible set. Thus, there must exist an SDP objective function such that this lowest-rank feasible solution uniquely optimizes it. −Ye, 2006 [222, 2.4] [223]

§

ˆ Rank of a sum of members F + G in Lemma 2.9.2.6.1 and location of a difference F − G in §2.9.2.9.1 similarly hold for S and S . ˆ Euclidean distance from any particular rank-3 positive semidefinite 3 +

3 +

matrix (in the cone interior) to the closest rank-2 positive semidefinite matrix (on the boundary) is generally less than the distance to the closest rank-1 positive semidefinite matrix. ( 7.1.2)

ˆ distance from any point in ∂ S distance from any point in

3 + ∂S+3

§

§

to int S3+ is infinitesimal ( 2.1.7.1.1) to int S+3 is infinitesimal

268

CHAPTER 4. SEMIDEFINITE PROGRAMMING

ˆ faces of S

3 +

correspond to faces of S+3 (confer Table 2.9.2.3.1)

k 0 boundary 1 2 interior 3

dim F(S+3 ) dim F(S3+ ) dim F(S3+ ∋ rank-k matrix) 0 0 0 1 1 1 2 3 3 3 6 6

Integer k indexes k-dimensional faces F of S+3 . Positive semidefinite cone S3+ has four kinds of faces, including cone itself (k = 3, boundary + interior), whose dimensions in isometrically isomorphic R6 are listed under dim F(S3+ ). Smallest face F(S3+ ∋ rank-k matrix) that contains a rank-k positive semidefinite matrix has dimension k(k + 1)/2 by (207).

ˆ For A equal to intersection of

m hyperplanes having independent normals, and for X ∈ S+3 ∩ A , we have rank X ≤ m ; the analogue to (247).

Proof. With reference to Figure 78: Assume one (m = 1) hyperplane A = ∂H intersects the polyhedral cone. Every intersecting plane contains at least one matrix having rank less than or equal to 1 ; id est, from all X ∈ ∂H ∩ S+3 there exists an X such that rank X ≤ 1. Rank 1 is therefore an upper bound in this case. Now visualize intersection of the polyhedral cone with two (m = 2) hyperplanes having linearly independent normals. The hyperplane intersection A makes a line. Every intersecting line contains at least one matrix having rank less than or equal to 2, providing an upper bound. In other words, there exists a positive semidefinite matrix X belonging to any line intersecting the polyhedral cone such that rank X ≤ 2. In the case of three independent intersecting hyperplanes (m = 3), the hyperplane intersection A makes a point that can reside anywhere in the polyhedral cone. The upper bound on a point in S+3 is also the greatest upper bound: rank X ≤ 3. ¨

4.1. CONIC PROBLEM

269

4.1.1.3.1 Example. Optimization on A ∩ S+3 . Consider minimization of the real linear function hC , X i on P , A ∩ S+3

(618)

a polyhedral feasible set; f0⋆ , minimize hC , X i

(619)

X

subject to X ∈ A ∩ S+3

As illustrated for particular vector C and hyperplane A = ∂H in Figure 78, this linear function is minimized on any X belonging to the face of P containing extreme points {Γ1 , Γ2 } and all the rank-2 matrices in between; id est, on any X belonging to the face of P F(P) = {X | hC , X i = f0⋆ } ∩ A ∩ S+3

(620)

exposed by the hyperplane {X | hC , X i = f0⋆ }. In other words, the set of all optimal points X ⋆ is a face of P {X ⋆ } = F(P) = Γ1 Γ2

(621)

comprising rank-1 and rank-2 positive semidefinite matrices. Rank 1 is the upper bound on existence in the feasible set P for this case m = 1 hyperplane constituting A . The rank-1 matrices Γ1 and Γ2 in face F(P) are extreme points of that face and (by transitivity ( 2.6.1.2)) extreme points of the intersection P as well. As predicted by analogy to Barvinok’s Proposition 2.9.3.0.1, the upper bound on rank of X existent in the feasible set P is satisfied by an extreme point. The upper bound on rank of an optimal solution X ⋆ existent in F(P) is thereby also satisfied by an extreme point of P precisely because {X ⋆ } constitutes F(P) ;4.5 in particular,

§

{X ⋆ ∈ P | rank X ⋆ ≤ 1} = {Γ1 , Γ2 } ⊆ F(P)

(622)

As all linear functions on a polyhedron are minimized on a face, [89] [234] [255] [259] by analogy we so demonstrate coexistence of optimal solutions X ⋆ 2 of (613P) having assorted rank. 4.5

and every face contains a subset of the extreme points of P by the extreme existence theorem (§2.6.0.0.2). This means: because the affine subset A and hyperplane {X | hC , X i = f0⋆ } must intersect a whole face of P , calculation of an upper bound on rank of X ⋆ ignores counting the hyperplane when determining m in (247).

270

CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.1.1.4

Previous work

§

Barvinok showed [24, 2.2] when given a positive definite matrix C and an arbitrarily small neighborhood of C comprising positive definite matrices, there exists a matrix C˜ from that neighborhood such that optimal solution n X ⋆ to (613P) (substituting C˜ ) is an extreme point of A ∩ S+ and satisfies 4.6 upper bound (247). Given arbitrary positive definite C , this means nothing inherently guarantees that an optimal solution X ⋆ to problem (613P) satisfies (247); certainly nothing given any symmetric matrix C , as the problem is posed. This can be proved by example: 4.1.1.4.1 Example. (Ye) Maximal Complementarity. Assume dimension n to be an even positive number. Then the particular instance of problem (613P), ¸ À ¿· I 0 , X minimize n 0 2I X∈ S (623) subject to X º 0 hI , X i = n has optimal solution ⋆

X =

·

2I 0 0 0

¸

∈ Sn

(624)

with an equal number of twos and zeros along the main diagonal. Indeed, optimal solution (624) is a terminal solution along the central path taken by the interior-point method as implemented in [366, 2.5.3]; it is also a solution of highest rank among all optimal solutions to (623). Clearly, rank of this primal optimal solution exceeds by far a rank-1 solution predicted by upper bound (247). 2

§

4.1.1.5

Later developments

This rational example (623) indicates the need for a more generally applicable and simple algorithm to identify an optimal solution X ⋆ satisfying Barvinok’s Proposition 2.9.3.0.1. We will review such an algorithm in 4.3, but first we provide more background.

§

4.6

Further, the set of all such C˜ in that neighborhood is open and dense.

4.2. FRAMEWORK

4.2

Framework

4.2.1

Feasible sets

271

Denote by C and C ∗ the convex sets of primal and dual points respectively satisfying the primal and dual constraints in (613), each assumed nonempty;     hA1 , X i   .. n n  = A ∩ S+ = b | C = X ∈ S+ .   hAm , X i (625) ½ ¾ m P n C ∗ = S ∈ S+ , y = [yi ] ∈ Rm | yi Ai + S = C i=1

These are the primal feasible set and dual feasible set. Geometrically, n primal feasible A ∩ S+ represents an intersection of the positive semidefinite n cone S+ with an affine subset A of the subspace of symmetric matrices S n in isometrically isomorphic Rn(n+1)/2 . The affine subset has dimension n(n + 1)/2 − m when the Ai are linearly independent. Dual feasible set C ∗ is the Cartesian product of the positive semidefinitePcone with its inverse image ( 2.1.9.0.1) under affine transformation4.7 C − yi Ai . Both feasible sets are convex, and the objective functions are linear on a Euclidean vector space. Hence, (613P) and (613D) are convex optimization problems.

§

4.2.1.1

n A ∩ S+ emptiness determination via Farkas’ lemma

4.2.1.1.1 Lemma. Semidefinite Farkas’ lemma. Given set {Ai ∈ S n , i = 1 . . . m} , vector b = [b i ] ∈ Rm , and affine subset

(616) A = {X ∈ S n |hAi , X i = b i , i = 1 . . . m} Ä {A svec X | X º 0} (352) is closed, n then primal feasible set A ∩ S+ is nonempty if and only if y T b ≥ 0 holds for m P each and every vector y = [yi ] ∈ Rm such that yi Ai º 0. i=1

n Equivalently, primal feasible set A ∩ S+ is nonempty if and only if m P y T b ≥ 0 holds for each and every vector kyk = 1 such that yi Ai º 0. ⋄ i=1

4.7

Inequality C − yi Ai º 0 follows directly from P (613D) (§2.9.0.1.1) and is known as a linear matrix inequality. (§2.13.5.1.1) Because yi Ai ¹ C , matrix S is known as a slack variable (a term borrowed from linear programming [89]) since its inclusion raises this inequality to equality. P

272

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Semidefinite Farkas’ lemma provides necessary and sufficient conditions n for a set of hyperplanes to have nonempty intersection A ∩ S+ with the positive semidefinite cone. Given   svec(A1 )T ..  ∈ Rm×n(n+1)/2 A = (614) . T svec(Am ) semidefinite Farkas’ lemma assumes that a convex cone K = {A svec X | X º 0}

(352)

§

is closed per membership relation (293) from which the lemma springs: [215, I] K closure is attained when matrix A satisfies Corollary 2.13.5.1.3. Given closed convex cone K and its dual from Example 2.13.5.1.1 ∗

K = {y |

m X j=1

yj Aj º 0}

(359)

then we can apply membership relation b ∈ K ⇔ hy , bi ≥ 0 ∀ y ∈ K

∗

(293)

to obtain the lemma b∈K

b∈K

⇔

⇔

∃ X º 0 Ä A svec X = b hy , bi ≥ 0 ∀ y ∈ K

∗

⇔

⇔

n A ∩ S+ 6= ∅

A∩

n S+

6= ∅

(626) (627)

The final equivalence synopsizes semidefinite Farkas’ lemma. While the lemma is correct as stated, a positive definite version is required for semidefinite programming [366, 1.3.8] because existence of a feasible n point in the relative interior A ∩ int S+ is required by Slater’s condition 4.8 to achieve 0 duality gap (primal−dual objective difference 4.2.3, Figure 56). Geometrically, a positive definite lemma is required to insure that a point of intersection closest to the origin is not at infinity; e.g., Figure 43. Then given

§

4.8

§

Slater’s sufficient condition is satisfied whenever any primal strictly feasible point exists; id est, any point feasible with the affine equality (or affine inequality) constraint functions and relatively interior to convex cone K . [308, §6.6] [38, p.325] If cone K were polyhedral, then Slater’s condition is satisfied when any feasible point exists relatively interior to K or on its relative boundary. [57, §5.2.3]

4.2. FRAMEWORK

273

A ∈ Rm×n(n+1)/2 having rank m , we wish to detect existence of a nonempty primal feasible set’s relative interior;4.9 by (14) and (355) b ∈ int K

∗

⇔

hy , bi > 0 ∀ y ∈ K , y 6= 0

⇔

n A ∩ int S+ 6= ∅

(628) ∗

Positive definite Farkas’ lemma is made from proper cones, K (352) and K (359), and membership relation (299) for which K closedness is unnecessary: 4.2.1.1.2 Lemma. Positive definite Farkas’ lemma. Given l.i. set {Ai ∈ S n , i = 1 . . . m} and vector b = [b i ] ∈ Rm , make affine set A = {X ∈ S n |hAi , X i = b i , i = 1 . . . m}

(616)

n Primal feasible set relative interior A ∩ int S+ is nonempty if and only if m P y T b > 0 holds for each and every vector y = [yi ] 6= 0 such that yi Ai º 0. i=1

n Equivalently, primal feasible set relative interior A ∩ int S+ is nonempty m P if and only if y T b > 0 holds for each and every vector kyk = 1 Ä yi Ai º 0. i=1 ⋄

§

4.2.1.1.3 Example. “New” Farkas’ lemma. Lasserre [215, III] presented an example in 1995, originally offered by Ben-Israel in 1969 [31, p.378], to support closedness in semidefinite Farkas’ Lemma 4.2.1.1.1: ¸ · ¸ · ¸ · svec(A1 )T 0 1 0 1 = , b= (629) A , T 0 0 1 0 svec(A2 ) n Intersection A ∩ S+ is practically empty because the solution set (" # ) α √12 {X º 0 | A svec X = b} = º 0 | α∈ R √1 0 2

(630)

is positive semidefinite only asymptotically (α → ∞). Yet the dual system m P yi Ai º 0 ⇒ y T b ≥ 0 erroneously indicates nonempty intersection because

i=1

K (352) violates a closedness condition of the lemma; videlicet, for kyk = 1 # " · ¸ · ¸ 0 √12 0 0 0 + y2 º 0 ⇔ y= ⇒ yTb = 0 (631) y1 √1 0 0 1 1 2 4.9

n Detection of A ∩ int S+ 6= ∅ by examining int K instead is a trick need not be lost.

274

CHAPTER 4. SEMIDEFINITE PROGRAMMING

On the other hand, positive definite Farkas’ Lemma 4.2.1.1.2 certifies that n A ∩ int S+ is empty; what we need to know for semidefinite programming. Lasserre suggested addition of another condition to semidefinite Farkas’ lemma ( 4.2.1.1.1) to make a new lemma having no closedness condition. But positive definite Farkas’ lemma ( 4.2.1.1.2) is simpler and obviates the additional condition proposed. 2

§

4.2.1.2

§

Theorem of the alternative for semidefinite programming

§

Because these Farkas’ lemmas follow from membership relations, we may construct alternative systems from them. Applying the method of 2.13.2.1.1, then from positive definite Farkas’ lemma we get n A ∩ int S+ 6= ∅

or in the alternative m P yTb ≤ 0 , yi Ai º 0 , y 6= 0

(632)

i=1

n Any single vector y satisfying the alternative certifies A ∩ int S+ is empty. Such a vector can be found as a solution to another semidefinite program: for linearly independent set {Ai ∈ S n , i = 1 . . . m}

yTb m P subject to yi Ai º 0 minimize y

(633)

i=1

kyk2 ≤ 1

If an optimal vector y ⋆ 6= 0 can be found such that y ⋆T b ≤ 0, then relative n interior of the primal feasible set A ∩ int S+ from (625) is empty. 4.2.1.3

Boundary-membership criterion

(confer (627) (628)) From boundary-membership relation (303) for linear ∗ matrix inequality proper cones K (352) and K (359) b ∈ ∂K

⇔

∗

∃ y 6= 0 Ä hy , bi = 0 , y ∈ K , b ∈ K

⇔

n n ∂ S+ ⊃ A ∩ S+ 6= ∅ (634)

4.2. FRAMEWORK

275

Whether vector b ∈ ∂K belongs to cone K boundary, that is a determination we can indeed make; one that is certainly expressible as a feasibility problem: Given linearly independent set4.10 {Ai ∈ S n , i = 1 . . . m} , for b ∈ K (626) find y 6= 0

subject to y T b = 0 m P yi Ai º 0

(635)

i=1

Any such feasible nonzero vector y certifies that affine subset A (616) n intersects the positive semidefinite cone S+ only on its boundary; in other n words, nonempty feasible set A ∩ S+ belongs to the positive semidefinite n . cone boundary ∂ S+

4.2.2

Duals

The dual objective function from (613D) evaluated at any feasible point represents a lower bound on the primal optimal objective value from (613P). n We can see this by direct substitution: Assume the feasible sets A ∩ S+ and ∗ C are nonempty. Then it is always true: ¿

hC , X i ≥ hb , yi À P yi Ai + S , X ≥ [ hA1 , X i · · · hAm , X i ] y

(636)

i

hS , X i ≥ 0

The converse also follows because X º 0 , S º 0 ⇒ hS , X i ≥ 0

(1426)

Optimal value of the dual objective thus represents the greatest lower bound on the primal. This fact is known as the weak duality theorem for semidefinite programming, [366, 1.3.8] and can be used to detect convergence in any primal/dual numerical method of solution.

§

4.10

From the results of Example 2.13.5.1.1, vector b on the boundary of K cannot be detected simply by looking for 0 eigenvalues in matrix X . We do not consider a n skinny-or-square matrix A because then feasible set A ∩ S+ is at most a single point.

276

CHAPTER 4. SEMIDEFINITE PROGRAMMING ˜ P

P duality

duality

D

˜ D

transformation

Figure 79: Connectivity indicates paths between particular primal and dual problems from Exercise 4.2.2.1.1. More generally, any path between primal ˜ and dual D (and equivalent D) ˜ is possible: problems P (and equivalent P) implying, any given path is not necessarily circuital; dual of a dual problem is not necessarily stated in precisely same manner as corresponding primal convex problem, in other words, although its solution set is equivalent to within some transformation.

4.2.2.1

Dual problem statement is not unique

Even subtle but equivalent restatements of a primal convex problem can lead to vastly different statements of a corresponding dual problem. This phenomenon is of interest because a particular instantiation of dual problem might be easier to solve numerically or it might take one of few forms for which analytical solution is known. Here is a canonical restatement of the prototypical dual semidefinite program (613D), equivalent by (179) for example: maximize

y∈Rm, S∈ Sn

(D)

hb , yi

≡ subject to S º 0 −1 T svec (A y) + S = C

maximize m y∈R

hb , yi

subject to svec−1 (AT y) ¹ C

˜ (613D)

4.2.2.1.1 Exercise. Primal prototypical semidefinite program. ˜ id est, Derive prototypical primal (613P) from its canonical dual (613D); H demonstrate that particular connectivity in Figure 79.

4.2. FRAMEWORK

4.2.3

277

Optimality conditions

n When any primal feasible point exists relatively interior to A ∩ S+ in S n , or when any dual feasible point exists relatively interior to C ∗ in S n × Rm , then by Slater’s sufficient condition (p.272) these two problems (613P) (613D) become strong duals. In other words, the primal optimal objective value becomes equivalent to the dual optimal objective value: there is no duality n or ∃ S, y ∈ rel int C ∗ then gap (Figure 56); id est, if ∃ X ∈ A ∩ int S+

¿ P i

hC , X ⋆ i = hb , y ⋆ i À ⋆ ⋆ ⋆ yi Ai + S , X = [ hA1 , X ⋆ i · · · hAm , X ⋆ i ] y ⋆

(637)

hS ⋆ , X ⋆ i = 0

where S ⋆ , y ⋆ denote a dual optimal solution.4.11 We summarize this:

§

§

4.2.3.0.1 Corollary. Optimality and strong duality. [334, 3.1] [366, 1.3.8] For semidefinite programs (613P) and (613D), assume primal n and dual feasible sets A ∩ S+ ⊂ S n and C ∗ ⊂ S n × Rm (625) are nonempty. Then

ˆ X is optimal for (P) ˆ S , y are optimal for (D) ˆ duality gap hC , X i−hb , y i is 0 ⋆

⋆

⋆

⋆

⋆

if and only if n i) ∃ X ∈ A ∩ int S+

or

∃ S , y ∈ rel int C ∗

and ii) hS ⋆ , X ⋆ i = 0

⋄

Optimality condition hS ⋆ , X ⋆ i = 0 is called a complementary slackness condition, in keeping with linear programming tradition [89], that forbids dual inequalities in (613) to simultaneously hold strictly. [284, §4] 4.11

278

CHAPTER 4. SEMIDEFINITE PROGRAMMING

§

For symmetric positive semidefinite matrices, requirement ii is equivalent to the complementarity ( A.7.4)

§

hS ⋆ , X ⋆ i = 0 ⇔ S ⋆ X ⋆ = X ⋆ S ⋆ = 0

(638)

Commutativity of diagonalizable matrices is a necessary and sufficient condition [188, 1.3.12] for these two optimal symmetric matrices to be simultaneously diagonalizable. Therefore rank X ⋆ + rank S ⋆ ≤ n

(639)

Proof. To see that, the product of symmetric optimal matrices X ⋆ , S ⋆ ∈ S n must itself be symmetric because of commutativity. (1418) The symmetric product has diagonalization [11, cor.2.11] S ⋆ X ⋆ = X ⋆ S ⋆ = QΛS ⋆ ΛX ⋆ QT = 0 ⇔ ΛX ⋆ ΛS ⋆ = 0

(640)

where Q is an orthogonal matrix. The product of the nonnegative diagonal Λ matrices can be 0 if their main diagonal zeros are complementary or coincide. Due only to symmetry, rank X ⋆ = rank ΛX ⋆ and rank S ⋆ = rank ΛS ⋆ for these optimal primal and dual solutions. (1404) So, because of the complementarity, the total number of nonzero diagonal entries from both Λ cannot exceed n . ¨ When equality is attained in (639) rank X ⋆ + rank S ⋆ = n

(641)

there are no coinciding main diagonal zeros in ΛX ⋆ ΛS ⋆ , and so we have what is called strict complementarity.4.12 Logically it follows that a necessary and sufficient condition for strict complementarity of an optimal primal and dual solution is X ⋆ + S⋆ ≻ 0 (642) 4.2.3.1

solving primal problem via dual

The beauty of Corollary 4.2.3.0.1 is its conjugacy; id est, one can solve either the primal or dual problem in (613) and then find a solution to the other via the optimality conditions. When a dual optimal solution is known, for example, a primal optimal solution belongs to the hyperplane {X | hS ⋆ , X i = 0}. 4.12

distinct from maximal complementarity (§4.1.1.1).

4.2. FRAMEWORK

279

§

4.2.3.1.1 Example. Minimum cardinality Boolean. [88] [32, 4.3.4] [322] (confer Example 4.5.1.2.1) Consider finding a minimum cardinality Boolean solution x to the classic linear algebra problem Ax = b given noiseless data A ∈ Rm×n and b ∈ Rm ; minimize x

kxk0

subject to Ax = b xi ∈ {0, 1} ,

(643) i=1 . . . n

where kxk0 denotes cardinality of vector x (a.k.a, 0-norm; not a convex function). A minimum cardinality solution answers the question: “Which fewest linear combination of columns in A constructs vector b ?” Cardinality problems have extraordinarily wide appeal, arising in many fields of science and across many disciplines. [296] [199] [156] [157] Yet designing an efficient algorithm to optimize cardinality has proved difficult. In this example, we also constrain the variable to be Boolean. The Boolean constraint forces an identical solution were the norm in problem (643) instead the 1-norm or 2-norm; id est, the two problems minimize (643)

x

kxk0

subject to Ax = b xi ∈ {0, 1} ,

minimize = i=1 . . . n

x

kxk1

(644) subject to Ax = b xi ∈ {0, 1} , i = 1 . . . n

are the same. The Boolean constraint makes the 1-norm problem nonconvex. Given data     −1 1 8 1 1 0 1   1  1 1 1  1 A =  −3 2 8 2 3 2 − 3  , b = 2  (645) 1 −9 4 8 41 19 41 − 19 4 the obvious and desired solution to the problem posed, x⋆ = e4 ∈ R6

(646)

has norm kx⋆ k2 = 1 and minimum cardinality; the minimum number of nonzero entries in vector x . The Matlab backslash command x=A\b ,

280

CHAPTER 4. SEMIDEFINITE PROGRAMMING

for example, finds 

xM

having norm kxM k2 = 0.7044 . id est, an optimal solution to

2 128



 0     5    =  128   0   90   128  0

(647)

Coincidentally, xM is a 1-norm solution; kxk1

minimize x

(488)

subject to Ax = b The pseudoinverse solution (rounded) 

−0.0456  −0.1881   0.0623 † xP = A b =   0.2668   0.3770 −0.1102

       

(648)

§

has least norm kxPk2 = 0.5165 ; id est, the optimal solution to ( E.0.1.0.1) minimize x

kxk2

subject to Ax = b

(649)

Certainly none of the traditional methods provide x⋆ = e4 (646) because, and in general, for Ax = b ° ° ° ° ° ° °arg inf kxk2 ° ≤ °arg inf kxk1 ° ≤ °arg inf kxk0 ° (650) 2 2 2 We can reformulate this minimum cardinality Boolean problem (643) as a semidefinite program: First transform the variable x , (ˆ x + 1) 21 so xˆi ∈ {−1, 1} ; equivalently, minimize x ˆ

k(ˆ x + 1) 21 k0

subject to A(ˆ x + 1) 12 = b δ(ˆ xxˆT ) = 1

(651)

(652)

4.2. FRAMEWORK

281

§

§

where δ is the main-diagonal linear operator ( A.1). By assigning ( B.1) G=

·

xˆ 1

¸

[ xˆT 1 ]

=

·

X xˆ xˆT 1

¸

,

·

xˆxˆT xˆ xˆT 1

¸

∈ S n+1

(653)

problem (652) becomes equivalent to: (Theorem A.3.1.0.7) minimize

X∈ Sn , x ˆ∈Rn

1T xˆ

subject to A(ˆ x + 1) 12 = b ¸ · X xˆ (º 0) G= xˆT 1 δ(X) = 1 rank G = 1

(654)

where solution is confined to rank-1 vertices of the elliptope in S n+1 ( 5.9.1.0.1) by the rank constraint, the positive semidefiniteness, and the equality constraints δ(X) = 1. The rank constraint makes this problem nonconvex; by removing it4.13 we get the semidefinite program

§

minimize 1T xˆ n n

X∈ S , x ˆ∈R

subject to A(ˆ x + 1) 21 = b · ¸ X xˆ G= º0 xˆT 1 δ(X) = 1

(655)

whose optimal solution x⋆ (651) is identical to that of minimum cardinality Boolean problem (643) if and only if rank G⋆ = 1. Hope4.14 of acquiring a rank-1 solution is not ill-founded because 2n elliptope vertices have rank 1, and we are minimizing an affine function on a subset of the elliptope (Figure 122) containing rank-1 vertices; id est, by assumption that the feasible set of minimum cardinality Boolean problem (643) is nonempty, 4.13

Relaxed problem (655) can also be derived via Lagrange duality; it is a dual of a dual program [sic] to (654). [282] [57, §5, exer.5.39] [352, §IV] [137, §11.3.4] The relaxed problem must therefore be convex having a larger feasible set; its optimal objective value represents a generally loose lower bound (1610) on the optimal objective of problem (654). 4.14 A more deterministic approach to constraining rank and cardinality is developed in §4.6.0.0.11.

282

CHAPTER 4. SEMIDEFINITE PROGRAMMING

a desired solution resides on the elliptope relative boundary at a rank-1 vertex.4.15 For that data given in (645), our semidefinite program solver [359] [360] (accurate in solution to approximately 1E-8)4.16 finds optimal solution to (655)   1 1 1 −1 1 1 −1  1 1 1 −1 1 1 −1      1 1 1 −1 1 1 −1   ⋆  1 −1 −1 1  (656) round(G ) =  −1 −1 −1    1 1 1 −1 1 1 −1    1 1 1 −1 1 1 −1  −1 −1 −1 1 −1 −1 1

§

near a rank-1 vertex of the elliptope in S n+1 ( 5.9.1.0.2); its sorted eigenvalues,   6.99999977799099  0.00000022687241     0.00000002250296     0.00000000262974 λ(G⋆ ) =  (657)    −0.00000000999738     −0.00000000999875  −0.00000001000000

Negative eigenvalues are undoubtedly finite-precision effects. Because the largest eigenvalue predominates by many orders of magnitude, we can expect to find a good approximation to a minimum cardinality Boolean solution by truncating all smaller eigenvalues. We find, indeed, the desired result (646)   0.00000000127947  0.00000000527369     0.00000000181001  ⋆   (658) x = round  0.99999997469044  = e4    0.00000001408950  0.00000000482903 2 4.15

Confinement to the elliptope can be regarded as a kind of normalization akin to matrix A column normalization suggested in [110] and explored in Example 4.2.3.1.2. 4.16 A typically ignored limitation of interior-point methods of solution is their relative accuracy of only about 1E-8 on a machine using 64-bit (double precision) floating-point arithmetic; id est, optimal solution x⋆ cannot be more accurate than square root of machine epsilon (ǫ = 2.2204E-16). Duality gap is not a good measure of solution accuracy.

4.2. FRAMEWORK

283

4.2.3.1.2 Example. Optimization on elliptope versus 1-norm polyhedron for minimum cardinality Boolean Example 4.2.3.1.1. A minimum cardinality problem is typically formulated via, what is by now, the standard practice [110] [64, 3.2, 3.4] of column normalization applied to a 1-norm problem surrogate like (488). Suppose we define a diagonal matrix   kA(: , 1)k2 0   kA(: , 2)k2   Λ , (659)  ∈ S6 . .   . 0 kA(: , 6)k2

§ §

used to normalize the columns (assumed nonzero) of given noiseless data matrix A . Then approximate the minimum cardinality Boolean problem minimize x

kxk0

subject to Ax = b xi ∈ {0, 1} ,

as

minimize y˜

where optimal solution

(643) i=1 . . . n

k˜ y k1

subject to AΛ−1 y˜ = b 1 º Λ−1 y˜ º 0

(660)

y ⋆ = round(Λ−1 y˜⋆ )

(661)

The inequality in (660) relaxes Boolean constraint yi ∈ {0, 1} from (643); bounding any solution y ⋆ to a nonnegative unit hypercube whose vertices are binary numbers. Convex problem (660) is justified by the convex envelope 1 cenv kxk0 on {x ∈ Rn | kxk∞ ≤ κ} = kxk1 (1311) κ Donoho concurs with this particular formulation, equivalently expressible as a linear program via (484). Approximation (660) is therefore equivalent to minimization of an affine function ( 3.3) on a bounded polyhedron, whereas semidefinite program

§

minimize 1T xˆ n n

X∈ S , x ˆ∈R

subject to A(ˆ x + 1) 21 = b ¸ · X xˆ º0 G= xˆT 1 δ(X) = 1

(655)

284

CHAPTER 4. SEMIDEFINITE PROGRAMMING

minimizes an affine function on an intersection of the elliptope with hyperplanes. Although the same Boolean solution is obtained from this approximation (660) as compared with semidefinite program (655), when given that particular data from Example 4.2.3.1.1, Singer confides a counter-example: Instead, given data # " # " 1 0 √12 1 , b = (662) A = 1 0 1 √2 1 then solving approximation (660) yields     1 − √12 0      ⋆ 1 y = round 1 − √2  =  0  1 1

(663)

(infeasible, with or without rounding, with respect to original problem (643)) whereas solving semidefinite program (655) produces   1 1 −1 1  1 1 −1 1   round(G⋆ ) =  (664)  −1 −1 1 −1  1 1 −1 1 with sorted eigenvalues



 3.99999965057264  0.00000035942736   λ(G⋆ ) =   −0.00000000000000  −0.00000001000000

(665)

Truncating all but the largest eigenvalue, from (651) we obtain (confer y ⋆ )     0.99999999625299 1 ⋆      0.99999999625299 x = round = 1  (666) 0.00000001434518 0

the desired minimum cardinality Boolean result.

2

4.2.3.1.3 Exercise. Minimum cardinality Boolean art. Assess general performance of standard-practice approximation (660) as compared with the proposed semidefinite program (655). H

4.3. RANK REDUCTION

285

4.2.3.1.4 Exercise. Conic independence. Matrix A from (645) is full-rank having three-dimensional nullspace. Find its four conically independent columns. ( 2.10) To what part of proper cone K = {Ax | x º 0} does vector b belong? H

§

4.3

Rank reduction . . . it is not clear generally how to predict rank X ⋆ or rank S ⋆ before solving the SDP problem. −Farid Alizadeh (1995) [11, p.22]

The premise of rank reduction in semidefinite programming is: an optimal solution found does not satisfy Barvinok’s upper bound (247) on rank. The particular numerical algorithm solving a semidefinite program may have instead returned a high-rank optimal solution ( 4.1.1.1; e.g., (624)) when a lower-rank optimal solution was expected.

§

4.3.1

Posit a perturbation of X ⋆

§

n Recall from 4.1.1.2, there is an extreme point of A ∩ S+ (616) satisfying upper bound (247) on rank. [24, 2.2] It is therefore sufficient to locate an extreme point of the intersection whose primal objective value (613P) is optimal:4.17 [103, 31.5.3] [222, 2.4] [223] [7, 3] [270] Consider again the affine subset

§

where for Ai ∈ S n

§

§

§

A = {X ∈ S n | A svec X = b}

 svec(A1 )T ..  ∈ Rm×n(n+1)/2 A , . svec(Am )T 

(616)

(614)

Given any optimal solution X ⋆ to

minimize hC , X i n X∈ S

n subject to X ∈ A ∩ S+ 4.17

(613)(P)

There is no known construction for Barvinok’s tighter result (252). −Monique Laurent

286

CHAPTER 4. SEMIDEFINITE PROGRAMMING

whose rank does not satisfy upper bound (247), we posit existence of a set of perturbations {tj Bj | tj ∈ R , Bj ∈ S n , j = 1 . . . n}

(667)

such that, for some 0 ≤ i ≤ n and scalars {tj , j = 1 . . . i} , ⋆

X +

i X

tj Bj

(668)

j=1

becomes an extreme point of A ∩ Sn+ and remains an optimal solution of (613P). Membership of (668) to affine subset A is secured for the i th perturbation by demanding hBi , Aj i = 0 ,

j =1 . . . m

(669)

n while membership to the positive semidefinite cone S+ is insured by small perturbation (678). In this manner feasibility is insured. Optimality is proved in 4.3.3. The following simple algorithm has very low computational intensity and locates an optimal extreme point, assuming a nontrivial solution:

§

4.3.1.0.1 Procedure. Rank reduction. initialize: Bi = 0 ∀ i for iteration i=1...n { 1. compute a nonzero perturbation matrix Bi of X ⋆ +

[Wıκımization]

i−1 P

tj Bj

j=1

2.

maximize

ti

subject to X ⋆ +

i P

j=1

}

n tj Bj ∈ S+

¶

4.3. RANK REDUCTION

287

A rank-reduced optimal solution is then ⋆

⋆

X ←X +

4.3.2

i X

tj Bj

(670)

j=1

Perturbation form

The perturbations are independent of constants C ∈ S n and b ∈ Rm in primal and dual problems (613). Numerical accuracy of any rank-reduced result, found by perturbation of an initial optimal solution X ⋆ , is therefore quite dependent upon initial accuracy of X ⋆ .

§

4.3.2.0.1 Definition. Matrix step function. (confer A.6.5.0.1) Define the signum-like quasiconcave real function ψ : S n → R ψ(Z ) ,

½

1, Z º 0 −1 , otherwise

(671)

The value −1 is taken for indefinite or nonzero negative semidefinite argument. △

§

Deza & Laurent [103, 31.5.3] prove: every perturbation matrix Bi , i = 1 . . . n , is of the form Bi = −ψ(Zi )Ri Zi RiT ∈ S n

(672)

where ⋆

X ,

R1 R1T

,

⋆

X +

i−1 X j=1

tj Bj , Ri RiT ∈ S n

(673)

where the tj are scalars and Ri ∈ Rn×ρ is full-rank and skinny where Ã

ρ , rank X ⋆ +

i−1 X j=1

!

tj Bj

(674)

288

CHAPTER 4. SEMIDEFINITE PROGRAMMING

and where matrix Zi ∈ S ρ is found at each iteration i by solving a very simple feasibility problem: 4.18 find Zi ∈ S ρ subject to hZi , RiTAj Ri i = 0 ,

j =1 . . . m

(675)

Were there a sparsity pattern common to each member of the set {RiTAj Ri ∈ S ρ , j = 1 . . . m} , then a good choice for Zi has 1 in each entry corresponding to a 0 in the pattern; id est, a sparsity pattern complement. At iteration i ⋆

X +

i−1 X j=1

tj Bj + ti Bi = Ri (I − ti ψ(Zi )Zi )RiT

(676)

By fact (1394), therefore

⋆

X +

i−1 X j=1

tj Bj + ti Bi º 0 ⇔ 1 − ti ψ(Zi )λ(Zi ) º 0

(677)

where λ(Zi ) ∈ Rρ denotes the eigenvalues of Zi . Maximization of each ti in step 2 of the Procedure reduces rank of (676) n and locates a new point on the boundary ∂(A ∩ S+ ) .4.19 Maximization of ti thereby has closed form; 4.18

A simple method of solution is closed-form projection of a random nonzero point on that proper subspace of isometrically isomorphic Rρ(ρ+1)/2 specified by the constraints. (§E.5.0.0.6) Such a solution is nontrivial assuming the specified intersection of hyperplanes is not the origin; guaranteed by ρ(ρ + 1)/2 > m . Indeed, this geometric intuition about forming the perturbation is what bounds any solution’s rank from below; m is fixed by the number of equality constraints in (613P) while rank ρ decreases with each iteration i . Otherwise, we might iterate indefinitely. 4.19 This holds because rank of a positive semidefinite matrix in S n is diminished below n by the number of its 0 eigenvalues (1404), and because a positive semidefinite matrix having one or more 0 eigenvalues corresponds to a point on the PSD cone boundary (178). Necessity and sufficiency are due to the facts: Ri can be completed to a nonsingular matrix (§A.3.1.0.5), and I − ti ψ(Zi )Zi can be padded with zeros while maintaining equivalence in (676).

4.3. RANK REDUCTION

289

(t⋆i )−1 = max {ψ(Zi )λ(Zi )j , j = 1 . . . ρ}

(678)

When Zi is indefinite, the direction of perturbation (determined by ψ(Zi )) is arbitrary. We may take an early exit from the Procedure were Zi to become 0 or were £ ¤ rank svec RiTA1 Ri svec RiTA2 Ri · · · svec RiTAm Ri = ρ(ρ + 1)/2

(679)

n which characterizes the rank ρ of any [sic] extreme point in A ∩ S+ . [222, 2.4] [223]

§

Proof. Assuming the form of every perturbation matrix is indeed (672), then by (675) ¤ £ svec Zi ⊥ svec(RiTA1 Ri ) svec(RiTA2 Ri ) · · · svec(RiTAm Ri )

(680)

By orthogonal complement we have £ ¤⊥ rank svec(RiTA1 Ri ) · · · svec(RiTAm Ri ) £ ¤ + rank svec(RiTA1 Ri ) · · · svec(RiTAm Ri ) = ρ(ρ + 1)/2

(681)

When Zi can only be 0, then the perturbation is null because an extreme point has been found; thus £

svec(RiTA1 Ri ) · · · svec(RiTAm Ri )

from which the stated result (679) directly follows.

¤⊥

= 0

(682) ¨

290

4.3.3

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Optimality of perturbed X ⋆

We show that the optimal objective value is unaltered by perturbation (672); id est, ⋆

hC , X +

i X j=1

tj Bj i = hC , X ⋆ i

(683)

Proof. From Corollary 4.2.3.0.1 we have the necessary and sufficient relationship between optimal primal and dual solutions under the assumption of existence of a relatively interior feasible point: S ⋆ X ⋆ = S ⋆ R1 R1T = X ⋆ S ⋆ = R1 R1T S ⋆ = 0

(684)

This means R(R1 ) ⊆ N (S ⋆ ) and R(S ⋆ ) ⊆ N (R1T ). From (673) and (676) we get the sequence: X ⋆ = R1 R1T X ⋆ + t1 B1 = R2 R2T = R1 (I − t1 ψ(Z1 )Z1 )R1T

X ⋆ + t1 B1 + t2 B2 = R3 R3T = R2 (I − t2 ψ(Z2 )Z2 )R2T = R1 (I − t1 ψ(Z1 )Z1 )(I − t2 ψ(Z2 )Z2 )R1T .. . Ã ! i i Q P (685) X⋆ + (I − tj ψ(Zj )Zj ) R1T tj Bj = R1 j=1

j=1

Substituting C = svec−1 (AT y ⋆ ) + S ⋆ from (613), hC , X ⋆ +

i P

j=1

*

Ã

*

+

tj Bj i = svec−1 (AT y ⋆ ) + S ⋆ , R1 =

m P

k=1

=

¿

m P

k=1

yk⋆ Ak , X ⋆ + yk⋆ Ak

⋆

i P

+S , X

À

because hBi , Aj i = 0 ∀ i , j by design (669).

j=1

!

+

(I − tj ψ(Zj )Zj ) R1T

tj Bj

j=1 ⋆

i Q

= hC , X ⋆ i

(686) ¨

4.3. RANK REDUCTION

291

4.3.3.0.1 Example. Aδ(X) = b . This academic example demonstrates that a solution found by rank reduction can certainly have rank less than Barvinok’s upper bound (247): Assume a given vector b ∈ Rm belongs to the conic hull of columns of a given matrix A ∈ Rm×n ;     −1 1 8 1 1 1     A =  −3 2 8 21 13  , b =  21  (687) −9

4

1 4

8

1 9

1 4

Consider the convex optimization problem minimize 5 X∈ S

tr X

subject to X º 0 Aδ(X) = b

(688)

that minimizes the 1-norm of the main diagonal; id est, problem (688) is the same as minimize kδ(X)k1 5 X∈ S

subject to X º 0 Aδ(X) = b

(689)

that finds a solution to Aδ(X) = b . Rank-3 solution X ⋆ = δ(xM ) is optimal, where (confer (647))  2  128

xM

 0   5   =  128   0 

(690)

90 128

Yet upper bound (247) predicts existence of at most a µ¹√ º ¶ 8m + 1 − 1 rank=2 2

(691)

feasible solution from m = 3 equality constraints. To find a lower rank ρ optimal solution to (688) (barring combinatorics), we invoke Procedure 4.3.1.0.1:

292

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Initialize: C = I , ρ = 3, Aj , δ(A(j, :)) , j = 1, 2, 3, X ⋆ = δ(xM ) , m = 3, n = 5. { Iteration i=1:  q

    Step 1: R1 =    

2 128

0 0

0 q0

0

0

0

0 0

5 128

0

0 q0

90 128



    .   

find Z1 ∈ S3 subject to hZ1 , R1TAj R1 i = 0 ,

j = 1, 2, 3

(692)

A nonzero randomly selected matrix Z1 having 0 main diagonal is feasible and yields a nonzero perturbation matrix. Choose, arbitrarily, Z1 = 11T − I ∈ S3 (693) then (rounding) 

  B1 =   

0 0 0.0247 0 0.1048

 0 0.0247 0 0.1048  0 0 0 0  0 0 0 0.1657    0 0 0 0 0 0.1657 0 0

Step 2: t⋆1 = 1 because λ(Z1 ) = [ −1 −1 2 ]T . So,  2 0 0.0247 128  0 0 0  5 0.0247 0 X ⋆ ← δ(xM ) + B1 =  128   0 0 0 0.1048 0 0.1657

 0 0.1048  0 0  0 0.1657    0 0 90 0 128

(694)

(695)

has rank ρ ← 1 and produces the same optimal objective value.

}

2

4.3. RANK REDUCTION

293

4.3.3.0.2 Exercise. Rank reduction of maximal complementarity. Apply rank reduction Procedure 4.3.1.0.1 to the maximal complementarity example ( 4.1.1.4.1). Demonstrate a rank-1 solution; which can certainly be found (by Barvinok’s Proposition 2.9.3.0.1) because there is only one equality constraint. H

§

4.3.4

thoughts regarding rank reduction

Because the rank reduction procedure is guaranteed only to produce another optimal solution conforming to Barvinok’s upper bound (247), the Procedure will not necessarily produce solutions of arbitrarily low rank; but if they exist, the Procedure can. Arbitrariness of search direction when matrix Zi becomes indefinite, mentioned on page 289, and the enormity of choices for Zi (675) are liabilities for this algorithm.

4.3.4.1

Inequality constraints

The question naturally arises: what to do when a semidefinite program (not in prototypical form (613))4.20 has linear inequality constraints of the form αiT svec X ¹ βi ,

i = 1...k

(696)

where {βi } are given scalars and {αi } are given vectors. One expedient way to handle this circumstance is to convert the inequality constraints to equality constraints by introducing a slack variable γ ; id est, αiT svec X + γi = βi ,

i = 1...k ,

γº0

(697)

thereby converting the problem to prototypical form. 4.20

Contemporary numerical packages for solving semidefinite programs can solve a range of problems wider than prototype (613). Generally, they do so by transforming a given problem into prototypical form by introducing new constraints and variables. [11] [360] We are momentarily considering a departure from the primal prototype that augments the constraint set with linear inequalities.

294

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Alternatively, we say the i th inequality constraint is active when it is met with equality; id est, when for particular i in (696), αiT svec X ⋆ = βi . An optimal high-rank solution X ⋆ is, of course, feasible satisfying all the constraints. But for the purpose of rank reduction, inactive inequality constraints are ignored while active inequality constraints are interpreted as equality constraints. In other words, we take the union of active inequality constraints (as equalities) with equality constraints A svec X = b to form a composite affine subset Aˆ substituting for (616). Then we proceed with rank reduction of X ⋆ as though the semidefinite program were in prototypical form (613P).

4.4

Rank-constrained semidefinite program

§

A generalization of the trace heuristic ( 7.2.2.1), for finding low-rank optimal solutions to SDPs of a more general form, was discovered in October 2005:

4.4.1

rank constraint by convex iteration

Consider a semidefinite feasibility problem of the form find

G∈ SN

G

subject to G ∈ C Gº0 rank G ≤ n

(698)

where C is a convex set presumed to contain positive semidefinite matrices of rank n or less; id est, C intersects the positive semidefinite cone boundary. We propose that this rank-constrained feasibility problem can be equivalently expressed as the problem sequence (699) (1627a) having convex constraints: minimize G∈ SN

hG , W i

subject to G ∈ C Gº0

(699)

where direction vector 4.21 W is an optimal solution to semidefinite program, for 0 ≤ n ≤ N − 1 4.21

Search direction W is a hyperplane-normal pointing opposite to direction of movement describing minimization of a real linear function hG , W i (p.74).

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM

N X

λ(G⋆ )i = minimize W ∈ SN

i=n+1

hG⋆ , W i

295

(1627a)

subject to 0 ¹ W ¹ I tr W = N − n

§

whose feasible set is a Fantope ( 2.3.2.0.1), and where G⋆ is an optimal solution to problem (699) given some iterate W . The idea is to iterate solution of (699) and (1627a) until convergence as defined in 4.4.1.2: 4.22 N X

§

λ(G⋆ )i = hG⋆ , W ⋆ i = λ(G⋆ )T λ(W ⋆ ) , 0

(700)

i=n+1

defines global convergence of the iteration; a vanishing objective that is a certificate of global optimality but cannot be guaranteed. Optimal direction vector W ⋆ is defined as any positive semidefinite matrix yielding optimal solution G⋆ of rank n or less to then convex equivalent (699) of feasibility problem (698): find

G∈ SN

(698)

G

subject to G ∈ C Gº0 rank G ≤ n

minimize ≡

G∈ SN

hG , W ⋆ i

subject to G ∈ C Gº0

(699)

id est, any direction vector for which the last N − n nonincreasingly ordered eigenvalues λ of G⋆ are zero. 4.22

Proposed iteration is neither dual projection (Figure 156) or alternating projection (Figure 158). Sum of eigenvalues follows from results of Ky Fan (page 631). Inner product of eigenvalues follows from (1522) and properties of commutative matrix products (page 583).

296

CHAPTER 4. SEMIDEFINITE PROGRAMMING

We emphasize that convex problem (699) is not a relaxation of rank-constrained feasibility problem (698); at convergence, convex iteration (699) (1627a) makes it instead an equivalent problem.4.23 4.4.1.1

direction interpretation; not a projection method

The feasible set of direction matrices in (1627a) is the convex hull of outer product of all rank-(N − n) orthonormal matrices; videlicet, © ª © ª conv U U T | U ∈ RN ×N −n , U T U = I = A ∈ SN | I º A º 0 , hI , A i = N − n (85) Set {U U T | U ∈ RN ×N −n , U T U = I } comprises the extreme points of this Fantope (85). An optimal solution W to (1627a), that is an extreme point, is known in closed form (p.631): Given ordered diagonalization N ⋆ T G = QΛQ ∈ S ( A.5.1), then direction matrix W = U ⋆ U ⋆T is optimal and extreme where U ⋆ = Q(: , n+1 : N ) ∈ RN ×N −n . Eigenvalue vector λ(W ) has 1 in each entry corresponding to the N − n smallest entries of δ(Λ) and has 0 elsewhere. By (208), polar direction −W can be regarded as pointing toward positive semidefinite rank-n matrices that are simultaneously diagonalizable with G⋆ . For that particular closed-form solution W , consequently

§

N X

λ(G⋆ )i = hG⋆ , W i = λ(G⋆ )T λ(W )

(701)

i=n+1

This is the connection to cardinality minimization of vectors;4.24 id est, eigenvalue λ cardinality (rank) is analogous to vector x cardinality via (716). 4.23

Terminology equivalent problem meaning, optimal solution to one problem can be derived from optimal solution to another. Terminology same problem means: optimal solution set for one problem is identical to the optimal solution set of another (without transformation). 4.24 not minimization of a diagonal matrix δ(x) as in [125, §1]. To make rank constrained problem (698) resemble cardinality problem (713), we could make C an affine subset: [278] find subject to

X ∈ SN A svec X = b Xº0 rank X ≤ n

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM

297

Sn (I −W )G⋆ (I −W )

G⋆

Sn⊥

W G⋆ W

Figure 80: Projection on subspace Sn of rank-n matrices simultaneously diagonalizable with G⋆ . This direction W is closed-form solution to (1627a).

So as not to be misconstrued under closed-form solution W to (1627a): Define Sn = {(I −W )G(I −W ) | G ∈ SN } as the symmetric subspace of rank-n matrices simultaneously diagonalizable with G⋆ . Then projection of G⋆ on Sn is (I −W )G⋆ (I −W ). ( E.7) Direction of projection is −W G⋆ W . (Figure 80) tr(W G⋆ W ) is a measure of proximity to Sn because its orthogonal complement is Sn⊥ = {W GW | G ∈ SN } ; the point being, convex iteration incorporating constrained tr(W GW ) = hG , W i minimization is not a projection method; certainly, not on these two subspaces. Closed-form solution W to problem (1627a), though efficient, comes with a caveat: there exist cases where this projection matrix solution W does not provide the most efficient route to an optimal rank-n solution G⋆ ; id est, direction W is not unique. When direction matrix W = I , as in the trace heuristic for example, then −W points directly at the origin (the rank-0 PSD matrix, Figure 81). Vector inner-product of an optimization variable with direction matrix W is therefore a generalization of the trace heuristic ( 7.2.2.1) for rank minimization; −W is instead trained toward the boundary of the positive semidefinite cone.

§

§

298

CHAPTER 4. SEMIDEFINITE PROGRAMMING

I

0

S2+

∂H = {G | hG , I i = κ} Figure 81: Trace heuristic can be interpreted as minimization of a hyperplane, with normal I , on the positive semidefinite cone drawn here in isometrically isomorphic R3 . Polar of direction vector W = I points toward the origin.

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 4.4.1.2

299

convergence

We study convergence to ascertain conditions under which a direction matrix will reveal a feasible G matrix of rank n or less in semidefinite program (699). Denote by W ⋆ a particular optimal direction matrix from semidefinite program (1627a) such that (700) holds (feasible rank G ≤ n found). Then we define global convergence of the iteration (699) (1627a) to correspond with this vanishing vector inner-product (700) of optimal solutions. Because this iterative technique for constraining rank is not a projection method, it can find a rank-n solution G⋆ ((700) will be satisfied) only if at least one exists in the feasible set of program (699).

4.4.1.2.1 Proof. Suppose hG⋆ , W i = τ is satisfied for some nonnegative constant τ after any particular iteration (699) (1627a) of the two minimization problems. Once a particular value of τ is achieved, it can never be exceeded by subsequent iterations because existence of feasible G and W having that vector inner-product τ has been established simultaneously in each problem. Because the infimum of vector inner-product of two positive semidefinite matrix variables is zero, the nonincreasing sequence of iterations is thus bounded below hence convergent because any bounded monotonic sequence in R is convergent. [240, 1.2] [39, 1.1] Local convergence to some nonnegative objective value τ is thereby established. ¨

§

§

Local convergence means convergence to a fixed point of possibly infeasible rank. Only local convergence can be established because objective hG , W i , when instead regarded simultaneously in two variables (G , W ) , is generally multimodal. When a rank-n feasible solution to (699) exists, it remains an open problem to state conditions under which hG⋆ , W ⋆ i = 0 (700) is achieved by iterative solution of semidefinite programs (699) and (1627a). Then pair (G⋆ , W ⋆ ) becomes a globally optimal fixed point of iteration, to which there can be no proof of convergence. A nonexistent feasible rank-n solution would mean certain failure to converge by definition (700) but, as proved, convex iteration always converges locally if not globally. Now, an application to optimal control of Euclidean dimension:

300

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Figure 82: Sensor-network localization in R2 , illustrating connectivity and circular radio-range per sensor. Smaller dark grey regions each hold an anchor at their center; known fixed sensor positions. Sensor/anchor distance is measurable with negligible uncertainty for sensor within those grey regions. Graphic by Geoff Merrett.

4.4.1.2.2 Example. Sensor-Network Localization and Wireless Location. Heuristic solution to a sensor-network localization problem, proposed by Carter, Jin, Saunders, & Ye in [68],4.25 is limited to two Euclidean dimensions and applies semidefinite programming (SDP) to little subproblems. There, a large network is partitioned into smaller subnetworks (as small as one sensor − a mobile point, whereabouts unknown) and then semidefinite programming and heuristics called spaseloc are applied to localize each and every partition by two-dimensional distance geometry. Their partitioning procedure is one-pass, yet termed iterative; a term 4.25

The paper constitutes Jin’s dissertation for University of Toronto [201] although her name appears as second author. Ye’s authorship is honorary.

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM

301

applicable only in so far as adjoining partitions can share localized sensors and anchors (absolute sensor positions known a priori). But there is no iteration on the entire network, hence the term “iterative” is perhaps inappropriate. As partitions are selected based on “rule sets” (heuristics, not geographics), they also term the partitioning adaptive. But no adaptation of a partition actually occurs once it has been determined. One can reasonably argue that semidefinite programming methods are unnecessary for localization of small partitions of large sensor networks. In the past, these nonlinear localization problems were solved algebraically and computed by least squares solution to hyperbolic equations; called multilateration.4.26 Indeed, practical contemporary numerical methods for global positioning by satellite (GPS) do not rely on semidefinite programming. Modern distance geometry is inextricably melded with semidefinite programming. The beauty of semidefinite programming, as relates to localization, lies in convex expression of classical multilateration: So & Ye showed [298] that the problem of finding unique solution, to a noiseless nonlinear system describing the common point of intersection of hyperspheres in real Euclidean vector space, can be expressed as a semidefinite program via distance geometry. But the need for SDP methods in Carter & Jin et alii is enigmatic for two more reasons: 1) guessing solution to a partition whose intersensor measurement data or connectivity is inadequate for localization by distance geometry, 2) reliance on complicated and extensive heuristics for partitioning a large network that could instead be solved whole by one semidefinite program. While partitions range in size between 2 and 10 sensors, 5 sensors optimally, heuristics provided are only for two spatial dimensions (no higher-dimensional heuristics are proposed). For these small numbers it remains unclarified as to precisely what advantage is gained over traditional least squares.

4.26

Multilateration − literally, having many sides; shape of a geometric figure formed by nearly intersecting lines of position. In navigation systems, therefore: Obtaining a fix from multiple lines of position.

302

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Partitioning of large sensor networks is a compromise to rapid growth of SDP computational intensity with problem size. But when impact of noise on distance measurement is of most concern, one is averse to a partitioning scheme because noise-effects vary inversely with problem size. [49, 2.2] ( 5.13.2) Since an individual partition’s solution is not iterated in Carter & Jin and is interdependent with adjoining partitions, we expect errors to propagate from one partition to the next; the ultimate partition solved, expected to suffer most. Heuristics often fail on real-world data because of unanticipated circumstances. When heuristics fail, generally they are repaired by adding more heuristics. Tenuous is any presumption, for example, that distance measurement errors have distribution characterized by circular contours of equal probability about an unknown sensor-location. (Figure 82) That presumption effectively appears within Carter & Jin’s optimization problem statement as affine equality constraints relating unknowns to distance measurements that are corrupted by noise. Yet in most all urban environments, this measurement noise is more aptly characterized by ellipsoids of varying orientation and eccentricity as one recedes from a sensor. Each unknown sensor must therefore instead be bound to its own particular range of distance, primarily determined by the terrain.4.27 The nonconvex problem we must instead solve is:

§

§

find

i,j ∈ I

{xi , xj }

subject to dij ≤ kxi − xj k2 ≤ dij

(702)

where xi represents sensor location, and where dij and dij respectively represent lower and upper bounds on measured distance-square from i th to j th sensor (or from sensor to anchor). Figure 87 illustrates contours of equal sensor-location uncertainty. By establishing these individual upper and lower bounds, orientation and eccentricity can effectively be incorporated into the problem statement. Generally speaking, there can be no unique solution to the sensor-network localization problem because there is no unique formulation; that is the art of Optimization. Any optimal solution obtained depends on whether or how a network is partitioned, whether distance data is complete, presence of noise, and how the problem is formulated. When a particular formulation is a 4.27

A distinct contour map corresponding to each anchor is required in practice.

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM

303

convex optimization problem, then the set of all optimal solutions forms a convex set containing the actual or true localization. Measurement noise precludes equality constraints representing distance. The optimal solution set is consequently expanded; necessitated by introduction of distance inequalities admitting more and higher-rank solutions. Even were the optimal solution set a single point, it is not necessarily the true localization because there is little hope of exact localization by any algorithm once significant noise is introduced.

§

Carter & Jin gauge performance of their heuristics to the SDP formulation of author Biswas whom they regard as vanguard to the art. [14, 1] Biswas posed localization as an optimization problem minimizing a distance measure. [44] [42] Intuitively, minimization of any distance measure yields compacted solutions; (confer 6.4.0.0.1) precisely the anomaly motivating Carter & Jin. Their two-dimensional heuristics outperformed Biswas’ localizations both in execution-time and proximity to the desired result. Perhaps, instead of heuristics, Biswas’ approach to localization can be improved: [41] [43].

§

§

The sensor-network localization problem is considered difficult. [14, 2] Rank constraints in optimization are considered more difficult. Control of Euclidean dimension in Carter & Jin is suboptimal because of implicit projection on R2 . In what follows, we present the localization problem as a semidefinite program (equivalent to (702)) having an explicit rank constraint which controls Euclidean dimension of an optimal solution. We show how to achieve that rank constraint only if the feasible set contains a matrix of desired rank. Our problem formulation is extensible to any spatial dimension.

proposed standardized test Jin proposes an academic test in two-dimensional real Euclidean space R2 that we adopt. In essence, this test is a localization of sensors and anchors arranged in a regular triangular lattice. Lattice connectivity is solely determined by sensor radio range; a connectivity graph is assumed incomplete. In the interest of test standardization, we propose adoption of a few small examples: Figure 83 through Figure 86 and their particular connectivity represented by matrices (703) through (706) respectively.

304

CHAPTER 4. SEMIDEFINITE PROGRAMMING 2

3

1

4

Figure 83: 2-lattice in R2 , hand-drawn. Nodes 3 and 4 are anchors; remaining nodes are sensors. Radio range of sensor 1 indicated by arc.

0 • ? •

• 0 • •

? • 0 ◦

• • ◦ 0

(703)

Matrix entries dot • indicate measurable distance between nodes while unknown distance is denoted by ? (question mark ). Matrix entries hollow dot ◦ represent known distance between anchors (to very high accuracy) while zero distance is denoted 0. Because measured distances are quite unreliable in practice, our solution to the localization problem substitutes a distinct range of possible distance for each measurable distance; equality constraints exist only for anchors. Anchors are chosen so as to increase difficulty for algorithms dependent on existence of sensors in their convex hull. The challenge is to find a solution in two dimensions close to the true sensor positions given incomplete noisy intersensor distance information.

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 5

6

7

3

4

8

2

9

1

305

Figure 84: 3-lattice in R2 , hand-drawn. Nodes 7, 8, and 9 are anchors; remaining nodes are sensors. Radio range of sensor 1 indicated by arc.

0 • • ? • ? ? • •

• 0 • • ? • ? • •

• • 0 • • • • • •

? • • 0 ? • • • •

• ? • ? 0 • • • •

? • • • • 0 • • •

? ? • • • • 0 ◦ ◦

• • • • • • ◦ 0 ◦

• • • • • • ◦ ◦ 0

(704)

306

CHAPTER 4. SEMIDEFINITE PROGRAMMING 10

11

13

7

8

14

4

9

6

5

15

1

12

3

2

16

Figure 85: 4-lattice in R2 , hand-drawn. Nodes 13, 14, 15, and 16 are anchors; remaining nodes are sensors. Radio range of sensor 1 indicated by arc.

0 ? ? • ? ? • ? ? ? ? ? ? ? • •

? 0 • • • • ? • ? ? ? ? ? • • •

? • 0 ? • • ? ? • ? ? ? ? ? • •

• • ? 0 • ? • • ? • ? ? • • • •

? • • • 0 • ? • • ? • • • • • •

? • • ? • 0 ? • • ? • • ? ? ? ?

• ? ? • ? ? 0 ? ? • ? ? • • • •

? • ? • • • ? 0 • • • • • • • •

? ? • ? • • ? • 0 ? • • • ? • ?

? ? ? • ? ? • • ? 0 • ? • • • ?

? ? ? ? • • ? • • • 0 • • • • ?

? ? ? ? • • ? • • ? • 0 ? ? ? ?

? ? ? • • ? • • • • • ? 0 ◦ ◦ ◦

? • ? • • ? • • ? • • ? ◦ 0 ◦ ◦

• • • • • ? • • • • • ? ◦ ◦ 0 ◦

• • • • • ? • • ? ? ? ? ◦ ◦ ◦ 0

(705)

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 17

21

18

19

9

23

10

5

12

7

24

2

16

15

11

6

1

20

22

14

13

8

4

3

25

307

Figure 86: 5-lattice in R2 . Nodes 21 through 25 are anchors. 0 • ? ? • • ? ? • ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

• 0 ? ? • • ? ? ? • ? ? ? ? ? ? ? ? ? ? ? ? ? • •

? ? 0 • ? • • • ? ? • • ? ? ? ? ? ? ? ? ? ? • • •

? ? • 0 ? ? • • ? ? ? • ? ? ? ? ? ? ? ? ? ? ? • ?

• • ? ? 0 • ? ? • • ? ? • • ? ? • ? ? ? ? ? • ? •

• • • ? • 0 • ? • • • ? ? • ? ? ? ? ? ? ? ? • • •

? ? • • ? • 0 • ? ? • • ? ? • • ? ? ? ? ? ? • • •

? ? • • ? ? • 0 ? ? • • ? ? • • ? ? ? ? ? ? ? • ?

• ? ? ? • • ? ? 0 • ? ? • • ? ? • • ? ? ? ? ? ? ?

? • ? ? • • ? ? • 0 • ? • • ? ? ? • ? ? • • • • •

? ? • ? ? • • • ? • 0 • ? • • • ? ? • ? ? • • • •

? ? • • ? ? • • ? ? • 0 ? ? • • ? ? • • ? • • • ?

? ? ? ? • ? ? ? • • ? ? 0 • ? ? • • ? ? • • ? ? ?

? ? ? ? • • ? ? • • • ? • 0 • ? • • • ? • • • • ?

? ? ? ? ? ? • • ? ? • • ? • 0 • ? ? • • • • • • ?

? ? ? ? ? ? • • ? ? • • ? ? • 0 ? ? • • ? • ? ? ?

? ? ? ? • ? ? ? • ? ? ? • • ? ? 0 • ? ? • ? ? ? ?

? ? ? ? ? ? ? ? • • ? ? • • ? ? • 0 • ? • • • ? ?

? ? ? ? ? ? ? ? ? ? • • ? • • • ? • 0 • • • • ? ?

? ? ? ? ? ? ? ? ? ? ? • ? ? • • ? ? • 0 • • ? ? ?

? ? ? ? ? ? ? ? ? • ? ? • • • ? • • • • 0 ◦ ◦ ◦ ◦

? ? ? ? ? ? ? ? ? • • • • • • • ? • • • ◦ 0 ◦ ◦ ◦

? ? • ? • • • ? ? • • • ? • • ? ? • • ? ◦ ◦ 0 ◦ ◦

? • • • ? • • • ? • • • ? • • ? ? ? ? ? ◦ ◦ ◦ 0 ◦

? • • ? • • • ? ? • • ? ? ? ? ? ? ? ? ? ◦ ◦ ◦ ◦ 0

(706)

308

CHAPTER 4. SEMIDEFINITE PROGRAMMING

T KE

AR

M .

St

Figure 87: Location uncertainty ellipsoid in R2 for each of 15 sensors • within three city blocks in downtown San Francisco. Data by Polaris Wireless. [300]

problem statement Ascribe points in a list {xℓ ∈ Rn , ℓ = 1 . . . N } to the columns of a matrix X ; X = [ x1 · · · xN ] ∈ Rn×N

(71)

§

where N is regarded as cardinality of list X . Positive semidefinite matrix X TX , formed from inner product of the list, is a Gram matrix ; [232, 3.6]   kx1 k2 xT xT · · · xT 1 x2 1 x3 1 xN   T · · · xT  x2 x1 kx2 k2 xT 2 x3 2 xN     ∈ SN T T 2 ... (846) G = X TX =  xT +  x3 x1 x3 x2 kx3 k 3 xN    . . . ... ... .. ..   .. T T T 2 xN x1 xN x2 xN x3 · · · kxN k

where SN + is the convex cone of N × N positive semidefinite matrices in the symmetric matrix subspace SN . Existence of noise precludes measured distance from the input data. We instead assign measured distance to a range estimate specified by individual upper and lower bounds: dij is an upper bound on distance-square from i th

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM

309

to j th sensor, while dij is a lower bound. These bounds become the input data. Each measurement range is presumed different from the others because of measurement uncertainty; e.g., Figure 87. Our mathematical treatment of anchors and sensors is not dichotomized.4.28 A known sensor position to high accuracy xˇi is an anchor. Then the sensor-network localization problem (702) can be expressed equivalently: Given a number of anchors m , and I a set of indices (corresponding to all measurable distances • ), for 0 < n < N find

G∈ SN , X∈ R n×N

subject to

X dij ≤ hG , (ei − ej )(ei − ej )T i ≤ dij

hG , ei eT i i T hG , (ei eT j + ej ei )/2i X(: , N − m + 1 : N ) · ¸ I X Z= XT G rank Z

∀(i, j) ∈ I

= kˇ xi k2 , i = N − m + 1 . . . N = xˇT ˇj , i < j , ∀ i, j ∈ {N − m + 1 . . . N } ix = [ xˇN −m+1 · · · xˇN ] º 0

(707)

= n

where ei is the i th member of the standard basis for RN . Distance-square dij = kxi − xj k22 = hxi − xj , xi − xj i

(833)

is related to Gram matrix entries G , [gij ] by vector inner-product dij = gii + gjj − 2gij

= hG , (ei − ej )(ei − ej )T i = tr(G T (ei − ej )(ei − ej )T )

(848)

hence the scalar inequalities. Each linear equality constraint in G ∈ SN represents a hyperplane in isometrically isomorphic Euclidean vector space RN (N +1)/2 , while each linear inequality pair represents a convex Euclidean body known as slab (an intersection of two parallel but opposing halfspaces, Figure 11). By Schur complement ( A.4), any feasible G and X provide a comparison with respect to the positive semidefinite cone

§

G º X TX 4.28

(883)

Wireless location problem thus stated identically; difference being: fewer sensors.

310

CHAPTER 4. SEMIDEFINITE PROGRAMMING

which is a convex relaxation of the desired equality constraint ¸ · ¸ · I X I [I X ] (884) = XT G XT The rank constraint insures this equality holds, by Theorem A.4.0.0.5, thus restricting solution to Rn . Assuming full-rank feasible list X rank Z = rank G = rank X

(708)

convex equivalent problem statement Problem statement (707) is nonconvex because of the rank constraint. We do not eliminate or ignore the rank constraint; rather, we find a convex way to enforce it: for 0 < n < N minimize

G∈ SN , X∈ R n×N

subject to

hZ , W i dij ≤ hG , (ei − ej )(ei − ej )T i ≤ dij

hG , ei eT i i T hG , (ei eT j + ej ei )/2i X(: , N − m + 1 : N ) ¸ · I X Z= XT G

∀(i, j) ∈ I

= kˇ xi k2 , i = N − m + 1 . . . N = xˇT ˇj , i < j , ∀ i, j ∈ {N − m + 1 . . . N } ix = [ xˇN −m+1 · · · xˇN ] º 0

(709)

§

Convex function tr Z is a well-known heuristic whose sole purpose is to represent convex envelope of rank Z . ( 7.2.2.1) In this convex optimization problem (709), a semidefinite program, we substitute a vector inner-product objective function for trace; tr Z = hZ , I i ← hZ , W i

(710)

a generalization of the trace heuristic for minimizing convex envelope of rank, +n where W ∈ SN is constant with respect to (709). Matrix W is normal to + a hyperplane in SN +n minimized over a convex feasible set specified by the constraints in (709). Matrix W is chosen so −W points in the direction of a feasible rank-n Gram matrix G . Thus the purpose of vector inner-product objective (710) is to locate a feasible rank-n Gram matrix assumed existent on the boundary of positive semidefinite cone SN + , as explained beginning in 4.4.1.

§

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM

311

direction matrix W Denote by Z ⋆ an optimal composite matrix from semidefinite program (709). Then for Z ⋆ ∈ SN +n whose eigenvalues λ(Z ⋆ ) ∈ RN +n are arranged in nonincreasing order, (Ky Fan) N +n X

λ(Z ⋆ )i = minimize N +n

i=n+1

W∈S

hZ ⋆ , W i

(1627a)

subject to 0 ¹ W ¹ I tr W = N

§

which has an optimal solution that is known in closed form (p.631, 4.4.1.1). This eigenvalue sum is zero when Z ⋆ has rank n or less. Foreknowledge of optimal Z ⋆ , to make possible this search for W , implies iteration; id est, semidefinite program (709) is solved for Z ⋆ initializing W = I or W = 0. Once found, Z ⋆ becomes constant in semidefinite program (1627a) where a new normal direction W is found as its optimal solution. Then this cycle (709) (1627a) iterates until convergence. When rank Z ⋆ = n , solution via this convex iteration is feasible for sensor-network localization problem (702) and its equivalent (707). numerical solution In all examples to follow, number of anchors √ m= N

(711)

equals square root of cardinality N of list X . Indices set I identifying all measurable distances • is ascertained from connectivity matrix (703), (704), (705), or (706). We solve iteration (709) (1627a) in dimension n = 2 for each respective example illustrated in Figure 83 through Figure 86. In presence of negligible noise, true position is reliably localized for every standardized example; noteworthy in so far as each example represents an incomplete graph. This implies that the set of all optimal solutions having lowest rank must be small. To make the examples interesting and consistent with previous work, we randomize each range of distance-square that bounds hG , (ei − ej )(ei − ej )T i in (709); id est, for each and every (i, j) ∈ I √ dij = dij (1 +√ 3 η χl )2 (712) dij = dij (1 − 3 η χl+1 )2

312

CHAPTER 4. SEMIDEFINITE PROGRAMMING

where η = 0.1 is a constant noise factor, χl is the l th sample of a noise process realization uniformly distributed in the interval [0, 1] like rand(1) from Matlab, and dij is actual distance-square from i th to j th sensor. Because of distinct function calls to rand() , each range of distance-square [ dij , dij ] is not necessarily centered on actual distance-square dij . Unit √ stochastic variance is provided by factor 3. Figure 88 through Figure 91 each illustrate one realization of numerical solution to the standardized lattice problems posed by Figure 83 through Figure 86 respectively. Exact localization is impossible because of measurement noise. Certainly, by inspection of their published graphical data, our new results are competitive with those of Carter & Jin. Obviously our solutions do not suffer from those compaction-type errors (clustering of localized sensors) exhibited by Biswas’ graphical results for the same noise factor η ; which is all we intended to demonstrate. localization example conclusion Solution to this sensor-network localization problem became apparent by understanding geometry of optimization. Trace of a matrix, to a student of linear algebra, is perhaps a sum of eigenvalues. But to us, trace represents the normal I to some hyperplane in Euclidean vector space. Numerical solution to noisy problems containing sensor variables in excess of 100 becomes difficult via the holistic semidefinite program we propose. When problem size is within reach of contemporary general-purpose semidefinite program solvers, then the equivalent program presented inherently overcomes limitations of Carter & Jin with respect to both noise performance and ability to constrain localization to any desired affine dimension. The legacy of Carter, Jin, Saunders, & Ye [68] is a sobering demonstration of the need for more efficient methods for solution of semidefinite programs, while that of So & Ye [298] is the bonding of distance geometry to semidefinite programming. Elegance of our semidefinite problem statement (709), for constraining affine dimension of sensor-network localization, should provide some impetus to focus more research on computational intensity of general-purpose semidefinite program solvers. An approach different from interior-point methods is required; higher speed and greater accuracy from a simplex-like solver is what is needed. 2

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM

313

1.2

1

0.8

0.6

0.4

0.2

0

−0.2 −0.5

0

0.5

1

1.5

2

Figure 88: Typical solution for 2-lattice in Figure 83 with noise factor η = 0.1 . Two red rightmost nodes are anchors; two remaining nodes are sensors. Radio range of sensor 1 indicated by arc; radius = 1.14 . Actual sensor indicated by target # while its localization is indicated by bullet • . Rank-2 solution found in 1 iteration (709) (1627a) subject to reflection error.

1.2

1

0.8

0.6

0.4

0.2

0

−0.2 −0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 89: Typical solution for 3-lattice in Figure 84 with noise factor η = 0.1 . Three red vertical middle nodes are anchors; remaining nodes are sensors. Radio range of sensor 1 indicated by arc; radius = 1.12 . Actual sensor indicated by target # while its localization is indicated by bullet • . Rank-2 solution found in 2 iterations (709) (1627a).

314

CHAPTER 4. SEMIDEFINITE PROGRAMMING 1.2

1

0.8

0.6

0.4

0.2

0

−0.2 −0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 90: Typical solution for 4-lattice in Figure 85 with noise factor η = 0.1 . Four red vertical middle-left nodes are anchors; remaining nodes are sensors. Radio range of sensor 1 indicated by arc; radius = 0.75 . Actual sensor indicated by target # while its localization is indicated by bullet • . Rank-2 solution found in 7 iterations (709) (1627a).

1.2

1

0.8

0.6

0.4

0.2

0

−0.2 −0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 91: Typical solution for 5-lattice in Figure 86 with noise factor η = 0.1 . Five red vertical middle nodes are anchors; remaining nodes are sensors. Radio range of sensor 1 indicated by arc; radius = 0.56 . Actual sensor indicated by target # while its localization is indicated by bullet • . Rank-2 solution found in 3 iterations (709) (1627a).

4.5. CONSTRAINING CARDINALITY

315

We numerically tested the foregoing technique for constraining rank on a wide range of problems including localization of randomized positions, stress ( 7.2.2.7.1), ball packing ( 5.4.2.3.4), and cardinality problems. We have had some success introducing the direction matrix inner-product (710) as a regularization term (a multiobjective optimization) whose purpose is to constrain rank, affine dimension, or cardinality:

§

4.5

§

Constraining cardinality

The convex iteration technique for constraining rank can be applied to cardinality problems:

4.5.1

nonnegative variable

Our goal is to reliably constrain rank in a semidefinite program. There is a direct analogy to linear programming that is simpler to present but, perhaps, more difficult to solve. In Optimization, that analogy is known as the cardinality problem. Consider a feasibility problem Ax = b , but with an upper bound k on cardinality kxk0 of a nonnegative solution x : for A ∈ Rm×n and vector b ∈ R(A) find x ∈ Rn subject to Ax = b xº0 kxk0 ≤ k

(713)

where kxk0 ≤ k means4.29 vector x has at most k nonzero entries; such a vector is presumed existent in the feasible set. Nonnegativity constraint x º 0 is analogous to positive semidefiniteness; the notation means vector x belongs to the nonnegative orthant Rn+ . Cardinality is quasiconcave on Rn+ n just as rank is quasiconcave on S+ . [57, 3.4.2] We propose that cardinality-constrained feasibility problem (713) can be equivalently expressed as a sequence of convex problems: for 0 ≤ k ≤ n−1

§

4.29

Although it is a metric (§5.2), cardinality kxk0 cannot be a norm (§3.3) because it is not positively homogeneous.

316

CHAPTER 4. SEMIDEFINITE PROGRAMMING hx , yi

minimize n x∈ R

subject to Ax = b xº0 n P

π(x⋆ )i =

hx⋆ , yi

minimize n y∈ R

i=k+1

subject to 0 ¹ y ¹ 1 yT1 = n − k

(146)

(495)

where π is the (nonincreasing) presorting function. This sequence is iterated until x⋆Ty ⋆ vanishes; id est, until desired cardinality is achieved. But this global convergence cannot always be guaranteed. Problem (495) is analogous to the rank constraint problem; (Ky Fan, confer p.295) N X

λ(G⋆ )i = minimize

i=k+1

W ∈ SN

hG⋆ , W i

(1627a)

subject to 0 ¹ W ¹ I tr W = N − k

Linear program (495) sums the n−k smallest entries from vector x . In context of problem (713), we want n−k entries of x to sum to zero; id est, we want a globally optimal objective x⋆Ty ⋆ to vanish: more generally, n X

π(|x⋆ |)i = h|x⋆ | , y ⋆ i , 0

(714)

i=k+1

defines global convergence for the iteration. Then n−k entries of x⋆ are themselves zero whenever their absolute sum is, and cardinality of x⋆ ∈ Rn is at most k . Optimal direction vector y ⋆ is defined as any nonnegative vector for which

(713)

find x ∈ Rn subject to Ax = b xº0 kxk0 ≤ k

minimize n ≡

x∈ R

hx , y ⋆ i

subject to Ax = b xº0

(146)

Existence of such a y ⋆ , whose nonzero entries are complementary to those of x⋆ , is obvious assuming existence of a feasible cardinality-k vector x⋆ .

4.5. CONSTRAINING CARDINALITY 4.5.1.1

317

direction vector interpretation

Vector y may be interpreted as a negative search direction; it points opposite to direction of movement of hyperplane {x | hx , yi = τ } in a minimization of real linear function hx , yi over the feasible set in linear program (146). (p.74) Direction vector y is not unique. The feasible set of direction vectors in (495) is the convex hull of all cardinality-(n−k) one-vectors; videlicet, conv{u ∈ Rn | card u = n−k , ui ∈ {0, 1}} = {a ∈ Rn | 1 º a º 0 , h1 , ai = n−k} (715) Set {u ∈ Rn | card u = n−k , ui ∈ {0, 1}} comprises the extreme points of set (715) which is a nonnegative hypercube slice. An optimal solution y to (495), that is an extreme point of its feasible set, is known in closed form: it has 1 in each entry corresponding to the n−k smallest entries of x⋆ and has 0 elsewhere. By Proposition 7.1.3.0.3, polar direction −y can be interpreted4.30 as pointing toward the set of all vectors having the same ordering as x⋆ and cardinality k . Consequently, for that particular direction n X

π(|x⋆ |)i = h|x⋆ | , yi

(716)

i=k+1

When y = 1, as in 1-norm minimization for example, then polar direction −y points directly at the origin (the cardinality-0 vector). 4.5.1.2

convergence can mean stalling

Convex iteration (146) (495) always converges to a locally optimal solution, a fixed point of possibly infeasible cardinality, by virtue of a monotonically nonincreasing real objective sequence. [240, 1.2] [39, 1.1] There can be no proof of global optimality, defined by (714). Constraining cardinality, solution to problem (713), can often be achieved but simple examples can be contrived that stall at a fixed point of infeasible cardinality; at a positive objective value hx⋆ , yi = τ > 0. The direction vector is then manipulated as countermeasure to steer out of local minima:

§

§

4.5.1.2.1 Example. Sparsest solution to Ax = b . [66] [108] (confer Example 4.5.1.5.1) Problem (645) has sparsest solution not easily 4.30

Convex iteration (146) (495) is not a projection method because there is no thresholding or discard of variable-vector x entries.

318

CHAPTER 4. SEMIDEFINITE PROGRAMMING

7 Donoho bound approximation x > 0 constraint

m/k 6

minimize kxk1 x

subject to Ax = b

(488)

5

m > k log2 (1+ n/k) minimize kxk1

4

x

subject to Ax = b (493) xº0

3

2

1

0

0.2

0.4

0.6

0.8

1

k/n

Figure 92: For Gaussian random matrix A ∈ Rm×n , graph illustrates Donoho/Tanner least lower bound on number of measurements m below which recovery of k-sparse n-length signal x by linear programming fails with overwhelming probability. Hard problems are below curve, but not the reverse; id est, failure above depends on proximity. Inequality demarcates approximation (dashed curve) empirically observed in [23]. Problems having nonnegativity constraint (dotted) are easier to solve. [112] [113]

4.5. CONSTRAINING CARDINALITY

319

recoverable by least 1-norm; id est, not by compressed sensing because of proximity to a theoretical lower bound on number of measurements m depicted in Figure 92: for A ∈ Rm×n

ˆ Given data from Example 4.2.3.1.1, for m = 3, n = 6, k = 1 

−1

1

8

1

1

0

2

8

−9

4

8

1 2 1 4

1 3 1 9

1 − 13 2 1 − 19 4

 A =  −3



 ,



 b =

1 1 2 1 4

  

(645)

the sparsest solution to classical linear equation Ax = b is x = e4 ∈ R6 (confer (658)). Although the sparsest solution is recoverable by inspection, we discern it instead by convex iteration; namely, by iterating problem sequence (146) (495) on page 316. From the numerical data given, cardinality kxk0 = 1 is expected. Iteration continues until xTy vanishes (to within some numerical precision); id est, until desired cardinality is achieved. But this comes not without a stall. Stalling, whose occurrence is sensitive to initial conditions of convex iteration, is a consequence of finding a local minimum of a multimodal objective hx , yi when regarded as simultaneously variable in x and y . Stalls are simply detected as fixed points x of infeasible cardinality, sometimes remedied by reinitializing direction vector y to a random positive state. Bolstered by success in breaking out of a stall, we then apply convex iteration to 22,000 randomized problems:

ˆ Given random data for m = 3, n = 6, k = 1, in Matlab notation

A = randn(3 , 6),

index = round(5∗rand(1)) + 1,

the sparsest solution x ∝ eindex

b = rand(1)∗ A(: , index) (717) is a scaled standard basis vector.

Without convex iteration or a nonnegativity constraint x º 0, rate of failure for this minimum cardinality problem Ax = b by 1-norm minimization of x is 22%. That failure rate drops to 6% with a nonnegativity constraint. If we then engage convex iteration, detect stalls, and randomly reinitialize the direction vector, failure rate drops to 0% but the amount of computation is approximately doubled. 2

320

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Stalling is not an inevitable behavior. For some problem types (beyond mere Ax = b), convex iteration succeeds nearly all the time. Here is a cardinality problem, with noise, whose statement is just a bit more intricate but easy to solve in a few convex iterations:

§

4.5.1.2.2 Example. Signal dropout. [111, 6.2] Signal dropout is an old problem; well studied from both an industrial and academic perspective. Essentially dropout means momentary loss or gap in a signal, while passing through some channel, caused by some man-made or natural phenomenon. The signal lost is assumed completely destroyed somehow. What remains within the time-gap is system or idle channel noise. The signal could be voice over Internet protocol (VoIP), for example, audio data from a compact disc (CD) or video data from a digital video disc (DVD), a television transmission over cable or the airwaves, or a typically ravaged cell phone communication, etcetera. Here we consider signal dropout in a discrete-time signal corrupted by additive white noise assumed uncorrelated to the signal. The linear channel is assumed to introduce no filtering. We create a discretized windowed signal for this example by positively combining k randomly chosen vectors from a discrete cosine transform (DCT) basis denoted Ψ ∈ Rn×n . Frequency increases, in the Fourier sense, from DC toward Nyquist as column index of basis Ψ increases. Otherwise, details of the basis are unimportant except for its orthogonality ΨT = Ψ−1 . Transmitted signal is denoted s = Ψz ∈ Rn

(718)

whose upper bound on DCT basis coefficient cardinality card z ≤ k is assumed known;4.31 hence a critical assumption: transmitted signal s is sparsely supported (k < n) on the DCT basis. It is further assumed that nonzero signal coefficients in vector z place each chosen basis vector above the noise floor. We also assume that the gap’s beginning and ending in time are precisely localized to within a sample; id est, index ℓ locates the last sample prior to the gap’s onset, while index n−ℓ+1 locates the first sample subsequent to the gap: for rectangularly windowed received signal g possessing a time-gap 4.31

This simplifies exposition, although it may be an unrealistic assumption in many applications.

4.5. CONSTRAINING CARDINALITY

321

loss and additive noise η ∈ Rn 

g =

s1:ℓ sn−ℓ+1:n

 + η1:ℓ ηℓ+1:n−ℓ  ∈ Rn + ηn−ℓ+1:n

(719)

The window is thereby centered on the gap and short enough so that the DCT spectrum of signal s can be assumed static over the window’s duration n . Signal to noise ratio within this window is defined °· ¸° ° ° s 1:ℓ ° ° ° sn−ℓ+1:n ° (720) SNR , 20 log kηk In absence of noise, knowing the signal DCT basis and having a good estimate of basis coefficient cardinality makes perfectly reconstructing gap-loss easy: it amounts to solving a linear system of equations and requires little or no optimization; with caveat, number of equations exceeds cardinality of signal representation (roughly ℓ ≥ k) with respect to DCT basis. But addition of a significant amount of noise η increases level of difficulty dramatically; a 1-norm based method of reducing cardinality, for example, almost always returns DCT basis coefficients numbering in excess of minimum cardinality. We speculate that is because signal cardinality 2ℓ becomes the predominant cardinality. DCT basis coefficient cardinality is an explicit constraint to the optimization problem we shall pose: In presence of noise, constraints equating reconstructed signal f to received signal g are not possible. We can instead formulate the dropout recovery problem as a best approximation: °· ¸° ° ° f1:ℓ − g1:ℓ ° ° minimize ° fn−ℓ+1:n − gn−ℓ+1:n ° x∈ Rn (721) subject to f = Ψ x xº0 card x ≤ k

§

We propose solving this nonconvex problem (721) by moving the cardinality constraint to the objective as a regularization term as explained in 4.5;

322

CHAPTER 4. SEMIDEFINITE PROGRAMMING

0.25

flatline and g

s+η

0.2 0.15

s+η

0.1

η

0.05

(a)

0 −0.05 −0.1 −0.15

dropout (s = 0)

−0.2 −0.25

0

100

200

300

400

500

0.25

f and g 0.2 0.15

f

0.1 0.05

(b)

0 −0.05 −0.1 −0.15 −0.2 −0.25

0

100

200

300

400

500

Figure 93: (a) Signal dropout in signal s corrupted by noise η (SNR = 10dB, g = s + η). Flatline indicates duration of signal dropout. (b) Reconstructed signal f (red) overlaid with corrupted signal g .

4.5. CONSTRAINING CARDINALITY

f−s

323

0.25 0.2 0.15 0.1 0.05

(a)

0 −0.05 −0.1 −0.15 −0.2 −0.25

0

100

200

300

400

500

0.25

f and s 0.2 0.15 0.1 0.05

(b)

0 −0.05 −0.1 −0.15 −0.2 −0.25

0

100

200

300

400

500

Figure 94: (a) Error signal power (reconstruction f less original noiseless signal s) is 36dB below s . (b) Original signal s overlaid with reconstruction f (red) from signal g having dropout plus noise.

324

CHAPTER 4. SEMIDEFINITE PROGRAMMING

id est, by iteration of two convex problems until convergence: °· ° f1:ℓ − g1:ℓ ° hx , yi + minimize ° n f x∈ R n−ℓ+1:n − gn−ℓ+1:n subject to f = Ψ x xº0

¸° ° ° °

(722)

and minimize n y∈ R

hx⋆ , yi

subject to 0 ¹ y ¹ 1 yT1 = n − k

(495)

Signal cardinality 2ℓ is implicit to the problem statement. When number of samples in the dropout region exceeds half the window size, then that deficient cardinality of signal remaining becomes a source of degradation to reconstruction in presence of noise. Thus, by observation, we divine a reconstruction rule for this signal dropout problem to attain good noise suppression: ℓ must exceed a maximum of cardinality bounds; 2ℓ ≥ max{2k , n/2}. Figure 93 and Figure 94 show one realization of this dropout problem. Original signal s is created by adding four (k = 4) randomly selected DCT basis vectors, from Ψ (n = 500 in this example), whose amplitudes are randomly selected from a uniform distribution above the noise floor; in the interval [10−10/20 , 1]. Then a 240-sample dropout is realized (ℓ = 130) and Gaussian noise η added to make corrupted signal g (from which a best approximation f will be made) having 10dB signal to noise ratio (720). The time gap contains much noise, as apparent from Figure 93a. But in only a few iterations (722) (495), original signal s is recovered with relative error power 36dB down; illustrated in Figure 94. Correct cardinality is also recovered (card x = card z) along with the basis vector indices used to make original signal s . Approximation error is due to DCT basis coefficient estimate error. When this experiment is repeated 1000 times on noisy signals averaging 10dB SNR, the correct cardinality and indices are recovered 99% of the time with average relative error power 30dB down. Without noise, we get perfect reconstruction in one iteration. [Matlab code: Wıκımization] 2

4.5. CONSTRAINING CARDINALITY

325

R3

c S = {s | s º 0 , 1Ts ≤ c} A

F

Figure 95: Line A intersecting two-dimensional (cardinality-2) face F of nonnegative simplex S from Figure 51 emerges at a (cardinality-1) vertex. S is first quadrant of 1-norm ball B1 from Figure 67. Kissing point (• on edge) is achieved as ball contracts or expands under optimization.

326

CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.5.1.3

Compressed sensing geometry with a nonnegative variable

It is well known that cardinality problem (713), on page 315, is easier to solve by linear programming when variable x is nonnegatively constrained than when not; e.g., Figure 68, Figure 92. We postulate a simple geometrical explanation: Figure 67 illustrates 1-norm ball B1 in R3 and affine subset A = {x | Ax = b}. Prototypical compressed sensing problem, for A ∈ Rm×n minimize x

kxk1

subject to Ax = b

(488)

is solved when the 1-norm ball B1 kisses affine subset A .

ˆ If variable x is constrained to the nonnegative orthant minimize x

kxk1

subject to Ax = b xº0

(493)

then 1-norm ball B1 (Figure 67) becomes simplex S in Figure 95.

Whereas 1-norm ball B1 has only six vertices in R3 corresponding to cardinality-1 solutions, simplex S has three edges (along the Cartesian axes) containing an infinity of cardinality-1 solutions. And whereas B1 has twelve edges containing cardinality-2 solutions, S has three (out of total four) facets constituting cardinality-2 solutions. In other words, likelihood of a low-cardinality solution is higher by kissing nonnegative simplex S (493) than by kissing 1-norm ball B1 (488) because facial dimension (corresponding to given cardinality) is higher in S . Empirically, this conclusion also holds in other Euclidean dimensions (Figure 68, Figure 92). 4.5.1.4

cardinality-1 compressed sensing problem always solvable

In the special case of cardinality-1 solution to prototypical compressed sensing problem (493), there is a geometrical interpretation that leads to an algorithm more efficient than convex iteration. Figure 95 shows a vertex-solution to problem (493) when desired cardinality is 1. But first-quadrant S of 1-norm ball B1 does not kiss line A ; which requires explanation: Under the assumption that nonnegative cardinality-1 solutions exist in feasible set A , it so happens,

4.5. CONSTRAINING CARDINALITY

327

ˆ columns of measurement matrix A , corresponding to any particular solution (to (493)) of cardinality greater than 1, may be deprecated and the problem solved again with those columns missing. Such columns are recursively removed from A until a cardinality-1 solution is found.

Either a solution to problem (493) is cardinality-1 or column indices of A , corresponding to a higher cardinality solution, do not intersect that index corresponding to the cardinality-1 solution. When problem (493) is first solved, in the example of Figure 95, solution is cardinality-2 at the kissing point • on the indicated edge of simplex S . Imagining that the corresponding cardinality-2 face F has been removed, then the simplex collapses to a line segment along one of the Cartesian axes. When that line segment kisses line A , then the cardinality-1 vertex-solution illustrated has been found. A similar argument holds for any orientation of line A and point of entry to simplex S . Because signed compressed sensing problem (488) can be equivalently expressed in a nonnegative variable, as we learned in Example 3.3.0.0.1 (p.220), and because a cardinality-1 constraint in (488) transforms to a cardinality-1 constraint in its nonnegative equivalent (492), then this cardinality-1 recursive reconstruction algorithm continues to hold for a signed variable as in (488). This cardinality-1 reconstruction algorithm also holds more generally when affine subset A has any higher dimension n−m . Although it is more efficient to search over the columns of matrix A for a cardinality-1 solution known a priori to exist, a combinatorial approach, tables are turned when cardinality exceeds 1 : 4.5.1.5

cardinality-k compressed sensing geometric presolver

§

This idea of deprecating columns has theoretical basis from convex cone theory. ( 2.13.4.3) Removing columns (and rows)4.32 from A ∈ Rm×n in a linear program like (493) ( 3.3) is known in the industry as presolving;4.33 the elimination of redundant constraints and identically zero variables prior to numerical optimal solution. Two interpretations of the constraints from problem (493) are realized in Figure 96. Assuming that a cardinality-k solution exists and matrix A 4.32

§

Rows of matrix A are removed based upon linear dependence. Assuming b ∈ R(A) , corresponding entries of vector b may also be removed without loss of generality. 4.33 The conic technique proposed here can exceed performance of the best industrial presolvers in terms of number of columns removed, but not in execution time.

328

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Rn+

Rn

A

(a)

P 0

Rm

b K

(b)

0 Figure 96: Constraint interpretations: (a) Halfspace-description of feasible set in problem (493) is a polyhedron P formed by intersection of nonnegative orthant Rn+ with hyperplanes A prescribed by equality constraint. (Drawing by Pedro S´anchez) (b) Vertex-description of feasible set in problem (493): point b belongs to polyhedral cone K = {Ax | x º 0}. Number of extreme directions in K may exceed dimensionality of ambient space.

4.5. CONSTRAINING CARDINALITY

329

describes a proper polyhedral cone K = {Ax | x º 0} , columns are removed from A if they do not belong to the smallest face of K containing vector b . Columns so removed correspond to 0-entries in variable vector x . Benefit accrues when vector b does not belong to the interior of K ; there would be no columns to remove were b ∈ int K since the smallest face becomes cone K itself (Example 4.5.1.5.1). Were b an extreme direction, at the other end of our spectrum, then the smallest face is an edge that is a ray containing b ; this is a geometrical description of a cardinality-1 case where all columns, save one, would be removed from A . When vector b resides in a face of K that is not K itself, benefit is realized as a reduction in computational intensity for a problem of smaller dimension. Number of columns removed depends completely on particular geometry of a given problem; particularly, location of b within K . There are always exactly n linear feasibility problems to solve in order to discern generators of the smallest face of K containing b ; the topic of 2.13.4.3. So comparison of computational intensity for this conic approach µ ¶ n pits combinatorial complexity ∝ a binomial coefficient versus numerical k solution of the presolved problem plus n more linear feasibility problems.

§

4.5.1.5.1 Example. Cardinality-2 solution to Ax = b . (confer Example 4.5.1.2.1) Again taking data from Example 4.2.3.1.1, for m = 3, n = 6, k = 2 (A ∈ Rm×n , desired cardinality of x is k) 

−1

 A =  −3 −9

1

8

1

1

0

2

8

4

8

1 2 1 4

1 3 1 9

1 − 31 2 1 − 91 4



 ,



 b =

1 1 2 1 4

  

(645)

§

proper cone K = {Ax | x º 0} is pointed as proven by method of 2.12.2.2. A cardinality-2 solution is known to exist; sum of the last two columns of matrix A . Generators of the smallest face containing vector b , found by the method in Example 2.13.4.3.1, comprise the entire A matrix because b ∈ int K ( 2.13.4.2.4). Geometry of this particular problem does not permit number of generators to be reduced below n by discerning the smallest face. But circumstances would change dramatically given new b instead on ∂K . 2

§

330

CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.5.2

constraining cardinality of signed variable

Now consider a feasibility problem equivalent to the classical problem from linear algebra Ax = b , but with an upper bound k on cardinality kxk0 : for vector b ∈ R(A) find x ∈ Rn subject to Ax = b (723) kxk0 ≤ k

§

where kxk0 ≤ k means vector x has at most k nonzero entries; such a vector is presumed existent in the feasible set. Convex iteration ( 4.5.1) works with a nonnegative variable; so absolute value |x| is needed here. We propose that nonconvex problem (723) can be equivalently written as a sequence of convex problems that move the cardinality constraint to the objective: minimize n x∈ R

h|x| , yi

subject to Ax = b

minimize

x∈ Rn , t∈ Rn

≡

ht , y + ε1i

(724)

subject to Ax = b −t ¹ x ¹ t

minimize n y∈ R

ht⋆ , y + ε1i

subject to 0 ¹ y ¹ 1 yT1 = n − k

(495)

where ε is a relatively small positive constant. This sequence is iterated until a direction vector y is found that makes |x⋆ |T y ⋆ vanish. The term ht , ε1i in (724) is necessary to determine absolute value |x⋆ | = t⋆ ( 3.3) because vector y can have zero-valued entries. By initializing y to (1−ε)1, the first iteration of problem (724) is a 1-norm problem (484); id est,

§

minimize ht , 1i n n

x∈ R , t∈ R

subject to Ax = b −t ¹ x ¹ t

≡

minimize n x∈ R

kxk1

subject to Ax = b

(488)

Subsequent iterations of problem (724) engaging cardinality term ht , yi can be interpreted as corrections to this 1-norm problem leading to a 0-norm solution; vector y can be interpreted as a direction of search.

4.5. CONSTRAINING CARDINALITY 4.5.2.1

331

local convergence

§

As before ( 4.5.1.2), convex iteration (724) (495) always converges to a locally optimal solution; a fixed point of possibly infeasible cardinality.

4.5.2.2

simple variations on a signed variable

Several useful equivalents to linear programs (724) (495) are easily devised, but their geometrical interpretation is not as apparent: e.g., equivalent in the limit ε → 0+ minimize ht , yi n n x∈ R , t∈ R

(725)

subject to Ax = b −t ¹ x ¹ t minimize n y∈ R

h|x⋆ | , yi

subject to 0 ¹ y ¹ 1 yT1 = n − k

(495)

We get another equivalent to linear programs (724) (495), in the limit, by interpreting problem (488) as infimum to a vertex-description of the 1-norm ball (Figure 67, Example 3.3.0.0.1, confer (487)): ha , yi minimize 2n a∈ R

(726)

subject to [ A −A ]a = b aº0 minimize 2n y∈ R

ha⋆ , yi

subject to 0 ¹ y ¹ 1 y T 1 = 2n − k

(495)

where x⋆ = [ I −I ]a⋆ ; from which it may be construed that any vector 1-norm minimization problem has equivalent expression in a nonnegative variable.

332

CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.6

Cardinality and rank constraint examples

§

§

4.6.0.0.1 Example. Projection on ellipsoid boundary. [48] [135, 5.1] [230, 2] Consider classical linear equation Ax = b but with constraint on norm of solution x , given matrices C , fat A , and vector b ∈ R(A) find x ∈ RN subject to Ax = b kC xk = 1

(727)

The set {x | kC xk = 1} (2) describes an ellipsoid boundary (Figure 13). This is a nonconvex problem because solution is constrained to that boundary. Assign G=

·

Cx 1

¸

[ xTC T 1 ]

=

·

X Cx T T x C 1

§

¸

,

·

C xxTC T C x xTC T 1

¸

∈ SN +1 (728)

Any rank-1 solution must have this form. ( B.1.0.2) Ellipsoidally constrained feasibility problem (727) is equivalent to: find

X∈ SN

x ∈ RN

subject to Ax = b ¸ · X Cx (º 0) G= xTC T 1 rank G = 1 tr X = 1

(729)

This is transformed to an equivalent convex problem by moving the rank constraint to the objective: We iterate solution of minimize

hG , Y i

subject to

Ax = b ¸ · X Cx º0 G= xTC T 1 tr X = 1

X∈ SN , x∈RN

(730)

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

333

with minimize Y ∈ SN +1

hG⋆ , Y i

(731)

subject to 0 ¹ Y ¹ I tr Y = N

until convergence. Direction matrix Y ∈ SN +1 , initially 0, controls rank. +1 ( A.6) provides a (1627a) Singular value decomposition G⋆ = U ΣQT ∈ SN + T new direction matrix Y = U (: , 2 : N + 1)U (: , 2 : N + 1) that optimally solves (731) at each iteration. An optimal solution to (727) is thereby found in a few iterations, making convex problem (730) its equivalent. It remains possible for the iteration to stall; were a rank-1 G matrix not found. In that case, the current search direction is momentarily reversed with an added randomized element:

§

Y = −U(: , 2 : N + 1) ∗ (U(: , 2 : N + 1)′ + randn(N , 1) ∗ U(: , 1)′ )

(732)

in Matlab notation. This heuristic is quite effective for problem (727) which is exceptionally easy to solve by convex iteration. When b ∈ / R(A) then problem (727) must be restated as a projection: minimize kAx − bk x∈RN

subject to kC xk = 1

(733)

This is a projection of point b on an ellipsoid boundary because any affine transformation of an ellipsoid remains an ellipsoid. Problem (730) in turn becomes minimize

X∈ SN , x∈RN

subject to

hG , Y i + kAx − bk · ¸ X Cx G= º0 xTC T 1 tr X = 1

(734)

We iterate this with calculation (731) of direction matrix Y as before until a rank-1 G matrix is found. 2

334

CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.6.0.0.2 Example. Procrustes problem. [48] n×p is Example 4.6.0.0.1 is extensible. An orthonormal matrix Q ∈ R T completely characterized by Q Q = I . Consider the particular case Q = [ x y ] ∈ Rn×2 as variable to a Procrustes problem ( C.3): given A ∈ Rm×n and B ∈ Rm×2

§

minimize kAQ − BkF n×2 Q∈ R

subject to QTQ = I

(735)

which is nonconvex. By vectorizing matrix Q we can make the assignment:      x [ xT y T 1 ] xxT xy T x X Z x G = y  =  Z T Y y  ,  yxT yy T y ∈ S2n+1 (736) xT y T 1 xT y T 1 1 

Now orthonormal Procrustes problem (735) can be equivalently restated: minimize

X , Y ∈S, Z , x, y

subject to

kA[ x y ] − BkF   X Z x G =  Z T Y y (º 0) xT y T 1 rank G = 1 tr X = 1 tr Y = 1 tr Z = 0

(737)

To solve this, we form the convex problem sequence: kA[ x y ] − BkF  X Z  subject to G = Z T Y xT y T tr X = 1 tr Y = 1 tr Z = 0 minimize

X , Y , Z , x, y

+ hG , W i  x y º 0 1

(738)

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES and minimize 2n+1 W∈S

hG⋆ , W i

335

(739)

subject to 0 ¹ W ¹ I tr W = 2n

which has an optimal solution W that is known in closed form (p.631). These two problems are iterated until convergence and a rank-1 G matrix is found. A good initial value for direction matrix W is 0. Optimal Q⋆ equals [ x⋆ y ⋆ ]. Numerically, this Procrustes problem is easy to solve; a solution seems always to be found in one or few iterations. This problem formulation is extensible, of course, to orthogonal (square) matrices Q . 2 4.6.0.0.3 Example. Combinatorial Procrustes problem. In case A , B ∈ Rn , when vector A = ΞB is known to be a permutation of vector B , solution to orthogonal Procrustes problem minimize n×n X∈ R

kA − X BkF

subject to X T = X −1

(1639)

§

is not necessarily a permutation matrix Ξ even though an optimal objective value of 0 is found by the known analytical solution ( C.3). The simplest method of solution finds permutation matrix X ⋆ = Ξ simply by sorting vector B with respect to A . Instead of sorting, we design two different convex problems each of whose optimal solution is a permutation matrix: one design is based on rank constraint, the other on cardinality. Because permutation matrices are sparse by definition, we depart from a traditional Procrustes problem by instead demanding a vector 1-norm which is known to produce solutions more sparse than Frobenius norm. There are two principal facts exploited by the first convex iteration design ( 4.4.1) we propose. Permutation matrices Ξ constitute:

§

1) the set of all nonnegative orthogonal matrices, 2) all points extreme to the polyhedron (94) of doubly stochastic matrices. That means:

336

CHAPTER 4. SEMIDEFINITE PROGRAMMING

{X(: , i) | X(: , i)T X(: , i) = 1}

{X(: , i) | 1T X(: , i) = 1} Figure 97: Permutation matrix i th column-sum and column-norm constraint in two dimensions when rank constraint is satisfied. Optimal solutions reside at intersection of hyperplane with unit circle.

1) norm of each row and column is 1 ,4.34 kΞ(: , i)k = 1 ,

kΞ(i , :)k = 1 ,

i=1 . . . n

(740)

§

2) sum of each nonnegative row and column is 1, ( 2.3.2.0.4) ΞT 1 = 1 ,

Ξ1 = 1 ,

Ξ≥0

(741)

solution via rank constraint The idea is to individually constrain each column of variable matrix X to have unity norm. Matrix X must also belong to that polyhedron, (94) in the nonnegative orthant, implied by constraints (741); so each row-sum and 4.34

This fact would be superfluous were the objective of minimization linear, because the permutation matrices reside at the extreme points of a polyhedron (94) implied by (741). But as posed, only either rows or columns need be constrained to unit norm because matrix orthogonality implies transpose orthogonality. (§B.5.1) Absence of vanishing inner product constraints that help define orthogonality, like tr Z = 0 from Example 4.6.0.0.2, is a consequence of nonnegativity; id est, the only orthogonal matrices having exclusively nonnegative entries are permutations of the identity.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

337

column-sum of X must also be unity. It is this combination of nonnegativity, sum, and sum square constraints that extracts the permutation matrices: (Figure 97) given nonzero vectors A , B n P minimize kA − X Bk + w hGi , Wi i 1 X∈ Rn×n, Gi ∈ Sn+1 i=1 · ¸ Gi (1 : n , 1 : n) X(: , i) Gi = º X(: , i)T 1 subject to tr Gi = 2

X T1 = 1 X1 = 1 X≥0

0

 

,

i=1 . . . n



(742)

where w ≈ 10 positively weights the rank regularization term. Optimal solutions G⋆i are key to finding direction matrices Wi for the next iteration of semidefinite programs (742) (743): hG⋆i , Wi i

minimize n+1 Wi ∈ S

   

subject to 0 ¹ Wi ¹ I  ,  tr Wi = n 

i=1 . . . n

(743)

Direction matrices thus found lead toward rank-1 matrices G⋆i on subsequent iterations. Constraint on trace of G⋆i normalizes the i th column of X ⋆ to unity because (confer p.397) G⋆i

=

·

¸ X ⋆ (: , i) [ X ⋆ (: , i)T 1 ] 1

(744)

at convergence. Binary-valued X ⋆ column entries result from the further sum constraint X1 = 1. Columnar orthogonality is a consequence of the further transpose-sum constraint X T 1 = 1 in conjunction with nonnegativity constraint X ≥ 0 ; but we leave proof of orthogonality an exercise. The optimal objective value is 0 for both semidefinite programs when vectors A and B are related by permutation. In any case, optimal solution X ⋆ becomes a permutation matrix Ξ .

338

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Because there are n direction matrices Wi to find, it can be advantageous to invoke a known closed-form solution for each from page 631. What makes this combinatorial problem more tractable are relatively small semidefinite constraints in (742). (confer (738)) When a permutation A of vector B exists, number of iterations can be as small as 1. But this combinatorial Procrustes problem can be made even more challenging when vector A has repeated entries. solution via cardinality constraint Now the idea is to force solution at a vertex of permutation polyhedron (94) by finding a solution of desired sparsity. Because permutation matrix X is n-sparse by assumption, this combinatorial Procrustes problem may instead be formulated as a compressed sensing problem with convex iteration on cardinality of vectorized X ( 4.5.1): given nonzero vectors A , B

§

minimize n×n X∈ R

kA − X Bk1 + whX , Y i

subject to X T 1 = 1 X1 = 1 X≥0

(745)

where direction vector Y is an optimal solution to minimize n×n Y ∈R

hX ⋆ , Y i

subject to 0 ≤ Y ≤ 1 1T Y 1 = n2 − n

(495)

each a linear program. In this circumstance, use of closed-form solution for direction vector Y is discouraged. When vector A is a permutation of B , both linear programs have objectives that converge to 0. When vectors A and B are permutations and no entries of A are repeated, optimal solution X ⋆ can be found as soon as the first iteration. In any case, X ⋆ = Ξ is a permutation matrix. 2 4.6.0.0.4 Exercise. Combinatorial Procrustes constraints. Assume that the objective of semidefinite program (742) is 0 at optimality. Prove that the constraints in program (742) are necessary and sufficient to produce a permutation matrix as optimal solution. Alternatively and equivalently, prove those constraints necessary and sufficient to optimally produce a nonnegative orthogonal matrix. H

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

339

4.6.0.0.5 Example. Tractable polynomial constraint. The ability to handle rank constraints makes polynomial constraints (generally nonconvex) transformable to convex constraints. All optimization problems having polynomial objective and polynomial constraints can be reformulated as a semidefinite program with a rank-1 constraint. [265] Suppose we require 3 + 2x − xy ≤ 0 (746) Identify   2 x [x y 1] x xy    y xy y2 G= = x y 1 

 x y  ∈ S3 1

(747)

Then nonconvex polynomial constraint (746) is equivalent to constraint set tr(GA) ≤ 0 G33 = 1 (G º 0) rank G = 1

(748)

§

with direct correspondence to sense of trace inequality where G is assumed symmetric ( B.1.0.2) and   0 − 21 1 0 0  ∈ S3 A =  − 21 (749) 1 0 3

§

Then the method of convex iteration from 4.4.1 is applied to implement the rank constraint. 2 4.6.0.0.6 Exercise. Binary Pythagorean theorem. Technique of Example 4.6.0.0.5 is extensible to any quadratic constraint; e.g., xTA x + 2bTx + c ≤ 0 , xTA x + 2bTx + c ≥ 0 , and xTA x + 2bTx + c = 0. Write a rank constrained semidefinite program to solve (Figure 97) ½ x+y =1 (750) x2 + y 2 = 1 whose feasible set is not connected. Implement this system in cvx [153] by convex iteration. H

340

CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.6.0.0.7 Example. High-order polynomials. Consider nonconvex problem from Canadian Mathematical Olympiad 1999: find

x , y , z∈ R

x, y, z

subject to x2 y + y 2 z + z 2 x = x+y+z =1 x, y, z ≥ 0

22 33

(751)

2

We wish to solve for what is known to be a tight upper bound 323 on the constrained polynomial x2 y + y 2 z + z 2 x by transformation to a rank constrained semidefinite program; identify    2  x [x y z 1] x xy zx x  y   xy y 2 yz y  4    G = = (752)  z   zx yz z 2 z  ∈ S 1 x y z 1   4  2  x x2 y 2 z 2 x2 x3 x2 y zx2 x2 x [ x2 y 2 z 2 x y z 1 ]  x2 y 2 y 4 y 2 z 2 xy 2 y 3 y 2 z y 2   y2    2 2 2 2  2  4 2 2 3 2   z x y z  z  z z x yz z z   3    xy 2 z 2 x x2 xy zx x  ∈ S7 = X=   x2  x   xy  y  y3 yz 2 xy y 2 yz y      2 2  zx  z  y z z3 zx yz z 2 z  x2 y2 z2 x y z 1 1 (753) then apply convex iteration ( 4.4.1) to implement rank constraints:

§

find

A , C∈ S , b

b 2

subject to tr(XE) = 323 · ¸ A b G= (º 0) bT 1 ¸ ·  δ(A) C (º 0) b X = £ ¤ T T δ(A) b 1 1T b = 1 bº0 rank G = 1 rank X = 1

(754)

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

341

where 

    E =    

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 1 0 0 0

0 0 1 0 0 0 0

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 0 0 0 0 0



   1  ∈ S7 2   

(755)

(Matlab code on Wıκımization) Positive semidefiniteness is optional only when rank-1 constraints are explicit by Theorem A.3.1.0.7. Optimal solution 2 to problem (751) is not unique: (x , y , z) = (0 , 32 , 13 ).

§

4.6.0.0.8 Example. Boolean vector feasible to Ax ¹ b . (confer 4.2.3.1.1) Now we consider solution to a discrete problem whose only known analytical method of solution is combinatorial in complexity: given A ∈ RM ×N and b ∈ RM find x ∈ RN (756) subject to Ax ¹ b T δ(xx ) = 1 This nonconvex problem demands a Boolean solution [ xi = ±1, i = 1 . . . N ]. Assign a rank-1 matrix of variables; symmetric variable matrix X and solution vector x : ¸ ¸ · · · ¸ T xxT x X x x [x 1] ∈ SN +1 (757) , = G= xT 1 xT 1 1 Then design an equivalent semidefinite feasibility problem to find a Boolean solution to Ax ¹ b : find

X∈ SN

x ∈ RN

subject to Ax ¹ b · ¸ X x G= (º 0) xT 1 rank G = 1 δ(X) = 1

(758)

342

CHAPTER 4. SEMIDEFINITE PROGRAMMING

where x⋆i ∈ {−1, 1} , i = 1 . . . N . The two variables X and x are made dependent via their assignment to rank-1 matrix G . By (1551), an optimal rank-1 matrix G⋆ must take the form (757). As before, we regularize the rank constraint by introducing a direction matrix Y into the objective: minimize

hG , Y i

subject to

Ax ¹ b · ¸ X x º0 G= xT 1 δ(X) = 1

X∈ SN , x∈RN

(759)

Solution of this semidefinite program is iterated with calculation of the direction matrix Y from semidefinite program (731). At convergence, in the sense (700), convex problem (759) becomes equivalent to nonconvex Boolean problem (756). Direction matrix Y can be an orthogonal projector having closed-form expression, by (1627a), although convex iteration is not a projection method. Given randomized data A and b for a large problem, we find that stalling becomes likely (convergence of the iteration to a positive objective hG⋆ , Y i). To overcome this behavior, we introduce a heuristic into the implementation on Wıκımization that momentarily reverses direction of search (like (732)) upon stall detection. We find that rate of convergence can be sped significantly by detecting stalls early. 2 4.6.0.0.9 Example. Variable-vector normalization. Suppose, within some convex optimization problem, we want vector variables x , y ∈ RN constrained by a nonconvex equality: xkyk = y

(760)

id est, kxk = 1 and x points in the same direction as y 6= 0 ; e.g., minimize x, y

f (x , y)

subject to (x , y) ∈ C xkyk = y

(761)

where f is some convex function and C is some convex set. We can realize the nonconvex equality by constraining rank and adding a regularization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

343

term to the objective. Make the assignment:    T T    x [x y 1] xxT xy T x X Z x G = y  =  Z Y y  ,  yxT yy T y ∈ S2N +1 (762) xT y T 1 xT y T 1 1

where X , Y ∈ SN , also Z ∈ SN [sic] . Any rank-1 solution must take the form of (762). ( B.1) The problem statement equivalent to (761) is then written minimize f (x , y) + kX − Y kF

§

X , Y ∈S, Z , x, y

subject to

(x , y) ∈ C   X Z x G =  Z Y y (º 0) xT y T 1 rank G = 1 tr(X) = 1 δ(Z) º 0

(763)

The trace constraint on X normalizes vector x while the diagonal constraint on Z maintains sign between respective entries of x and y . Regularization term kX − Y kF then makes x equal to y to within a real scalar; ( C.2.0.0.2) in this case, a positive scalar. To make this program solvable by convex iteration, as explained in Example 4.4.1.2.2 and other previous examples, we move the rank constraint to the objective

§

minimize

X , Y , Z , x, y

f (x , y) + kX − Y kF + hG , W i

subject to (x , y) ∈ C   X Z x G =  Z Y y º 0 xT y T 1 tr(X) = 1 δ(Z) º 0

(764)

by introducing a direction matrix W found from (1627a): minimize W ∈ S 2N +1

hG⋆ , W i

subject to 0 ¹ W ¹ I tr W = 2N

(765)

344

CHAPTER 4. SEMIDEFINITE PROGRAMMING

This semidefinite program has an optimal solution that is known in closed form. Iteration (764) (765) terminates when rank G = 1 and linear regularization hG , W i vanishes to within some numerical tolerance in (764); typically, in two iterations. If function f competes too much with the regularization, positively weighting each regularization term will become required. At convergence, problem (764) becomes a convex equivalent to 2 the original nonconvex problem (761). 4.6.0.0.10

Example.

[103]

fast max cut.

Let Γ be an n-node graph, and let the arcs (i , j) of the graph be associated with [ ] weights aij . The problem is to find a cut of the largest possible weight, i.e., to partition the set of nodes into two parts S , S ′ in such a way that the total weight of all arcs linking S and S ′ (i.e., with one incident node in S and the other one in S ′ [Figure 98]) is as large as possible. [32, 4.3.3]

§

Literature on the max cut problem is vast because this problem has elegant primal and dual formulation, its solution is very difficult, and there exist many commercial applications; e.g., semiconductor design [115], quantum computing [362]. Our purpose here is to demonstrate how iteration of two simple convex problems can quickly converge to an optimal solution of the max cut problem with a 98% success rate, on average.4.35 max cut is stated: P maximize aij (1 − xi xj ) 21 x 1≤i<j≤n (766) subject to δ(xxT ) = 1 where [aij ] are real arc weights, and binary vector x = [xi ] ∈ Rn corresponds to the n nodes; specifically, node i ∈ S ⇔ xi = 1 node i ∈ S ′ ⇔ xi = −1

(767)

If nodes i and j have the same binary value xi and xj , then they belong to the same partition and contribute nothing to the cut. Arc (i , j) traverses the cut, otherwise, adding its weight aij to the cut. 4.35

We term our solution to max cut fast because we sacrifice a little accuracy to achieve speed; id est, only about two or three convex iterations, achieved by heavily weighting a rank regularization term.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

2

3

345

S

4

1 CUT

5 16 S′ 6

1 -1

15

-3 4

7

-3

5 2 1

14 13

8 9 10 12

11

Figure 98: A cut partitions nodes {i = 1 . . . 16} of this graph into S and S ′ . Linear arcs have circled weights. The problem is to find a cut maximizing total weight of all arcs linking partitions made by the cut.

346

CHAPTER 4. SEMIDEFINITE PROGRAMMING

max cut statement (766) is the same as, for A = [aij ] ∈ S n maximize x

1 h11T − 4 T

xxT , Ai

(768)

subject to δ(xx ) = 1 Because of Boolean assumption δ(xxT ) = 1 h11T − xxT , Ai = hxxT , δ(A1) − Ai

(769)

so problem (768) is the same as maximize x

1 hxxT , 4 T

δ(A1) − Ai

(770)

subject to δ(xx ) = 1

This max cut problem is combinatorial (nonconvex). Because an estimate of upper bound to max cut is needed to ascertain convergence when vector x has large dimension, we digress to derive the dual problem: Directly from (770), its Lagrangian is [57, 5.1.5] (1362)

§

L(x , ν) = 41 hxxT , δ(A1) − Ai + hν , δ(xxT ) − 1i = 41 hxxT , δ(A1) − Ai + hδ(ν) , xxT i − hν , 1i = 41 hxxT , δ(A1 + 4ν) − Ai − hν , 1i

(771)

where quadratic xT(δ(A1 + 4ν)−A)x has supremum 0 if δ(A1 + 4ν)−A is negative semidefinite, and has supremum ∞ otherwise. The finite supremum of dual function ½ −hν , 1i , A − δ(A1 + 4ν) º 0 g(ν) = sup L(x , ν) = (772) ∞ otherwise x is chosen to be the objective of minimization to dual (convex) problem minimize ν

−ν T 1

subject to A − δ(A1 + 4ν) º 0

(773)

whose optimal value provides a least upper bound to max cut, but is not tight ( 14 hxxT , δ(A1)−Ai < g(ν) , duality gap is nonzero). [144] In fact, we find that the bound’s variance with problem instance is too large to be useful for this problem; thus ending our digression.4.36 4.36

Taking the dual of dual problem (773) would provide (774) but without the rank constraint. [137] Dual of a dual of even a convex primal problem is not necessarily the same primal problem; although, optimal solution of one can be obtained from the other.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

347

To transform max cut to its convex equivalent, first define X = xxT ∈ S n

(778)

then max cut (770) becomes 1 hX 4

maximize n X∈ S

, δ(A1) − Ai

subject to δ(X) = 1 (X º 0) rank X = 1

(774)

whose rank constraint can be regularized as in maximize n X∈ S

1 hX 4

, δ(A1) − Ai − whX , W i

(775)

subject to δ(X) = 1 Xº0

where w ≈ 1000 is a nonnegative fixed weight, and W is a direction matrix determined from n X i=2

λ(X ⋆ )i = minimize n W∈S

hX ⋆ , W i

(1627a)

subject to 0 ¹ W ¹ I

tr W = n − 1

which has an optimal solution that is known in closed form. These two problems (775) and (1627a) are iterated until convergence as defined on page 295. Because convex problem statement (775) is so elegant, it is numerically solvable for large binary vectors within reasonable time.4.37 To test our convex iterative method, we compare an optimal convex result to an actual solution of the max cut problem found by performing a brute force combinatorial search of (770)4.38 for a tight upper bound. Search-time limits binary vector lengths to 24 bits (about five days CPU time). Accuracy 4.37

We solved for a length-250 binary vector in only a few minutes and convex iterations on a Dell Precision computer, model M90. 4.38 more computationally intensive than the proposed convex iteration by many orders of magnitude. Solving max cut by searching over all binary vectors of length 100, for example, would occupy a contemporary supercomputer for a million years.

348

CHAPTER 4. SEMIDEFINITE PROGRAMMING

obtained, 98%, is independent of binary vector length (12, 13, 20, 24) when averaged over more than 231 problem instances including planar, randomized, and toroidal graphs.4.39 When failure occurred, large and small errors were manifest. That same 98% average accuracy is presumed maintained when binary vector length is further increased. A Matlab program is provided on Wıκımization. 2 4.6.0.0.11 Example. Cardinality/rank problem. d’Aspremont, El Ghaoui, Jordan, & Lanckriet [90] propose approximating a positive semidefinite matrix A ∈ SN + by a rank-one matrix having constraint on cardinality c : for 0 < c < N minimize z

kA − zz T kF

subject to card z ≤ c

(776)

which, they explain, is a hard problem equivalent to maximize xTA x x

subject to kxk = 1 card x ≤ c

(777)

X , xxT ∈ SN

(778)

√ where z , λ x and where optimal solution x⋆ is a principal eigenvector (1621) ( A.5) of A and λ = x⋆TA x⋆ is the principal eigenvalue [146, p.331] when c is true cardinality of that eigenvector. This is principal component analysis with a cardinality constraint which controls solution sparsity. Define the matrix variable

§

whose desired rank is 1, and whose desired diagonal cardinality card δ(X) ≡ card x 4.39

(779)

Existence of a polynomial-time approximation to max cut with accuracy better than 94.11% would refute proof of NP-hardness, which H˚ astad believes to be highly unlikely. [167, thm.8.2] [168]

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

349

is equivalent to cardinality c of vector x . Then we can transform cardinality problem (777) to an equivalent problem in new variable X :4.40 maximize hX , Ai X∈ SN

subject to hX , I i = 1 (X º 0) rank X = 1 card δ(X) ≤ c

(780)

We transform problem (780) to an equivalent convex problem by introducing two direction matrices into regularization terms: W to achieve desired cardinality card δ(X) , and Y to find an approximating rank-one matrix X : maximize hX , A − w1 Y i − w2 hδ(X) , δ(W )i X∈ SN

subject to hX , I i = 1 Xº0

(781)

where w1 and w2 are positive scalars respectively weighting tr(XY ) and δ(X)T δ(W ) just enough to insure that they vanish to within some numerical precision, where direction matrix Y is an optimal solution to semidefinite program minimize hX ⋆ , Y i Y ∈ SN

subject to 0 ¹ Y ¹ I tr Y = N − 1

(782)

and where diagonal direction matrix W ∈ SN optimally solves linear program minimize 2 W = δ (W )

hδ(X ⋆ ) , δ(W )i

subject to 0 ¹ δ(W ) ¹ 1 tr W = N − c

(783)

Both direction matrix programs are derived from (1627a) whose analytical solution is known but is not necessarily unique. We emphasize (confer p.295): 4.40

A semidefiniteness constraint X º 0 is not required, theoretically, because positive semidefiniteness of a rank-1 matrix is enforced by symmetry. (Theorem A.3.1.0.7)

350

CHAPTER 4. SEMIDEFINITE PROGRAMMING

phantom(256)

Figure 99: Shepp-Logan phantom from Matlab image processing toolbox.

because this iteration (781) (782) (783) (initial Y, W = 0) is not a projection method, success relies on existence of matrices in the feasible set of (781) having desired rank and diagonal cardinality. In particular, the feasible set of convex problem (781) is a Fantope (86) whose extreme points constitute the set of all normalized rank-one matrices; among those are found rank-one matrices of any desired diagonal cardinality. Convex problem (781) is neither a relaxation of cardinality problem (777); instead, problem (781) is a convex equivalent to (777) at convergence of iteration (781) (782) (783). Because the feasible set of convex problem (781) contains all normalized ( B.1) symmetric rank-one matrices of every nonzero diagonal cardinality, a constraint too low or high in cardinality c will not prevent solution. An optimal rank-one solution X ⋆ , whose diagonal cardinality is equal to cardinality of a principal eigenvector of matrix A , will produce the lowest residual Frobenius norm (to within machine noise processes) in the original problem statement (776). 2

§

4.6.0.0.12 Example. Compressive sampling of a phantom. In summer of 2004, Cand`es, Romberg, & Tao [66] and Donoho [108] released papers on perfect signal reconstruction from samples that stand in violation of Shannon’s classical sampling theorem. These defiant signals are assumed sparse inherently or under some sparsifying affine transformation. Essentially, they proposed sparse sampling theorems asserting average sample rate independent of signal bandwidth and less than Shannon’s rate.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

351

Minimum sampling rate:

ˆ of Ω-bandlimited signal: 2Ω ˆ of k-sparse length-n signal: k log (1+ n/k) 2

(Shannon) (Cand`es/Donoho)

Certainly, much was already known about nonuniform (Figure 92). or random sampling [34] [193] and about subsampling or multirate systems [86] [332]. Vetterli, Marziliano, & Blu [341] had congealed a theory of noiseless signal reconstruction, in May 2001, from samples that violate the Shannon rate. They anticipated the sparsifying transform by recognizing: it is the innovation (onset) of functions constituting a (not necessarily bandlimited) signal that determines minimum sampling rate for perfect reconstruction. Average onset (sparsity) Vetterli et alii call rate of innovation. The vector inner-products that Cand`es/Donoho call samples or measurements, Vetterli calls projections. From those projections Vetterli demonstrates reconstruction (by digital signal processing and “root finding”) of a Dirac comb, the very same prototypical signal from which Cand`es probabilistically derives minimum sampling rate (Compressive Sampling and Frontiers in Signal Processing, University of Minnesota, June 6, 2007). Combining their terminology, we paraphrase a sparse sampling theorem:

ˆ Minimum sampling rate, asserted by Cand`es/Donoho, is proportional to Vetterli’s rate of innovation (a.k.a, information rate, degrees of freedom, ibidem June 5, 2007).

What distinguishes these researchers are their methods of reconstruction. Properties of the 1-norm were also well understood by June 2004 finding applications in deconvolution of linear systems [79], penalized linear regression (Lasso) [296], and basis pursuit [198]. But never before had there been a formalized and rigorous sense that perfect reconstruction were possible by convex optimization of 1-norm when information lost in a subsampling process became nonrecoverable by classical methods. Donoho named this discovery compressed sensing to describe a nonadaptive perfect reconstruction method by means of linear programming. By the time Cand`es’ and Donoho’s landmark papers were finally published by IEEE in 2006, compressed sensing was old news that had spawned intense research which still persists; notably, from prominent members of the wavelet community.

352

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Reconstruction of the Shepp-Logan phantom (Figure 99), from a severely aliased image (Figure 101) obtained by Magnetic Resonance Imaging (MRI), was the impetus driving Cand`es’ quest for a sparse sampling theorem. He realized that line segments appearing in the aliased image were regions of high total variation. There is great motivation, in the medical community, to apply compressed sensing to MRI because it translates to reduced scan-time which brings great technological and physiological benefits. MRI is now about 35 years old, beginning in 1973 with Nobel laureate Paul Lauterbur from Stony Brook. There has been much progress in MRI and compressed sensing since 2004, but there have also been indications of 1-norm abandonment (indigenous to reconstruction by compressed sensing) in favor of criteria closer to 0-norm because of a correspondingly smaller number of measurements required to accurately reconstruct a sparse signal:4.41 5481 complex samples (22 radial lines, ≈ 256 complex samples per) were required in June 2004 to reconstruct a noiseless 256×256-pixel Shepp-Logan phantom by 1-norm minimization of an image-gradient integral estimate called total variation; id est, 8.4% subsampling of 65536 data. [66, 1.1] [65, 3.2] It was soon discovered that reconstruction of the Shepp-Logan phantom were possible with only 2521 complex samples (10 radial lines, Figure 100); 3.8% subsampled data input to a (nonconvex) 12 -norm total-variation minimization. [72, IIIA] The closer to 0-norm, the fewer the samples required for perfect reconstruction. Passage of a few years witnessed an algorithmic speedup and dramatic reduction in minimum number of samples required for perfect reconstruction of the noiseless Shepp-Logan phantom. But minimization of total variation is ideally suited to recovery of any piecewise-constant image, like a phantom, because the gradient of such images is highly sparse by design. There is no inherent characteristic of real-life MRI images that would make reasonable an expectation of sparse gradient. Sparsification of a discrete image-gradient tends to preserve edges. Then minimization of total variation seeks an image having fewest edges. There is no deeper theoretical foundation than that. When applied to human brain scan or angiogram, with as much as 20% of 256×256 Fourier samples, we have observed4.42 a

§

§

§

4.41

Efficient techniques continually emerge urging 1-norm criteria abandonment; [76] [331] [330, §IID] e.g., five techniques for compressed sensing are compared in [35] demonstrating that 1-norm performance limits for cardinality minimization can be reliably exceeded. 4.42 Experiments with real-life images were performed by Christine Law at Lucas Center for Imaging, Stanford University.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

353

30dB image/reconstruction-error ratio4.43 barrier that seems impenetrable by the total-variation objective. Total-variation minimization has met with moderate success, in retrospect, only because some medical images are moderately piecewise-constant signals. One simply hopes a reconstruction, that is in some sense equal to a known subset of samples and whose gradient is most sparse, is that unique image we seek.4.44 The total-variation objective, operating on an image, is expressible as norm of a linear transformation (802). It is natural to ask whether there exist other sparsifying transforms that might break the real-life 30dB barrier (any sampling pattern @20% 256×256 data) in MRI. There has been much research into application of wavelets, discrete cosine transform (DCT), randomized orthogonal bases, splines, etcetera, but with suspiciously little focus on objective measures like image/error or illustration of difference images; the predominant basis of comparison instead being subjectively visual (Duensing & Huang, ISMRM Toronto 2008).4.45 Despite choice of transform, there seems yet to have been a breakthrough of the 30dB barrier. Application of compressed sensing to MRI, therefore, remains fertile in 2008 for continued research. Lagrangian form of compressed sensing in imaging We now repeat Cand`es’ image reconstruction experiment from 2004 which led to discovery of sparse sampling theorems. [66, 1.2] But we achieve perfect reconstruction with an algorithm based on vanishing gradient of a

§

4.43

Noise considered here is due only to the reconstruction process itself; id est, noise in excess of that produced by the best reconstruction of an image from a complete set of samples in the sense of Shannon. At less than 30dB image/error, artifacts generally remain visible to the naked eye. We estimate about 50dB is required to eliminate noticeable distortion in a visual A/B comparison. 4.44 In vascular radiology, diagnoses are almost exclusively based on morphology of vessels and, in particular, presence of stenoses. There is a compelling argument for total-variation reconstruction of magnetic resonance angiogram because it helps isolate structures of particular interest. 4.45 I have never calculated the PSNR of these reconstructed images [of Barbara]. −Jean-Luc Starck The sparsity of the image is the percentage of transform coefficients sufficient for diagnostic-quality reconstruction. Of course the term “diagnostic quality” is subjective. . . . I have yet to see an “objective” measure of image quality. Difference images, in my experience, definitely do not tell the whole story. Often I would show people some of my results and get mixed responses, but when I add artificial Gaussian noise to an image, often people say that it looks better. −Michael Lustig

354

CHAPTER 4. SEMIDEFINITE PROGRAMMING

compressed sensing problem’s Lagrangian, which is computationally efficient. Our contraction method (p.360) is fast also because matrix multiplications are replaced by fast Fourier transforms and number of constraints is cut in half by sampling symmetrically. Convex iteration for cardinality minimization ( 4.5) is incorporated which allows perfect reconstruction of a phantom at 4.1% subsampling rate; 50% Cand`es’ rate. By making neighboring-pixel selection adaptive, convex iteration reduces discrete image-gradient sparsity of the Shepp-Logan phantom to 1.9% ; 33% lower than previously reported. We demonstrate application of discrete image-gradient sparsification to the n×n = 256×256 Shepp-Logan phantom, simulating idealized acquisition of MRI data by radial sampling in the Fourier domain (Figure 100).4.46 Define a Nyquist-centric discrete Fourier transform (DFT) matrix

§



   F ,   

1 1 1 1 − 2π/n − 4π/n − 6π/n 1 e e e 1 e− 4π/n e− 8π/n e−12π/n 1 e− 6π/n e−12π/n e−18π/n .. .. .. .. . . . . −(n−1)2π/n −(n−1)4π/n −(n−1)6π/n 1 e e e

 ··· 1 · · · e−(n−1)2π/n   · · · e−(n−1)4π/n  n×n 1 −(n−1)6π/n √ ∈ C ··· e n  .. ...  . −(n−1)2 2π/n ··· e (784)

a symmetric (nonHermitian) unitary matrix characterized F = FT F −1 = F H

(785)

Denoting an unknown image U ∈ Rn×n , its two-dimensional discrete Fourier transform F is F(U) , F UF (786) hence the inverse discrete transform

§

U = F H F(U)F H

(787)

From A.1.1 no.31 we have a vectorized two-dimensional DFT via Kronecker product ⊗ vec F(U) , (F ⊗F ) vec U (788) and from (787) its inverse [152, p.24] vec U = (F H ⊗ F H )(F ⊗F ) vec U = (F HF ⊗ F HF ) vec U

(789)

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

355

Φ

Figure 100: MRI radial sampling pattern, in DC-centric Fourier domain, representing 4.1% (10 lines) subsampled data. Only half of these complex samples, in any halfspace about the origin in theory, need be acquired for a real image because of conjugate symmetry. Due to MRI machine imperfections, samples are generally taken over full extent of each radial line segment. MRI acquisition time is proportional to number of lines.

Idealized radial sampling in the Fourier domain can be simulated by Hadamard product ◦ with a binary mask Φ ∈ Rn×n whose nonzero entries could, for example, correspond with the radial line segments in Figure 100. To make the mask Nyquist-centric, like DFT matrix F , define a circulant [154] symmetric permutation matrix4.47 ¸ · 0 I ∈ Sn (790) Θ , I 0 Then given subsampled Fourier domain (MRI k-space) measurements in incomplete K ∈ C n×n , we might constrain F(U) thus: ΘΦΘ ◦ F UF = K

(791)

and in vector form, (37) (1711) δ(vec ΘΦΘ)(F ⊗F ) vec U = vec K 4.46

(792)

k-space is conventional acquisition terminology indicating domain of the continuous raw data provided by an MRI machine. An image is reconstructed by inverse discrete Fourier transform of that data interpolated on a Cartesian grid in two dimensions. 4.47 Matlab fftshift()

356

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Because measurements K are complex, there are actually twice the number of equality constraints as there are measurements. We can cut that number of constraints in half via vertical and horizontal mask Φ symmetry which forces the imaginary inverse transform to 0 : The inverse subsampled transform in matrix form is F H (ΘΦΘ ◦ F UF )F H = F HKF H

(793)

(F H ⊗ F H )δ(vec ΘΦΘ)(F ⊗F ) vec U = (F H ⊗ F H ) vec K

(794)

and in vector form

later abbreviated P vec U = f

(795)

where 2 × n2

P , (F H ⊗ F H )δ(vec ΘΦΘ)(F ⊗F ) ∈ C n

(796)

Because of idempotence P = P 2 , P is a projection matrix. Because of its Hermitian symmetry [152, p.24] P = (F H ⊗ F H )δ(vec ΘΦΘ)(F ⊗F ) = (F ⊗F )H δ(vec ΘΦΘ)(F H ⊗ F H )H = P H (797) 4.48 P vec U is real when P is real; id est, when P is an orthogonal projector. for positive even integer n Φ=

·

Φ11 Φ(1 , 2 : n)Ξ ΞΦ(2 : n , 1) ΞΦ(2 : n , 2 : n)Ξ

¸

∈ Rn×n

(798)

where Ξ ∈ S n−1 is the order-reversing permutation matrix (1654). In words, this necessary and sufficient condition on Φ (for a real inverse subsampled transform [264, p.53]) demands vertical symmetry about row n2 +1 and horizontal symmetry4.49 about column n2 +1. 4.48

(796) is a diagonalization of matrix P whose binary eigenvalues are δ(vec ΘΦΘ) while the corresponding eigenvectors constitute the columns of unitary matrix F H ⊗ F H . 4.49 This condition on Φ applies to both DC- and Nyquist-centric DFT matrices.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

357

vec−1 f

Figure 101: Aliasing of Shepp-Logan phantom in Figure 99 resulting from k-space subsampling pattern in Figure 100. This image is real because binary mask Φ is vertically and horizontally symmetric. It is remarkable that the phantom can be reconstructed, by convex iteration, given only U 0 = vec−1 f . Define 

1

0

0

 1 0  −1  .  −1 1 . .  ∆ , ... ... ...   ...  1  0T

−1



      ∈ Rn×n    0 

(799)

1

Express an image-gradient estimate   U∆  U ∆T  4n×n  ∇U ,   ∆U ∈ R ∆T U

(800)

§

that is a simple first-order difference of neighboring pixels (Figure 102) to the right, left, above, and below.4.50 By A.1.1 no.31, its vectorization: 4.50

There is significant improvement in reconstruction quality by augmentation of a

358

CHAPTER 4. SEMIDEFINITE PROGRAMMING 2 × n2

for Ψi ∈ Rn

   Ψ1 ∆T ⊗ I  T  ∆⊗I  vec U ,  Ψ1 vec U , Ψ vec U ∈ R4n2 vec ∇U =   Ψ2   I ⊗∆  T ΨT I ⊗∆ 2 

2

(801)

2

where Ψ ∈ R4n ×n . A total-variation minimization for reconstructing MRI image U , that is known suboptimal [192] [67], may be concisely posed minimize U

kΨ vec Uk1

(802)

subject to P vec U = f where 2

f = (F H ⊗ F H ) vec K ∈ C n

(803)

is the known inverse subsampled Fourier data (a vectorized aliased image, Figure 101), and where a norm of discrete image-gradient ∇U is equivalently expressed as norm of a linear transformation Ψ vec U . Although this simple problem statement (802) is equivalent to a linear program ( 3.3), its numerical solution is beyond the capability of even the most highly regarded of contemporary commercial solvers.4.51 Our only recourse is to recast the problem in Lagrangian form ( 3.2) and write customized code to solve it:

§

§

minimize U

h|Ψ vec U| , yi

subject to P vec U = f ≡ minimize h|Ψ vec U| , yi + 21 λkP vec U − f k22 U

(a) (804) (b)

nominally two-point discrete image-gradient estimate to four points per pixel by inclusion of two polar directions. Improvement is due to centering; symmetry of discrete differences about a central pixel. We find small improvement on real-life images, ≈ 1dB empirically, by further augmentation with diagonally adjacent pixel differences. 4.51 for images as small as 128×128 pixels. Obstacle to numerical solution is not a computer resource: e.g., execution time, memory. The obstacle is, in fact, inadequate numerical precision. Even when all dependent equality constraints are manually removed, the best commercial solvers fail simply because computer numerics become nonsense; id est, numerical errors enter significant digits and the algorithm exits prematurely, loops indefinitely, or produces an infeasible solution.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

359

Figure 102: Neighboring-pixel stencil [331] for image-gradient estimation on Cartesian grid. Implementation selects adaptively from darkest four • about central. Continuous image-gradient from two pixels holds only in a limit. For discrete differences, better practical estimates are obtained when centered.

§

where multiobjective parameter λ ∈ R+ is quite large (λ ≈1E8) so as to enforce the equality constraint: P vec U −f = 0 ⇔ kP vec U −f k22 = 0 ( A.7.1). We 2 introduce a direction vector y ∈ R4n as part of a convex iteration ( 4.5.2) + to overcome that known suboptimal minimization of discrete image-gradient cardinality: id est, there exists a vector y ⋆ with entries yi⋆ ∈ {0, 1} such that minimize U

kΨ vec Uk0

subject to P vec U = f

≡

§

minimize h|Ψ vec U| , y ⋆ i + 12 λkP vec U − f k22 U (805)

Existence of such a y ⋆ , complementary to an optimal vector Ψ vec U ⋆ , is obvious by definition of global optimality h|Ψ vec U ⋆ | , y ⋆ i = 0 (714) under which a cardinality-c optimal objective is assumed to exist. Because (804b) is an unconstrained convex problem, a zero in the objective function’s gradient is necessary and sufficient for optimality ( 2.13.3); id est, ( D.2.1)

§

§

ΨT δ(y) sgn(Ψ vec U ) + λP H (P vec U − f ) = 0

(806)

Because of P idempotence and Hermitian symmetry and sgn(x) = x/|x| ,

360

CHAPTER 4. SEMIDEFINITE PROGRAMMING

this is equivalent to ¡ ¢ lim ΨT δ(y)δ(|Ψ vec U| + ǫ1)−1 Ψ + λP vec U = λP f ǫ→0

(807)

where small positive constant ǫ ∈ R+ has been introduced for invertibility. The analytical effect of introducing ǫ is to make the objective’s gradient valid everywhere in the function domain; id est, it transforms absolute value in (804b) from a function differentiable almost everywhere into a differentiable function. An example of such a transformation in one dimension is illustrated in Figure 103. When small enough for practical purposes4.52 (ǫ ≈1E-3), we may ignore the limiting operation. Then the mapping, for 0 ¹ y ¹ 1 ¡ ¢−1 vec U t+1 = ΨT δ(y)δ(|Ψ vec U t | + ǫ1)−1 Ψ + λP λP f

(808)

is a contraction in U t that can be solved recursively in t for its unique fixed point; id est, until U t+1 → U t . [211, p.300] [189, p.155] Calculating this inversion directly is not possible for large matrices on contemporary computers because of numerical precision, so instead we apply the conjugate gradient method of solution to ¡

¢ ΨT δ(y)δ(|Ψ vec U t | + ǫ1)−1 Ψ + λP vec U t+1 = λP f

(809)

which is linear in U t+1 at each recursion in the Matlab program.4.53 Observe that P (796), in the equality constraint from problem (804a), is not a fat matrix.4.54 Although number of Fourier samples taken is equal to the number of nonzero entries in binary mask Φ , matrix P is square but never actually formed during computation. Rather, a two-dimensional fast Fourier transform of U is computed followed by masking with ΘΦΘ and then an inverse fast Fourier transform. This technique significantly reduces memory requirements and, together with contraction method of solution, is the principal reason for relatively fast computation. 4.52

We are looking for at least 50dB image/error ratio from only 4.1% subsampled data (10 radial lines in k-space). With this setting of ǫ , we actually attain in excess of 100dB from a simple Matlab program in about a minute on a 2006 vintage laptop. By trading execution time and treating discrete image-gradient cardinality as a known quantity for this phantom, over 160dB is achievable. 4.53 Conjugate gradient method requires positive definiteness. [140, §4.8.3.2] 4.54 Fat is typical of compressed sensing problems; e.g., [65] [72].

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

361

x y dy −1 |y|+ǫ

R

|x|

Figure 103: Real absolute value function f2 (x) = |x| on x ∈ [−1, 1] from Figure 65b superimposed upon integral of its derivative at ǫ = 0.05 which smooths objective function.

convex iteration By convex iteration we mean alternation of solution to (804a) and (810) until convergence. Direction vector y is initialized to 1 until the first fixed point is found; which means, the contraction recursion begins calculating a (1-norm) solution U ⋆ to (802) via problem (804b). Once U ⋆ is found, vector y is updated according to an estimate of discrete image-gradient 2 cardinality c : Sum of the 4n2 − c smallest entries of |Ψ vec U ⋆ | ∈ R4n is the optimal objective value from a linear program, for 0 ≤ c ≤ 4n2 − 1 (495) 4n P2

π(|Ψ vec U ⋆ |)i =

i=c+1

minimize 2 y∈ R4n

h|Ψ vec U ⋆ | , yi

subject to 0 ¹ y ¹ 1 y T 1 = 4n2 − c

(810)

where π is the nonlinear permutation-operator sorting its vector argument into nonincreasing order. An optimal solution y to (810), that is an extreme point of its feasible set, is known in closed form: it has 1 in each entry corresponding to the 4n2 − c smallest entries of |Ψ vec U ⋆ | and has 0 elsewhere (page 317). Updated image U ⋆ is assigned to U t , the contraction is recomputed solving (804b), direction vector y is updated again, and so on until convergence which is guaranteed by virtue of a monotonically nonincreasing real sequence of objective values in (804a) and (810).

362

CHAPTER 4. SEMIDEFINITE PROGRAMMING

There are two features that distinguish problem formulation (804b) and our particular implementation of it [Matlab on Wıκımization]: 1) An image-gradient estimate may engage any combination of four adjacent pixels. In other words, the algorithm is not locked into a four-point gradient estimate (Figure 102); number of points constituting an estimate is directly determined by direction vector y . 4.55 Indeed, we find only c = 5092 zero entries in y ⋆ for the Shepp-Logan phantom; meaning, discrete image-gradient sparsity is actually closer to 1.9% than the 3% reported elsewhere; e.g., [330, IIB].

§

2) Numerical precision of the fixed point of contraction (808) (≈1E-2 for almost perfect reconstruction @−103dB error) is a parameter to the implementation; meaning, direction vector y is updated after contraction begins but prior to its culmination. Impact of this idiosyncrasy tends toward simultaneous optimization in variables U and y while insuring y settles on a boundary point of the feasible set (nonnegative hypercube slice) in (810) at every iteration; for only a boundary point4.56 can yield the sum of smallest entries in |Ψ vec U ⋆ |. Almost perfect reconstruction of the Shepp-Logan phantom at 103dB image/error is achieved in a Matlab minute with 4.1% subsampled data (2671 complex samples); well below an 11% least lower bound predicted by the sparse sampling theorem. Because reconstruction approaches optimal solution to a 0-norm problem, minimum number of Fourier-domain samples is bounded below by cardinality of discrete image-gradient at 1.9%. 2

4.6.0.0.13 Exercise. Contraction operator. Determine conditions on λ and ǫ under which (808) is a contraction and ΨT δ(y)δ(|Ψ vec U t | + ǫ1)−1 Ψ + λP from (809) is positive definite. H 4.55

This adaptive gradient was not contrived. It is an artifact of the convex iteration method for minimum cardinality solution; in this case, cardinality minimization of a discrete image-gradient. 4.56 Simultaneous optimization of these two variables U and y should never be a pinnacle of aspiration; for then, optimal y might not attain a boundary point.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES

363

Figure 104: Massachusetts Institute of Technology (MIT) logo, including its white boundary, may be interpreted as a rank-5 matrix. (Stanford University logo rank is much higher;) This constitutes Scene Y observed by the one-pixel camera in Figure 105 for Example 4.6.0.0.14.

4.6.0.0.14 Example. Compressed sensing, compressive sampling. [278] As our modern technology-driven civilization acquires and exploits ever-increasing amounts of data, everyone now knows that most of the data we acquire can be thrown away with almost no perceptual loss − witness the broad success of lossy compression formats for sounds, images, and specialized technical data. The phenomenon of ubiquitous compressibility raises very natural questions: Why go to so much effort to acquire all the data when most of what we get will be thrown away? Can’t we just directly measure the part that won’t end up being thrown away? −David Donoho [108] Lossy data compression techniques are popular, but it is also well known that compression artifacts become quite perceptible with signal postprocessing that goes beyond mere playback of a compressed signal. [205] [228] Spatial or audio frequencies presumed masked by a simultaneity, for example, become perceptible with significant postfiltering of the compressed signal. Further, there can be no universally acceptable and unique metric of perception for gauging exactly how much data can be tossed. For these reasons, there will always be need for raw (uncompressed) data. In this example we throw out only so much information as to leave perfect reconstruction within reach. Specifically, the MIT logo in Figure 104 is perfectly reconstructed from 700 time-sequential samples {yi } acquired by the one-pixel camera illustrated in Figure 105. The MIT-logo image in this example impinges a 46×81 array micromirror device. This mirror array

364

CHAPTER 4. SEMIDEFINITE PROGRAMMING Y

yi

Figure 105: One-pixel camera. [321] [345] Compressive imaging camera block diagram. Incident lightfield (corresponding to the desired image Y ) is reflected off a digital micromirror device (DMD) array whose mirror orientations are modulated in the pseudorandom pattern supplied by the random number generators (RNG). Each different mirror pattern produces a voltage at the single photodiode that corresponds to one measurement yi .

is modulated by a pseudonoise source that independently positions all the individual mirrors. A single photodiode (one pixel) integrates incident light from all mirrors. After stabilizing the mirrors to a fixed but pseudorandom pattern, light so collected is then digitized into one sample yi by analog-to-digital (A/D) conversion. This sampling process is repeated with the micromirror array modulated to a new pseudorandom pattern. The most important questions are: How many samples do we need for perfect reconstruction? Does that number of samples represent compression of the original data? We claim that perfect reconstruction of the MIT logo can be reliably achieved with as few as 700 samples y = [yi ] ∈ R700 from this one-pixel camera. That number represents only 19% of information obtainable from 3726 micromirrors.4.57 (Figure 106) 4.57

That number (700 samples) is difficult to achieve, as reported in [278, §6]. If a minimal basis for the MIT logo were instead constructed, only five rows or columns worth of data (from a 46×81 matrix) are linearly independent. This means a lower bound on achievable compression is about 5×46 = 230 samples plus 81 samples column encoding; which corresponds to 8% of the original information. (Figure 106)

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 1

2

3

4

365 5

samples 0

1000

2000

3000

3726

Figure 106: Estimates of compression for various encoding methods: 1) linear interpolation (140 samples), 2) minimal columnar basis (311 samples), 3) convex iteration (700 samples) can achieve lower bound predicted by compressed sensing (670 samples, n = 46×81, k = 140, Figure 92) whereas nuclear norm minimization alone does not [278, 6], 4) JPEG @100% quality (2588 samples), 5) no compression (3726 samples).

§

Our approach to reconstruction is to look for low-rank solution to an underdetermined system: find

X ∈ R46×81

X

subject to A vec X = y rank X ≤ 5

(811)

where vec X is the vectorized (32) unknown image matrix. Each row of fat matrix A is one realization of a pseudorandom pattern applied to the micromirrors. Since these patterns are deterministic (known), then the i th sample yi equals A(i , :) vec Y ; id est, y = A vec Y . Perfect reconstruction here means optimal solution X ⋆ equals scene Y ∈ R46×81 to within machine precision. Because variable matrix X is generally not square or positive semidefinite, we constrain its rank by rewriting the problem equivalently find

W1 ∈ R46×46 , W2 ∈ R81×81 , X ∈ R46×81

subject to

X A vec·X = y ¸ W1 X ≤5 rank X T W2

(812)

This rank constraint on the composite (block) matrix insures rank X ≤ 5 for

366

CHAPTER 4. SEMIDEFINITE PROGRAMMING

any choice of dimensionally compatible matrices W1 and W2 .4.58 But to solve this problem by convex iteration, we alternate solution of semidefinite program ¸ ¶ µ· W1 X Z minimize tr X T W2 W1 ∈ S46 , W2 ∈ S81 , X ∈ R46×81 subject to A vec X = y (813) ¸ · W1 X º0 X T W2 with semidefinite program ¸⋆ ¶ µ· W1 X minimize tr Z X T W2 Z ∈ S46+81 subject to 0 ¹ Z ¹ I tr Z = 46 + 81 − 5

(814)

(which has an optimal solution that is known in closed form, p.631) until a rank-5 composite matrix is found. With 1000 samples {yi } , convergence occurs in two iterations; 700 samples require more than ten iterations but reconstruction remains perfect. Iterating more admits taking of fewer samples. Reconstruction is independent of pseudorandom sequence parameters; e.g., binary sequences succeed with the same efficiency as Gaussian or uniformly distributed sequences. 2

4.6.1

rank-constraint midsummary

We find that this direction matrix idea works well and quite independently of desired rank or affine dimension. This idea of direction matrix is good principally because of its simplicity: When confronted with a problem otherwise convex if not for a rank or cardinality constraint, then that constraint becomes a linear regularization term in the objective. 4.58

which holds because X is a projection of the composite matrix on subspace R46×81 W1 and W2 , any rank-k positive semidefinite composite matrix · . For symmetric ¸ W1 X G= can be factored into rank-k terms R ; G = RTR where R , [ B C ] X T W2 and rank B , rank C ≤ rank R and B ∈ Rk×46 and C ∈ Rk×81 . Because W1 and W2 and X = B T C are variable, (1409) rank X ≤ rank B , rank C ≤ rank R = rank G is tight.

4.7. CONVEX ITERATION RANK-1

367

There exists a common thread through all these Examples; that being, convex iteration with a direction matrix as normal to a linear regularization (a generalization of the well-known trace heuristic). But each problem type (per Example) possesses its own idiosyncrasies that slightly modify how a rank-constrained optimal solution is actually obtained: The ball packing problem in Chapter 5.4.2.3.4, for example, requires a problem sequence in a progressively larger number of balls to find a good initial value for the direction matrix, whereas many of the examples in the present chapter require an initial value of 0. Finding a feasible Boolean vector in Example 4.6.0.0.8 requires a procedure to detect stalls, while other problems have no such requirement. The combinatorial Procrustes problem in Example 4.6.0.0.3 allows use of a known closed-form solution for direction vector when solved via rank constraint, but not when solved via cardinality constraint. Some problems require a careful weighting of the regularization term, whereas other problems do not, and so on. It would be nice if there were a universally applicable method for constraining rank; one that is less susceptible to quirks of a particular problem type. Poor initialization of the direction matrix from the regularization can lead to an erroneous result. We speculate one reason to be a simple dearth of optimal solutions of desired rank or cardinality;4.59 an unfortunate choice of initial search direction leading astray. Ease of solution by convex iteration occurs when optimal solutions abound. With this speculation in mind, we now propose a further generalization of convex iteration for constraining rank that attempts to ameliorate quirks and unify problem types:

4.7

Convex Iteration rank-1

§

We now develop a general method for constraining rank that first decomposes a given problem via standard diagonalization of matrices ( A.5). This method is motivated by observation ( 4.4.1.1) that an optimal direction matrix can be diagonalizable simultaneously with an optimal variable matrix. This suggests minimization of an objective function directly in terms of eigenvalues. A second motivating observation is that variable orthogonal matrices seem easily found by convex iteration; e.g., Procrustes Example 4.6.0.0.2.

§

4.59

Recall that, in Convex Optimization, an optimal solution generally comes from a convex set of optimal solutions that can be large.

368

CHAPTER 4. SEMIDEFINITE PROGRAMMING

It turns out that this general method always requires solution to a rank-1 constrained problem regardless of desired rank from the original problem. To demonstrate, we pose a semidefinite feasibility problem find X ∈ Sn subject to A svec X = b Xº0 rank X ≤ ρ

(815)

given an upper bound 0 < ρ < n on rank, vector b ∈ Rm , and typically fat full-rank   svec(A1 )T ..  ∈ Rm×n(n+1)/2 A = (614) . T svec(Am )

where Ai ∈ S n , i = 1 . . . m are also given. Symmetric matrix vectorization svec is defined in (51). Thus   tr(A1 X ) ..  (615) A svec X =  . tr(Am X )

This program (815) states the classical problem of finding a matrix of specified rank in the intersection of the positive semidefinite cone with a number m of hyperplanes in the subspace of symmetric matrices S n . [26, II.13] [24, 2.2] Express the nonincreasingly ordered diagonalization of variable matrix

§

§

X , QΛQT =

n X i=1

λ i Qii ∈ S n

(816)

which is a sum of rank-1 orthogonal projection matrices Qii weighted by eigenvalues λ i where Qij , qi qjT ∈ Rn×n , Q = [ q1 · · · qn ] ∈ Rn×n , QT = Q−1 , Λii = λ i ∈ R , and   λ1 0   λ2   (817) Λ =  ∈ Sn ...   0T λn

4.7. CONVEX ITERATION RANK-1

369

The fact Λº0 ⇔ Xº0

(1394)

allows splitting semidefinite feasibility problem (815) into two parts: ° ¶ µ ρ ° P ° λ i Qii − minimize °A svec Λ i=1 · ρ ¸ R+ subject to δ(Λ) ∈ 0 ° ¶ µ ρ ° P ⋆ ° λ i Qii − minimize °A svec Q

i=1

T

−1

subject to Q = Q

° ° b° °

(818)

° ° b° °

(819)

The linear equality constraint A svec X = b has been moved to the objective within a norm because these two problems (818) (819) are iterated; equality might only become feasible near convergence. This iteration always converges to a local minimum because the sequence of objective values is monotonic and nonincreasing; any monotonically nonincreasing real sequence converges. [240, 1.2] [39, 1.1] A rank-ρ matrix X feasible to the original problem (815) is found when the objective converges to 0 ; a certificate of global optimality for the iteration. Positive semidefiniteness of matrix X with an upper bound ρ on rank is assured by the constraint on eigenvalue matrix Λ in convex problem (818); it means, the main diagonal of Λ must belong to the nonnegative orthant in a ρ-dimensional subspace of Rn .

§

§

The second problem (819) in the iteration is not convex. We propose solving it by convex iteration: Make the assignment  q1 [ q1T · · · qρT ] G =  ...  ∈ S nρ qρ     Q11 · · · Q1ρ q1 q1T · · · q1 qρT . ..   .. ..  ... ... =  .. , . . . T T q q · · · q QT · · · Q ρρ ρ 1 ρ qρ 1ρ 

(820)

370

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Given ρ eigenvalues λ⋆i , then a problem equivalent to (819) is ° ° ¶ µ ρ ° ° P ⋆ ° ° − b λ Q A svec minimize ii i ° ° n n×n Qii ∈ S , Qij ∈ R

subject to

i=1

tr Qii = 1 , i=1 . . . ρ tr Qij = 0 , i<j = 2 . . . ρ   Q11 · · · Q1ρ . ..  ... G =  .. (º 0) . T Q1ρ · · · Qρρ rank G = 1

(821)

This new problem always enforces a rank-1 constraint on matrix G ; id est, regardless of upper bound on rank ρ of variable matrix X , this equivalent problem (821) always poses a rank-1 constraint. Given some fixed weighting factor w , we propose solving (821) by iteration of the convex sequence ° ° ¶ µ ρ ° ° P ⋆ ° + w tr(G W ) °A svec λ − b Q minimize ii i ° ° n×n Qij ∈ R

i=1

subject to tr Qii = 1 , tr Qij = 0 ,   Q11 · · · Q1ρ . ..  ... G =  .. º0 . T Q1ρ · · · Qρρ with

minimize nρ W∈S

i=1 . . . ρ i<j = 2 . . . ρ

(822)

tr(G⋆ W )

subject to 0 ¹ W ¹ I tr W = nρ − 1

(823)

the latter providing direction of search W for a rank-1 matrix G in (822). An optimal solution to (823) has closed form (p.631). These convex problems (822) (823) are iterated with (818) until a rank-1 G matrix is found and the objective of (822) vanishes. For a fixed number of equality constraints m and upper bound ρ , the feasible solution set in rank-constrained semidefinite feasibility problem (815) grows with matrix dimension n . Empirically, this semidefinite feasibility problem becomes easier to solve by method of convex iteration as n increases.

4.7. CONVEX ITERATION RANK-1

371

We find a good value of weight w ≈ 5 for randomized problems. Initial value of direction matrix W is not critical. With dimension n = 26 and number of equality constraints m = 10, convergence to a rank-ρ = 2 solution to semidefinite feasibility problem (815) is achieved for nearly all realizations in less than 10 iterations. (This finding is significant in so far as Barvinok’s Proposition 2.9.3.0.1 predicts existence of matrices having rank 4 or less in this intersection, but his upper bound can be no tighter than 3.) For small n , stall detection is required; one numerical implementation is disclosed on Wıκımization. This diagonalization decomposition technique is extensible to other problem types, of course, including optimization problems having nontrivial objectives. Because of a greatly expanded feasible set, for example, find X ∈ Sn subject to A svec X º b Xº0 rank X ≤ ρ

§

(824)

this relaxation of (815) [209, III] is more easily solved by convex iteration rank-1. (The inequality in A remains in the constraints after diagonalization.) Because of the nonconvex nature of a rank constrained problem, more generally, there can be no proof of global convergence of convex iteration from an arbitrary initialization; although, local convergence is guaranteed by virtue of monotonically nonincreasing real objective sequences. [240, 1.2] [39, 1.1]

§

§

372

CHAPTER 4. SEMIDEFINITE PROGRAMMING

Chapter 5 Euclidean Distance Matrix These results [(856)] were obtained by Schoenberg (1935), a surprisingly late date for such a fundamental property of Euclidean geometry. −John Clifford Gower [148, 3]

§

By itself, distance information between many points in Euclidean space is lacking. We might want to know more; such as, relative or absolute position or dimension of some hull. A question naturally arising in some fields (e.g., geodesy, economics, genetics, psychology, biochemistry, engineering) [95] asks what facts can be deduced given only distance information. What can we know about the underlying points that the distance information purports to describe? We also ask what it means when given distance information is incomplete; or suppose the distance information is not reliable, available, or specified only by certain tolerances (affine inequalities). These questions motivate a study of interpoint distance, well represented in any spatial dimension by a simple matrix from linear algebra.5.1 In what follows, we will answer some of these questions via Euclidean distance matrices.

√ e.g., ◦ D ∈ RN ×N , a classical two-dimensional matrix representation of absolute interpoint distance because its entries (in ordered rows and columns) can be written neatly on a piece of paper. Matrix D will be reserved throughout to hold distance-square. 5.1

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

373

374

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

1



3

 0 1 5 D = 1 0 4  5 4 0

2 √

2

1 2 3

5 1

Figure 107: Convex hull of three points (N = 3) is shaded in R3 (n = 3). Dotted lines are imagined vectors to points whose affine dimension is 2.

5.1

EDM

Euclidean space Rn is a finite-dimensional real vector space having an inner product defined on it, inducing a metric. [211, 3.1] A Euclidean distance ×N matrix, an EDM in RN , is an exhaustive table of distance-square dij + between points taken by pair from a list of N points {xℓ , ℓ = 1 . . . N } in Rn ; the squared metric, the measure of distance-square:

§

dij = kxi − xj k22 , hxi − xj , xi − xj i

(825)

Each point is labelled ordinally, hence the row or column index of an EDM, i or j = 1 . . . N , individually addresses all the points in the list. Consider the following example of an EDM for the case N = 3 :       d11 d12 d13 0 d12 d13 0 1 5 D = [dij ] =  d21 d22 d23  =  d12 0 d23  =  1 0 4  (826) d13 d23 0 d31 d32 d33 5 4 0

Matrix D has N 2 entries but only N (N − 1)/2 pieces of information. In Figure 107 are shown three points in R3 that can be arranged in a list

5.2. FIRST METRIC PROPERTIES

375

§

to correspond to D in (826). But such a list is not unique because any rotation, reflection, or translation ( 5.5) of those points would produce the same EDM D .

5.2

First metric properties

§

§

For i,j = 1 . . . N , distance between points xi and xj must satisfy the defining requirements imposedpupon any metric space: [211, 1.1] [240, 1.7] namely, for Euclidean metric dij ( 5.4) in Rn

§

1.

2. 3. 4.

p dij ≥ 0 , i 6= j

nonnegativity

p dij = 0 ⇔ xi = xj

self-distance

p p dij = dji

symmetry

p p p dij ≤ dik + dkj , i 6= j 6= k

triangle inequality

Then all entries of an EDM must be in concord with these Euclidean metric properties: specifically, each entry must be nonnegative,5.2 the main diagonal must be 0 ,5.3 and an EDM must be symmetric. The fourth property provides upper and lower bounds for each entry. Property 4 is true more generally when there are no restrictions on indices i,j,k , but furnishes no new information.

5.3

∃ fifth Euclidean metric property

The four properties of the Euclidean metric provide information insufficient to certify that a bounded convex polyhedron more complicated than a triangle has a Euclidean realization. [148, 2] Yet any list of points or the vertices of any bounded convex polyhedron must conform to the properties.

§

p dij ≥ 0 ⇔ dij ≥ 0 is always assumed. Implicit from the terminology, 5.3 What we call self-distance, Marsden calls nondegeneracy. [240, §1.6] Kreyszig calls these first metric properties axioms of the metric; [211, p.4] Blumenthal refers to them as postulates. [46, p.15] 5.2

376

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.3.0.0.1 Example. Triangle. Consider the EDM in (826), but missing one of its entries:   0 1 d13 D= 1 0 4  d31 4 0

(827)

Can we determine unknown √ √entries of D by applying the metric properties? Property 1 demands d13 , d√ property 2 requires the main diagonal 31 ≥ 0, √ be 0, while property 3 makes d31 = d13 . The fourth property tells us p 1 ≤ d13 ≤ 3 (828) Indeed, described over that closed interval [1, 3] is a family of triangular polyhedra whose angle at vertex x2 varies from 0 to π radians. So, yes we can determine the unknown entries of D , but they are not unique; nor should they be from the information given for this example. 2 5.3.0.0.2 Example. Small completion problem, I. Now consider the polyhedron in Figure 108(b) formed from an unknown list {x1 , x2 , x3 , x4 }. The corresponding EDM less one critical piece of information, d14 , is given by   0 1 5 d14  1 0 4 1   (829) D=  5 4 0 1  d14 1 1 0

From metric property 4 we may write a few inequalities for the two triangles common to d14 ; we find p √ 5−1 ≤ d14 ≤ 2 (830) √ We cannot further narrow those bounds on d14 using only the √ four metric properties ( 5.8.3.1.1). Yet there is only one possible choice √ for d14 because points x2 , x3 , x4 must be collinear. All other values of d14 in the interval √ [ 5−1, 2] specify impossible distances in any dimension; id est, in√this particular example the triangle inequality does not yield an interval for d14 over which a family of convex polyhedra can be reconstructed. 2

§

5.3. ∃ FIFTH EUCLIDEAN METRIC PROPERTY

377

x3

x3 1 √

(a)

5

x4

x4

2

(b)

1 x1

x2

1

x1

x2

Figure 108: (a) Complete dimensionless EDM graph. (b) Emphasizing obscured segments x2 x4 , x4 x3 , and x2 x3 , now only five (2N − 3) absolute distances are specified. EDM so represented is incomplete, missing d14 as in (829), yet the isometric reconstruction ( 5.4.2.3.7) is unique as proved in 5.9.3.0.1 and 5.14.4.1.1. First four properties of Euclidean metric are not a recipe for reconstruction of this polyhedron.

§

§

§

§

§

§

We will return to this simple Example 5.3.0.0.2 to illustrate more elegant methods of solution in 5.8.3.1.1, 5.9.3.0.1, and 5.14.4.1.1. Until then, we can deduce some general principles from the foregoing examples:

ˆ Unknown d

ij

of an EDM are not necessarily uniquely determinable.

ˆ The triangle inequality does not produce necessarily tight bounds.

5.4

ˆ Four Euclidean metric properties are insufficient for reconstruction. 5.4

The term tight with reference to an inequality means equality is achievable.

378

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.3.1

Lookahead

There must exist at least one requirement more than the four properties of the Euclidean metric that makes them altogether necessary and sufficient to certify realizability of bounded convex polyhedra. Indeed, there are infinitely many more; there are precisely N + 1 necessary and sufficient Euclidean metric requirements for N points constituting a generating list ( 2.3.2). Here is the fifth requirement:

§

§

5.3.1.0.1 Fifth Euclidean metric property. Relative-angle inequality. (confer 5.14.2.1.1) Augmenting the four fundamental properties of the Euclidean metric in Rn , for all i, j, ℓ 6= k ∈ {1 . . . N } , i < j < ℓ , and for N ≥ 4 distinct points {xk } , the inequalities cos(θikℓ + θℓkj ) ≤ cos θikj ≤ cos(θikℓ − θℓkj ) 0 ≤ θikℓ , θℓkj , θikj ≤ π

(831)

where θikj = θjki represents angle between vectors at vertex xk (900) (Figure 109), must be satisfied at each point xk regardless of affine dimension. ⋄

§

We will explore this in 5.14. One of our early goals is to determine matrix criteria that subsume all the Euclidean metric properties and any further requirements. Looking ahead, we will find (1181) (856) (860)  −z TDz ≥ 0   1Tz = 0  ⇔ D ∈ EDMN (832) (∀ kzk = 1)    D ∈ SN h

where the convex cone of Euclidean distance matrices EDMN ⊆ SN h belongs 5.5 to the subspace of symmetric hollow matrices ( 2.2.3.0.1). [Numerical test isedm(D) is provided on Wıκımization.] Having found equivalent matrix criteria, we will see there is a bridge from bounded convex polyhedra to EDMs in 5.9 .5.6

§

§

5.5

0 main diagonal. From an EDM, a generating list (§2.3.2, §2.12.2) for a polyhedron can be found (§5.12) correct to within a rotation, reflection, and translation (§5.5). 5.6

5.3. ∃ FIFTH EUCLIDEAN METRIC PROPERTY

379

k

θi

kj

i

θikℓ

θj kℓ

ℓ

j Figure 109: Fifth Euclidean metric property nomenclature. Each angle θ is made by a vector pair at vertex k while i, j, k, ℓ index four points at the vertices of a generally irregular tetrahedron. The fifth property is necessary for realization of four or more points; a reckoning by three angles in any dimension. Together with the first four Euclidean metric properties, this fifth property is necessary and sufficient for realization of four points.

380

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Now we develop some invaluable concepts, moving toward a link of the Euclidean metric properties to matrix criteria.

5.4

EDM definition

Ascribe points in a list {xℓ ∈ Rn , ℓ = 1 . . . N } to the columns of a matrix X = [ x1 · · · xN ] ∈ Rn×N

(71)

where N is regarded as cardinality of list X . When matrix D = [dij ] is an EDM, its entries must be related to those points constituting the list by the Euclidean distance-square: for i, j = 1 . . . N ( A.1.1 no.33)

§

dij = kxi − xj k2 = (xi − xj )T (xi − xj ) = kxi k2 + kxj k2 − 2xT i xj ¸· ¸ · £ ¤ xi I −I = xT xT i j xj −I I

(833)

= vec(X)T (Φij ⊗ I ) vec X = hΦij , X TX i

where 

  vec X =  

x1 x2 .. . xN



   ∈ RnN 

(834)

§

and where ⊗ signifies Kronecker product ( D.1.2.1). Φij ⊗ I is positive semidefinite (1424) having I ∈ S n in its ii th and jj th block of entries while −I ∈ S n fills its ij th and ji th block; id est, N T T T Φij , δ((ei eT j + ej ei )1) − (ei ej + ej ei ) ∈ S+ T T T = ei eT i + ej ej − ei ej − ej ei

(835)

= (ei − ej )(ei − ej )T

where {ei ∈ RN , i = 1 . . . N } is the set of standard basis vectors. Thus each entry dij is a convex quadratic function ( A.4.0.0.1) of vec X (32). [285, 6]

§

§

5.4. EDM DEFINITION

381

The collection of all Euclidean distance matrices EDMN is a convex subset ×N of RN called the EDM cone ( 6, Figure 144 p.528); +

§

N ×N 0 ∈ EDMN ⊆ SN ⊂ SN h ∩ R+

(836)

An EDM D must be expressible as a function of some list X ; id est, it must have the form D(X) , δ(X TX)1T + 1δ(X TX)T − 2X TX ∈ EDMN = [vec(X)T (Φij ⊗ I ) vec X , i, j = 1 . . . N ]

(837) (838)

Function D(X) will make an EDM given any X ∈ Rn×N , conversely, but D(X) is not a convex function of X ( 5.4.1). Now the EDM cone may be described: © ª EDMN = D(X) | X ∈ RN −1×N (839)

§

Expression D(X) is a matrix definition of EDM and so conforms to the Euclidean metric properties: Nonnegativity of EDM entries (property 1, 5.2) is obvious from the distance-square definition (833), so holds for any D expressible in the form D(X) in (837). When we say D is an EDM, reading from (837), it implicitly means the main diagonal must be 0 (property 2, self-distance) and D must be symmetric (property 3); δ(D) = 0 and DT = D or, equivalently, D ∈ SN h are necessary matrix criteria.

§

5.4.0.1

homogeneity

Function D(X) is homogeneous in the sense, for ζ ∈ R p ◦

p D(ζX) = |ζ| ◦ D(X)

(840)

where the positive square root is entrywise ( ◦ ). Any nonnegatively scaled EDM remains an EDM; id est, the matrix class EDM is invariant to nonnegative scaling (α D(X) for α ≥ 0) because all EDMs of dimension N constitute a convex cone EDMN ( 6, Figure 131).

§

382

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.4.1

−VNT D(X)VN convexity

¸¶ xi are convex quadratic functions. Yet We saw that EDM entries dij xj −D(X) (837) is not a quasiconvex function of matrix X ∈ Rn×N because the second directional derivative ( 3.9) ¯ ¡ ¢ d 2 ¯¯ (841) − 2 ¯ D(X + t Y ) = 2 −δ(Y T Y )1T − 1δ(Y T Y )T + 2 Y T Y dt t=0 µ·

§

§

is indefinite for any Y ∈ Rn×N since its main diagonal is 0. [146, 4.2.8] [188, 7.1, prob.2] Hence −D(X) can neither be convex in X . The outcome is different when instead we consider

§

−VNT D(X)VN = 2VNTX TXVN

§

(842)

where we introduce the full-rank skinny Schoenberg auxiliary matrix ( B.4.2)   −1 −1 · · · −1  1 0  · ¸  1  1 −1T   1 VN , √  ∈ RN ×N −1 (843)  = √ I  2 2 . ..   0 1 (N (VN ) = 0) having range

R(VN ) = N (1T ) ,

VNT 1 = 0

§

(844)

Matrix-valued function (842) meets the criterion for convexity in 3.8.3.0.2 over its domain that is all of Rn×N ; videlicet, for any Y ∈ Rn×N d2 T − 2 VN D(X + t Y )VN = 4VNT Y T Y VN º 0 dt

(845)

§

Quadratic matrix-valued function −VNT D(X)VN is therefore convex in X achieving its minimum, with respect to a positive semidefinite cone ( 2.7.2.2), at X = 0. When the penultimate number of points exceeds the dimension of the space n < N − 1, strict convexity of the quadratic (842) becomes impossible because (845) could not then be positive definite.

5.4. EDM DEFINITION

5.4.2

383

Gram-form EDM definition

§

Positive semidefinite matrix X TX in (837), formed from inner product of list X , is known as a Gram matrix ; [232, 3.6]   kx1 k2 xT xT · · · xT 1 x2 1 x3 1 xN  T   x1 [ x1 · · · xN ]  xT kx2 k2 xT · · · xT 2 x1 2 x3 2 xN     .   ∈ SN T T 2 ... T G , X TX =  ..  = x x x x kx k x x + 3  3 1 3 2 3 N   .  T . . . . .. .. .. .. xN  ..  T T T 2 xN x1 xN x2 xN x3 · · · kxN k   1 cos ψ12 cos ψ13 · · · cos ψ1N       kx1 k kx1 k cos ψ12 1 cos ψ23 · · · cos ψ2N     kx2 k     kx2 k      .   . = δ  ..   cos ψ13 cos ψ23 . cos ψ3N  δ  ..  1  .     .  .. . . . . .. .. .. ..   . kxN k kxN k cos ψ1N cos ψ2N cos ψ3N · · · 1 ,

p

p δ 2(G) Ψ δ 2(G)

(846)

where ψij (865) is angle between vectors xi and xj , and where δ 2 denotes a diagonal matrix in this case. Positive semidefiniteness of interpoint angle matrix Ψ implies positive semidefiniteness of Gram matrix G ; Gº0 ⇐ Ψº0

§

(847)

When δ 2(G) is nonsingular, then G º 0 ⇔ Ψ º 0. ( A.3.1.0.5) Distance-square dij (833) is related to Gram matrix entries G T = G , [gij ] dij = gii + gjj − 2gij = hΦij , Gi

where Φij is defined in (835). Hence the linear EDM definition ) D(G) , δ(G)1T + 1δ(G)T − 2G ∈ EDMN ⇐ Gº0 = [hΦij , Gi , i, j = 1 . . . N ] The EDM cone may be described, (confer (935)(941)) © ª EDMN = D(G) | G ∈ SN +

(848)

(849)

(850)

384 5.4.2.1

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX First point at origin

Assume the first point x1 in an unknown list X resides at the origin; Xe1 = 0 ⇔ Ge1 = 0

(851)

T Consider the symmetric translation (I − 1eT 1 )D(G)(I − e1 1 ) that shifts the first row and column of D(G) to the origin; setting Gram-form EDM operator D(G) = D for convenience, ¡ ¢ T T T T 1 = G − (Ge1 1T + 1eT − D − (De1 1T + 1eT 1 G) + 1e1 Ge1 1 1 D) + 1e1 De1 1 2 (852) where   1  0  (853) e1 ,  ..  . 0 is the first vector from the standard basis. Then it follows for D ∈ SN h ¡ ¢ 1 G = − D − (De1 1T + 1eT x1 = 0 1 D) 2 , √ √ ¤T £ ¤1 £ 2VN D 0 2VN 2 =− 0 · ¸ (854) 0 0T = T 0 −VN DVN

where

VNT GVN = −VNTDVN 21

∀X

i h √ 2VN I − e1 1T = 0

(855)

§

is a projector nonorthogonally projecting ( E.1) on subspace N SN 1 = {G ∈ S | Ge1 = 0} o n£ √ √ ¤T £ ¤ N 2VN Y 0 2VN | Y ∈ S = 0

(1925)

in the Euclidean sense. From (854) we get sufficiency of the first matrix criterion for an EDM proved by Schoenberg in 1935; [290]5.7 ( −1 −VNTDVN ∈ SN + N D ∈ EDM ⇔ (856) D ∈ SN h From (844) we know R(VN ) = N (1T ) , so (856) is the same as (832). In fact, any matrix V in place of VN will satisfy (856) whenever R(V ) = R(VN ) = N (1T ). But VN is the matrix implicit in Schoenberg’s seminal exposition. 5.7

5.4. EDM DEFINITION

385

§

We provide a rigorous complete more geometric proof of this fundamental on Wıκımization] Schoenberg criterion in ·5.9.1.0.4. [isedm(D) ¸ T 0 0 By substituting G = (854) into D(G) (849), 0 −VNTDVN · ¸ · ¸ h ¡ ¢T i 0 0 0T T T ¡ ¢ 1 + 1 0 δ −VN DVN D= −2 (955) 0 −VNTDVN δ −VNTDVN

§

assuming x1 = 0. Details of this bijection are provided in 5.6.2. 5.4.2.2

0 geometric center

§

Assume the geometric center ( 5.5.1.0.1) of an unknown list X is the origin; X1 = 0 ⇔ G1 = 0

(857)

Now consider the calculation (I − N1 11T )D(G)(I − N1 11T ) , a geometric centering or projection operation. ( E.7.2.0.2) Setting D(G) = D for convenience as in 5.4.2.1, ¡ ¢ G = − D − N1 (D11T + 11TD) + N12 11TD11T 21 , X1 = 0

§

§

= −V D V

V G V = −V D V

1 2 1 2

(858)

∀X

where more properties of the auxiliary (geometric centering, projection) matrix 1 V , I − 11T ∈ SN (859) N are found in B.4. V G V may be regarded as a covariance matrix of means 0. From (858) and the assumption D ∈ SN h we get sufficiency of the more popular form of Schoenberg’s criterion: ( −V D V ∈ SN + N D ∈ EDM ⇔ (860) N D ∈ Sh

§

§

Of particular utility when D ∈ EDMN is the fact, ( B.4.2 no.20) (833) Ã ! ¢ ¡ P P 1 1 vec(X)T dij = 2N Φij ⊗ I vec X tr −V D V 21 = 2N i,j

= tr(V G V ) , = tr G

i,j

(861)

Gº0

=

N P

ℓ=1

kxℓ k2 = kXk2F ,

X1 = 0

386

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

P where Φij ∈ SN + (835), therefore convex in vec X . We will find this trace useful as a heuristic to minimize affine dimension of an unknown list arranged columnar in X ( 7.2.2), but it tends to facilitate reconstruction of a list configuration having least energy; id est, it compacts a reconstructed list by minimizing total norm-square of the vertices. By substituting G = −V D V 21 (858) into D(G) (849), assuming X1 = 0

§

¡ ¢ ¡ ¢T ¡ ¢ D = δ −V D V 12 1T + 1δ −V D V 12 − 2 −V D V 21

(945)

§

Details of this bijection can be found in 5.6.1.1. 5.4.2.3

hypersphere

These foregoing relationships allow combination of distance and Gram constraints in any optimization problem we might pose:

ˆ Interpoint angle Ψ can be constrained by fixing all the individual point lengths δ(G)1/2 ; then

1 Ψ = − δ 2(G)−1/2 V D V δ 2(G)−1/2 2

(862)

ˆ (confer §5.9.1.0.3, (1044) (1190)) Constraining all main diagonal entries gii of a Gram matrix to 1, for example, is equivalent to the constraint that all points lie on a hypersphere of radius 1 centered at the origin. D = 2(g11 11T − G) ∈ EDMN

(863)

Requiring 0 geometric center then becomes equivalent to the constraint: D1 = 2N 1. [84, p.116] Any further constraint on that Gram matrix applies only to interpoint angle matrix Ψ = G . Because any point list may be constrained to lie on a hypersphere boundary whose affine dimension exceeds that of the list, a Gram matrix may always be constrained to have equal positive values along its main diagonal. (Laura Klanfer 1933 [290, 3]) This observation renewed interest in the elliptope ( 5.9.1.0.1).

§

§

5.4. EDM DEFINITION

387

5.4.2.3.1 Example. List-member constraints via Gram matrix. Capitalizing on identity (858) relating Gram and EDM D matrices, a constraint set such as  ¢ ¡ = kxi k2  tr − 12 V D V ei eT i   ¡ 1 ¢ T T 1 T tr − 2 V D V (ei ej + ej ei ) 2 = xi xj (864)  ¢ ¡ 1   = kxj k2 tr − 2 V D V ej eT j

§

relates list member xi to xj to within an isometry through inner-product identity [355, 1-7] xT i xj (865) cos ψij = kxi k kxj k where ψij is angle between the two vectors as in (846). For M list members, there total M (M+1)/2 such constraints. Angle constraints are incorporated in Example 5.4.2.3.3 and Example 5.4.2.3.10. 2

Consider the academic problem of finding a Gram matrix subject to constraints on each and every entry of the corresponding EDM: find −V D V 12 ∈ SN D∈SN h ® ­ T 1 ˇ subject to D , (ei eT j + ej ei ) 2 = dij ,

i, j = 1 . . . N ,

i<j

(866)

−V D V º 0

where the dˇij are given nonnegative constants. EDM D can, of course, be replaced with the equivalent Gram-form (849). Requiring only the self-adjointness property (1362) of the main-diagonal linear operator δ we get, for A ∈ SN ­ ® hD , Ai = δ(G)1T + 1δ(G)T − 2G , A = hG , δ(A1) − Ai 2 (867) Then the problem equivalent to (866) becomes, for G ∈ SN c ⇔ G1 = 0

G ∈ SN D E ¡ ¢ T T T subject to G , δ (ei eT + e e )1 − (e e + e e ) = dˇij , j j i i j j i find

G∈SN c

Gº0

i, j = 1 . . . N ,

i<j

(868)

388

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Figure 110: Rendering of Fermat point in oil on canvas by Suman Vaze. Three circles intersect at Fermat point of minimum total distance from three vertices of (and interior to) red/black/white triangle.

Barvinok’s Proposition 2.9.3.0.1 predicts existence for either formulation (866) or (868) such that implicit equality constraints induced by subspace membership are ignored % $p 8(N (N − 1)/2) + 1 − 1 = N−1 (869) rank G , rank V D V ≤ 2

§

§

because, in each case, the Gram matrix is confined to a face of positive N −1 semidefinite cone SN ( 6.7.1). ( E.7.2.0.2) This bound + isomorphic with S+ is tight ( 5.7.1.1) and is the greatest upper bound.5.8

§

5.4.2.3.2 Example. First duality. Kuhn reports that the first dual optimization problem5.9 to be recorded in the literature dates back to 1755. [Wıκımization] Perhaps more intriguing 5.8

−V D V |N ← 1 = 0 (§B.4.1) By dual problem is meant: in the strongest sense, the optimal objective achieved by a maximization problem dual to a given minimization problem (related to each other by a Lagrangian function) is always equal to the optimal objective achieved by the minimization. (Figure 56 Example 2.13.1.0.3) A dual problem is always convex. 5.9

5.4. EDM DEFINITION

389

is the fact: this earliest instance of duality is a two-dimensional Euclidean distance geometry problem known as a Fermat point (Figure 110) named after the French mathematician. Given N distinct points in the plane {xi ∈ R2 , i = 1 . . . N } , the Fermat point y is an optimal solution to minimize y

N X i=1

ky − xi k

(870)

a convex minimization of total distance. The historically first dual problem formulation asks for the smallest equilateral triangle encompassing (N = 3) three points xi . Another problem dual to (870) (Kuhn 1967) maximize {zi }

subject to

N P

i=1 N P

hzi , xi i (871)

zi = 0

i=1

kzi k ≤ 1 ∀ i has interpretation as minimization of work required to balance potential energy in an N -way tug-of-war between equally matched opponents situated at {xi }. [349] It is not so straightforward to write the Fermat point problem (870) equivalently in terms of a Gram matrix from this section. Squaring instead minimize α

N X i=1

kα−xi k2 ≡

minimize D∈SN +1

h−V , Di

T 1 ˇ subject to hD , ei eT j + ej ei i 2 = dij ∀(i, j) ∈ I

−V D V º 0

(872)

yields an inequivalent convex geometric centering problem whose equality constraints comprise EDM D main-diagonal zeros and known distances-square.5.10 Going the other way, a problem dual to total distance-square maximization (Example 6.4.0.0.1) is a penultimate minimum eigenvalue problem having application to PageRank calculation by search engines [214, 4]. [317] Fermat function (870) is empirically compared with (872) in [57, 8.7.3], but for multiple unknowns in R2 , where propensity of (870) for producing

§

§

α⋆ is geometric center of points xi (919). For three points, I = {1, 2, 3} ; optimal affine dimension (§5.7) must be 2 because a third dimension can only increase total distance. Minimization of h−V, Di is a heuristic for rank minimization. (§7.2.2) 5.10

390

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX x3 x1 x6 x2 x4 x5

Figure 111: Arbitrary hexagon in R3 whose vertices are labelled clockwise.

§

zero distance between unknowns is revealed. An optimal solution to (870) gravitates toward gradient discontinuities ( D.2.1), as in Figure 69, whereas optimal solution to (872) is less compact in the unknowns.5.11 2

§

5.4.2.3.3 Example. Hexagon. Barvinok [25, 2.6] poses a problem in geometric realizability of an arbitrary hexagon (Figure 111) having: 1. prescribed (one-dimensional) face-lengths l 2. prescribed angles ϕ between the three pairs of opposing faces 3. a constraint on the sum of norm-square of each and every vertex x ; ten affine equality constraints in all on a Gram matrix G ∈ S6 (858). Let’s realize this as a convex feasibility problem (with constraints written in the same order) also assuming 0 geometric center (857): find6 −V D V 12 ∈ S6 D∈Sh ¡ ¢ 2 T 1 subject to tr D(ei eT j −1 = (i = 1 . . . 6) mod 6 j + ej ei ) 2 = lij , ¢ ¡ 1 1 tr − 2 V D V (Ai + AT i ) 2 = cos ϕi , i = 1, 2, 3 tr(− 12 V D V ) = 1 −V D V º 0 (873) 5.11

Optimal solution to (870) has mechanical interpretation in terms of interconnecting springs with constant force when distance is nonzero; otherwise, 0 force. Problem (872) is interpreted instead using linear springs.

5.4. EDM DEFINITION

391

Figure 112: Sphere-packing illustration from [348, kissing number ]. Translucent balls illustrated all have the same diameter.

where, for Ai ∈ R6×6 (865) A1 = (e1 − e6 )(e3 − e4 )T /(l61 l34 ) A2 = (e2 − e1 )(e4 − e5 )T /(l12 l45 ) A3 = (e3 − e2 )(e5 − e6 )T /(l23 l56 )

(874)

and where the first constraint on length-square lij2 can be equivalently written as a constraint on the Gram matrix −V D V 12 via (867). We show how to numerically solve such a problem by alternating projection in E.10.2.1.1. Barvinok’s Proposition 2.9.3.0.1 asserts existence of a list, corresponding to Gram matrix G solving this feasibility problem, whose affine dimension ( 5.7.1.1) does not exceed 3 because the convex feasible set is bounded by the third constraint tr(− 21 V D V ) = 1 (861). 2

§

§

5.4.2.3.4 Example. Kissing number of sphere packing. Two nonoverlapping Euclidean balls are said to kiss if they touch. An elementary geometrical problem can be posed: Given hyperspheres, each having the same diameter 1, how many hyperspheres can simultaneously kiss one central hypersphere? [369] The noncentral hyperspheres are allowed, but not required, to kiss. As posed, the problem seeks the maximal number of spheres K kissing a central sphere in a particular dimension. The total number of spheres is N = K + 1. In one dimension the answer to this kissing problem is 2. In two dimensions, 6. (Figure 7)

392

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

The question was presented in three dimensions to Isaac Newton in the context of celestial mechanics, and became controversy with David Gregory on the campus of Cambridge University in 1694. Newton correctly identified the kissing number as 12 (Figure 112) while Gregory argued for 13. Their dispute was finally resolved in 1953 by Sch¨ utte & van der Waerden. [276] In 2003, Oleg Musin tightened the upper bound on kissing number K in four dimensions from 25 to K = 24 by refining a method by Philippe Delsarte from 1973 providing an infinite number [16] of linear inequalities necessary for converting a rank-constrained semidefinite program5.12 to a linear program.5.13 [254] There are no proofs known for kissing number in higher dimension excepting dimensions eight and twenty four. Interest persists [80] because sphere packing has found application to error correcting codes from the fields of communications and information theory; specifically to quantum computing. [87] Translating this problem to an EDM graph realization (Figure 108, Figure 113) is suggested by Pfender & Ziegler. Imagine the centers of each sphere are connected by line segments. Then the distance between centers must obey simple criteria: Each sphere touching the central sphere has a line segment of length exactly 1 joining its center to the central sphere’s center. All spheres, excepting the central sphere, must have centers separated by a distance of at least 1. From this perspective, the kissing problem can be posed as a semidefinite program. Assign index 1 to the central sphere, and assume a total of N spheres: minimize − tr W VNTDVN D∈SN

subject to D1j = 1 , Dij ≥ 1 ,

D ∈ EDMN

j = 2...N 2 ≤ i < j = 3...N

(875)

Then the kissing number K = Nmax − 1 5.12

(876)

whose feasible set belongs to that subset of an elliptope (§5.9.1.0.1) bounded above by some desired rank. 5.13 Simplex-method solvers for linear programs produce numerically better results than contemporary log-barrier (interior-point method) solvers for semidefinite programs by about 7 orders of magnitude.

5.4. EDM DEFINITION

393

is found from the maximal number of spheres N that solve this semidefinite program in a given affine dimension r . Matrix W can be interpreted as the −1 direction of search through the positive semidefinite cone SN for a rank-r + optimal solution −VNTD⋆ VN ; it is constant, in this program, determined by a method disclosed in 4.4.1. In two dimensions,   4 1 2 −1 −1 1  1 4 −1 −1 2 1    1  2 −1 4 1 1 −1  W = (877) 6  −1 −1 1 4 1 2    −1 2 1 1 4 −1  1 1 −1 2 −1 4

§

In three dimensions, 

         W =         

9 1 −2 −1 3 −1 −1 1 2 1 −2 1

1 9 3 −1 −1 1 1 −2 1 2 −1 −1

−2 3 9 1 2 −1 −1 2 −1 −1 1 2

−1 −1 1 9 1 −1 1 −1 3 2 −1 1

3 −1 2 1 9 1 1 −1 −1 −1 1 −1

−1 1 −1 −1 1 9 2 −1 2 −1 2 3

−1 1 −1 1 1 2 9 3 −1 1 −2 −1

1 −2 2 −1 −1 −1 3 9 2 −1 1 1

2 1 −1 3 −1 2 −1 2 9 −1 1 −1

1 2 −1 2 −1 −1 1 −1 −1 9 3 1

−2 −1 1 −1 1 2 −2 1 1 3 9 −1

1 −1 2 1 −1 3 −1 1 −1 1 −1 9



        1   12        

(878)

A four-dimensional solution has rational direction matrix as well, but these direction matrices are not unique and their precision not critical. Here is an optimal point list5.14 in Matlab output format: 5.14

An optimal five-dimensional point list is known: The answer was known at least 175 years ago. I believe Gauss knew it. Moreover, Korkine & Zolotarev proved in 1882 that D5 is the densest lattice in five dimensions. So they proved that if a kissing arrangement in five dimensions can be extended to some lattice, then k(5) = 40. Of course, the conjecture in the general case also is: k(5) = 40. You would like to see coordinates? Easily. √ Let A = 2 . Then p(1) = (A, A, 0, 0, 0), p(2) = (−A, A, 0, 0, 0), p(3) = (A, −A, 0, 0, 0), . . . p(40) = (0, 0, 0, −A, −A) ; i.e., we are considering points with coordinates that have two A and three 0 with any choice of signs and any ordering of the coordinates; the same

394

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Columns 1 through 6 X = 0 -0.1983 -0.4584 0 0.6863 0.2936 0 -0.4835 0.8146 0 0.5059 0.2004 Columns 7 through 12 -0.4815 -0.9399 -0.7416 0 0.2936 -0.3927 -0.8756 -0.0611 0.4224 -0.0372 0.1632 -0.3427 Columns 13 through 18 0.2832 -0.2926 -0.6473 0.6863 0.9176 -0.6239 0.3922 0.1698 -0.2309 0.5431 -0.2088 0.3721 Columns 19 through 25 -0.0943 0.6473 -0.1657 0.2313 0.6239 -0.6239 0.6533 0.2309 0.6448 -0.7147 -0.3721 0.4093

0.1657 0.6239 -0.6448 -0.4093

0.9399 -0.2936 0.0611 -0.1632

0.7416 0.3927 -0.4224 0.3427

0.1983 -0.6863 0.4835 -0.5059

0.4584 -0.2936 -0.8146 -0.2004

-0.2832 -0.6863 -0.3922 -0.5431

0.0943 -0.2313 -0.6533 0.7147

0.3640 -0.0624 -0.1613 -0.9152

-0.3640 0.0624 0.1613 0.9152

0.2926 -0.9176 -0.1698 0.2088

-0.5759 0.2313 -0.2224 -0.7520

0.5759 -0.2313 0.2224 0.7520

0.4815 0 0.8756 0.0372

This particular optimal solution was found by solving a problem sequence in increasing number of spheres. Numerical problems begin to arise with matrices of this size due to interior-point methods of solution. By eliminating some equality constraints for this particular problem, matrix size can be reduced: From 5.8.3 we have · T ¸ 1 0 T T (879) −VN DVN = 11 − [ 0 I ] D I 2

§

(which does not hold more generally) where identity matrix I ∈ SN −1 has coordinates-expression in dimensions 3 and 4. The first miracle happens in dimension 6. There are better packings than D6 (Conjecture: k(6) = 72). It’s a real miracle how dense the packing is in eight dimensions (E8 =Korkine & Zolotarev packing that was discovered in 1880s) and especially in dimension 24, that is the so-called Leech lattice. Actually, people in coding theory have conjectures on the kissing numbers for dimensions up to 32 (or even greater? ). However, sometimes they found better lower bounds. I know that Ericson & Zinoviev a few years ago discovered (by hand, no computer ) in dimensions 13 and 14 better kissing arrangements than were known before. −Oleg Musin

5.4. EDM DEFINITION

395

one less dimension than EDM D . By defining an EDM principal submatrix · T ¸ 0 −1 ˆ D , [0 I ] D ∈ SN (880) h I we get a convex problem equivalent to (875) minimize ˆ N −1 D∈S

ˆ − tr(W D)

ˆ ij ≥ 1 , subject to D 1 ≤ i < j = 2... N−1 ˆ1 º0 11T − D 2 ˆ δ(D) = 0

§

ˆ 1 belongs to an elliptope ( 5.9.1.0.1). Any feasible matrix 11T − D 2

(881)

2

This next example shows how finding the common point of intersection for three circles in a plane, a nonlinear problem, has convex expression. 5.4.2.3.5 Example. Trilateration in wireless sensor network. Given three known absolute point positions in R2 (three anchors xˇ2 , xˇ3 , xˇ4 ) and only one unknown point (one sensor x1 ), the sensor’s absolute position is determined from its noiseless measured distance-square dˇi1 to each of three anchors (Figure 2, Figure 113a). This trilateration can be expressed as a convex optimization problem in terms of list X , [ x1 xˇ2 xˇ3 xˇ4 ] ∈ R2×4 and Gram matrix G ∈ S4 (846): minimize

G∈ S4 , X∈ R2×4

subject to

tr G tr(GΦi1 ) ¡ ¢ tr Gei eT i T tr(G(ei eT j + ej ei )/2) X(: , 2 : 4) ¸ · I X XT G

= = = =

dˇi1 , i = 2, 3, 4 2 kˇ xi k , i = 2, 3, 4 T xˇi xˇj , 2 ≤ i < j = 3, 4 [ xˇ2 xˇ3 xˇ4 ]

(882)

º 0

where Φij = (ei − ej )(ei − ej )T ∈ SN (835) + and where the constraint on distance-square dˇi1 is equivalently written as a constraint on the Gram matrix via (848). There are 9 independent affine

396

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

xˇ3 √

(a) √

xˇ2

d12

d13 √

x3

xˇ4

x4

(b) d14 x2

x1

x1

x5

x4

x5

x2

x6

x2 (d)

(c) x3

x3

x4

x1

x1

Figure 113: (a) Given three distances indicated with absolute point positions xˇ2 , xˇ3 , xˇ4 known and noncollinear, absolute position of x1 in R2 can be precisely and uniquely determined by trilateration; solution to a system of nonlinear equations. Dimensionless EDM graphs (b) (c) (d) represent EDMs in various states of completion. Line segments represent known absolute distances and may cross without vertex at intersection. (b) Four-point list must always be embeddable in affine subset having dimension rank VNTDVN not exceeding 3. To determine relative position of x2 , x3 , x4 , triangle inequality is necessary and sufficient ( 5.14.1). Knowing all distance information, then (by injectivity of D ( 5.6)) point position x1 is uniquely determined to within an isometry in any dimension. (c) When fifth point is introduced, only distances to x3 , x4 , x5 are required to determine relative position of x2 √in R2 . Graph represents first instance of√ missing information; d12 . (d) Three distances are absent √ distance √ ( d12 , d13 , d23 ) from complete set of interpoint distances, yet unique isometric reconstruction ( 5.4.2.3.7) of six points in R2 is certain.

§

§

§

5.4. EDM DEFINITION

397

equality constraints on that Gram matrix while the sensor is constrained only by dimensioning to lie in R2 . Although tr G the objective of 5.15 insures a solution on the boundary of positive semidefinite minimization cone S4+ , for this problem, we claim that the set of feasible Gram matrices forms a line ( 2.5.1.1) in isomorphic R10 tangent ( 2.1.7.2) to the positive semidefinite cone boundary. ( 5.4.2.3.6, confer 4.2.1.3) By Schur complement ( A.4, 2.9.1.0.1), any feasible G and X provide

§

§

§ §

§

§

G º X TX

(883)

which is a convex relaxation of the desired (nonconvex) equality constraint ¸ ¸ · · I X I [I X ] (884) = XT G XT expected positive semidefinite rank-2 under noiseless conditions. by (1431), the relaxation admits (3 ≥) rank G ≥ rank X

But, (885)

(a third dimension corresponding to an intersection of three spheres, not circles, were there noise). If rank of an optimal solution equals 2, · ¸ I X⋆ rank = 2 (886) X ⋆T G⋆ then G⋆ = X ⋆TX ⋆ by Theorem A.4.0.0.5. As posed, this localization problem does not require affinely independent (Figure 27, three noncollinear) anchors. Assuming the anchors exhibit no rotational or reflective symmetry in their affine hull ( 5.5.2) and assuming the sensor x1 lies in that affine hull, then sensor position solution x⋆1 = X ⋆ (: , 1) is unique under noiseless measurement. [298] 2

§

This preceding transformation of trilateration to a semidefinite program works all the time ((886) holds) despite relaxation (883) because the optimal solution set is a unique point. 5.15

Trace (tr G = hI , Gi) minimization is a heuristic for rank minimization. (§7.2.2.1) It may be interpreted as squashing G which is bounded below by X TX as in (883); id est, G−X TX º 0 ⇒ tr G ≥ tr X TX (1429). δ(G−X TX) = 0 ⇔ G = X TX (§A.7.2) ⇒ tr G = tr X TX which is a condition necessary for equality.

398

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.4.2.3.6 Proof (sketch). Only the sensor location x1 is unknown. The objective function together with the equality constraints make a linear system of equations in Gram matrix variable G tr G = kx1 k2 + kˇ x2 k2 + kˇ x3 k2 + kˇ x4 k2 tr(GΦi1 ) = dˇi1 , i = 2, 3, 4 (887) ¡ ¢ T 2 tr Gei ei = kˇ xi k , i = 2, 3, 4 T tr(G(ei eT ˇT ˇj , 2 ≤ i < j = 3, 4 j + ej ei )/2) = x ix which is invertible: −1    svec(I )T kx1 k2 + kˇ x2 k2 + kˇ x3 k2 + kˇ x4 k2    svec(Φ21 )T  dˇ21     T   svec(Φ ) ˇ   31 d31     T   svec(Φ41 )   ˇ41 d     T T   2 svec(e e )   2 2   kˇ x k 2  (888)  T T svec G =   2 svec(e e )   3 3   kˇ x k 3   T T   svec(e4 e4 ) 2     kˇ x k 4 ¡ ¢    svec (e eT + e eT )/2 T   T    x ˇ x ˇ 3 2 2 3   ¡ 2 3T ¢   T T     svec (e2 e4 + e4 eT x ˇ x ˇ )/2 4 2 2 ¡ ¢ T T T xˇ3 xˇ4 svec (e3 eT 4 + e4 e3 )/2

That line in the ambient space S4 of G , claimed on page 397, is traced by kx1 k2 ∈ R on the right-hand side, as it turns out. One must show this line to be tangential ( 2.1.7.2) to S4+ in order to prove uniqueness. Tangency is possible for affine dimension 1 or 2 while its occurrence depends completely on the known measurement data. ¥

§

But as soon as significant noise is introduced or whenever distance data is incomplete, such problems can remain convex although the set of all optimal solutions generally becomes a convex set bigger than a single point (and still containing the noiseless solution).

§

5.4.2.3.7 Definition. Isometric reconstruction. (confer 5.5.3) Isometric reconstruction from an EDM means building a list X correct to within a rotation, reflection, and translation; in other terms, reconstruction of relative position, correct to within an isometry, or to within a rigid transformation. △

5.4. EDM DEFINITION

399

How much distance information is needed to uniquely localize a sensor (to recover actual relative position)? The narrative in Figure 113 helps dispel any notion of distance data proliferation in low affine dimension (r < N − 2) .5.16 Huang, Liang, and Pardalos [191, 4.2] claim O(2N ) distances is a least lower bound (independent of affine dimension r) for unique isometric reconstruction; achievable under certain noiseless conditions on graph connectivity and point position. Alfakih shows how to ascertain uniqueness over all affine dimensions via Gale matrix. [9] [4] [5] Figure 108b (p.377, from small completion problem Example 5.3.0.0.2) is an example in R2 requiring only 2N − 3 = 5 known symmetric entries for unique isometric reconstruction, although the four-point example in Figure 113b will not yield a unique reconstruction when any one of the distances is left unspecified. The list represented by the particular dimensionless EDM graph in Figure 114, having only 2N − 3 = 9 absolute distances specified, has only one realization in R2 but has more realizations in higher dimensions. For sake of reference, we provide the complete corresponding EDM:   0 50641 56129 8245 18457 26645  50641 0 49300 25994 8810 20612     56129 49300  0 24202 31330 9160  D = (889)  8245 25994 24202 0 4680 5290     18457 8810 31330 4680 0 6658  26645 20612 9160 5290 6658 0

§

We consider paucity of distance information in this next example which shows it is possible to recover exact relative position given incomplete noiseless distance information. An ad hoc method for recovery of the lowest-rank optimal solution under noiseless conditions is introduced:

5.4.2.3.8 Example. Tandem trilateration in wireless sensor network. Given three known absolute point-positions in R2 (three anchors xˇ3 , xˇ4 , xˇ5 ), two unknown sensors x1 , x2 ∈ R2 have absolute position determinable from their noiseless distances-square (as indicated in Figure 115) assuming the anchors exhibit no rotational or reflective symmetry in their affine hull 5.16

When affine dimension r reaches N − 2, then all distances-square in the EDM must be known for unique isometric reconstruction in Rr ; going the other way, when r = 1 then the condition that the dimensionless EDM graph be connected is necessary and sufficient. [174, §2.2]

400

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

x1

x4 x5 x6 x3

x2

Figure 114: Incomplete EDM corresponding to this dimensionless EDM graph provides unique isometric reconstruction in R2 . (drawn freehand, no symmetry intended)

xˇ4 x1 x2 xˇ3

xˇ5

Figure 115: Two sensors • and three anchors ◦ in R2 . (Ye) Connecting line-segments denote known absolute distances. Incomplete EDM corresponding to this dimensionless EDM graph provides unique isometric reconstruction in R2 .

5.4. EDM DEFINITION

401

10

5

xˇ4 0

xˇ3

xˇ5

−5

x2 −10

x1 −15 −6

−4

−2

0

2

4

6

8

Figure 116: Given in red # are two discrete linear trajectories of sensors x1 and x2 in R2 localized by algorithm (890) as indicated by blue bullets • . Anchors xˇ3 , xˇ4 , xˇ5 corresponding to Figure 115 are indicated by ⊗ . When targets # and bullets • coincide under these noiseless conditions, localization is successful. On this run, two visible localization errors are due to rank-3 Gram optimal solutions. These errors can be corrected by choosing a different normal in objective of minimization.

402

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

§

( 5.5.2). This example differs from Example 5.4.2.3.5 in so far as trilateration of each sensor is now in terms of one unknown position: the other sensor. We express this localization as a convex optimization problem (a semidefinite program, 4.1) in terms of list X , [ x1 x2 xˇ3 xˇ4 xˇ5 ] ∈ R2×5 and Gram matrix G ∈ S5 (846) via relaxation (883):

§

minimize

G∈ S5 , X∈ R2×5

tr G

subject to tr(GΦi1 )

where

tr(GΦi2 ) ¡ ¢ tr Gei eT i T tr(G(ei eT j + ej ei )/2) X(: , 3 : 5) ¸ · I X XT G

= dˇi1 , = dˇi2 , = kˇ xi k2 , = xˇT ˇj , ix = [ xˇ3 xˇ4 xˇ5 ]

i = 2, 4, 5 i = 3, 5 i = 3, 4, 5 3 ≤ i < j = 4, 5

(890)

º 0

Φij = (ei − ej )(ei − ej )T ∈ SN +

(835)

This problem realization is fragile because of the unknown distances between sensors and anchors. Yet there is no more information we may include beyond the 11 independent equality constraints on the Gram matrix (nonredundant constraints not antithetical) to reduce the feasible set5.17 . (By virtue of their dimensioning, the sensors are already constrained to R2 the affine hull of the anchors.) Exhibited in Figure 116 are two mistakes in solution X ⋆ (: , 1 : 2) due to a rank-3 optimal Gram matrix G ⋆ . The trace objective is a heuristic minimizing convex envelope of quasiconcave function5.18 rank G . ( 2.9.2.6.2, 7.2.2.1) A rank-2 optimal Gram matrix can be found and the errors corrected by choosing a different normal for the linear objective function, now implicitly the identity matrix I ; id est,

§

§

tr G = hG , I i ← hG , δ(u)i

(891)

where vector u ∈ R5 is randomly selected. A random search for a good normal δ(u) in only a few iterations is quite easy and effective because 5.17

the presumably nonempty convex set of all points G and X satisfying the constraints. Projection on that nonconvex subset of all N × N -dimensional positive semidefinite matrices, in an affine subset, whose rank does not exceed 2 is a problem considered difficult to solve. [328, §4] 5.18

5.4. EDM DEFINITION

403

the problem is small, an optimal solution is known a priori to exist in two dimensions, a good normal direction is not necessarily unique, and (we speculate) because the feasible affine-subset slices the positive semidefinite 2 cone thinly in the Euclidean sense.5.19 We explore ramifications of noise and incomplete data throughout; their individual effect being to expand the optimal solution set, introducing more solutions and higher-rank solutions. Hence our focus shifts in 4.4 to discovery of a reliable means for diminishing the optimal solution set by introduction of a rank constraint. Now we illustrate how a problem in distance geometry can be solved without equality constraints representing measured distance; instead, we have only upper and lower bounds on distances measured:

§

5.4.2.3.9 Example. Wireless location in a cellular telephone network. Utilizing measurements of distance, time of flight, angle of arrival, or signal power, multilateration is the process of localizing (determining absolute position of) a radio signal source • by inferring geometry relative to multiple fixed base stations ◦ whose locations are known. We consider localization of a cellular telephone by distance geometry, so we assume distance to any particular base station can be inferred from received signal power. On a large open flat expanse of terrain, signal-power measurement corresponds well with inverse distance. But it is not uncommon for measurement of signal power to suffer 20 decibels in loss caused by factors such as multipath interference (signal reflections), mountainous terrain, man-made structures, turning one’s head, or rolling the windows up in an automobile. Consequently, contours of equal signal power are no longer circular; their geometry is irregular and would more aptly be approximated by translated ellipsoids of graduated orientation and eccentricity as in Figure 118. Depicted in Figure 117 is one cell phone x1 whose signal power is automatically and repeatedly measured by 6 base stations ◦ nearby.5.20 Those signal power measurements are transmitted from that cell phone to 5.19

The log det rank-heuristic from §7.2.2.4 does not work here because it chooses the wrong normal. Rank reduction (§4.1.1.2) is unsuccessful here because Barvinok’s upper bound (§2.9.3.0.1) on rank of G⋆ is 4. 5.20 Cell phone signal power is typically encoded logarithmically with 1-decibel increment and 64-decibel dynamic range.

404

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX xˇ3

xˇ7

xˇ2

xˇ4

x1 xˇ6

xˇ5

Figure 117: Regions of coverage by base stations ◦ in a cellular telephone network. The term cellular arises from packing of regions best covered by neighboring base stations. Illustrated is a pentagonal cell best covered by base station xˇ2 . Like a Voronoi diagram, cell geometry depends on base-station arrangement. In some US urban environments, it is not unusual to find base stations spaced approximately 1 mile apart. There can be as many as 20 base-station antennae capable of receiving signal from any given cell phone • ; practically, that number is closer to 6.

Figure 118: Some fitted contours of equal signal power in R2 transmitted from a commercial cellular telephone • over about 1 mile suburban terrain outside San Francisco in 2005. Data courtesy Polaris Wireless. [300]

5.4. EDM DEFINITION

405

base station xˇ2 who decides whether to transfer (hand-off or hand-over ) responsibility for that call should the user roam outside its cell.5.21 Due to noise, at least one distance measurement more than the minimum number of measurements is required for reliable localization in practice; 3 measurements are minimum in two dimensions, 4 in three.5.22 Existence of noise precludes measured distance from the input data. We instead assign measured distance to a range estimate specified by individual upper and lower bounds: di1 is the upper bound on distance-square from the cell phone to i th base station, while di1 is the lower bound. These bounds become the input data. Each measurement range is presumed different from the others. Then convex problem (882) takes the form: minimize

tr G

G∈ S7 , X∈ R2×7

subject to

di1 ≤ tr(GΦi1 ) ¡ ¢ T tr Gei ei T tr(G(ei eT j + ej ei )/2) X(: , 2 : 7) · ¸ I X XT G

≤ = = =

di1 , i=2...7 2 kˇ xi k , i=2...7 T xˇi xˇj , 2≤ i< j = 3...7 [ xˇ2 xˇ3 xˇ4 xˇ5 xˇ6 xˇ7 ]

º 0

(892)

where Φij = (ei − ej )(ei − ej )T ∈ SN +

(835)

This semidefinite program realizes the wireless location problem illustrated in Figure 117. Location X ⋆ (: , 1) is taken as solution, although measurement noise will often cause rank G⋆ to exceed 2. Randomized search for a rank-2 optimal solution is not so easy here as in Example 5.4.2.3.8. We introduce a method in 4.4 for enforcing the stronger rank-constraint (886). To formulate this same problem in three dimensions, point list X is simply redimensioned in the semidefinite program. 2

§

5.21

Because distance to base station is quite difficult to infer from signal power measurements in an urban environment, localization of a particular cell phone • by distance geometry would be far easier were the whole cellular system instead conceived so cell phone x1 also transmits (to base station x ˇ2 ) its signal power as received by all other cell phones within range. 5.22 In Example 4.4.1.2.2, we explore how this convex optimization algorithm fares in the face of measurement noise.

406

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Figure 119: A depiction of molecular conformation. [106]

5.4.2.3.10 Example. (Biswas, Nigam, Ye) Molecular Conformation. The subatomic measurement technique called nuclear magnetic resonance spectroscopy (NMR) is employed to ascertain physical conformation of molecules; e.g., Figure 3, Figure 119. From this technique, distance, angle, and dihedral angle measurements can be obtained. Dihedral angles arise consequent to a phenomenon where atom subsets are physically constrained to Euclidean planes. In the rigid covalent geometry approximation, the bond lengths and angles are treated as completely fixed, so that a given spatial structure can be described very compactly indeed by a list of torsion angles alone. . . These are the dihedral angles between the planes spanned by the two consecutive triples in a chain of four covalently bonded atoms.

§

−G. M. Crippen & T. F. Havel (1988) [83, 1.1] Crippen & Havel recommend working exclusively with distance data because they consider angle data to be mathematically cumbersome. The present example shows instead how inclusion of dihedral angle data into a problem statement can be made elegant and convex.

5.4. EDM DEFINITION

407

As before, ascribe position information to the matrix X = [ x1 · · · xN ] ∈ R3×N

(71)

and introduce a matrix ℵ holding normals η to planes respecting dihedral angles ϕ : ℵ , [ η 1 · · · ηM ] ∈ R3×M (893) As in the other examples, we preferentially work with Gram matrices G because of the bridge they provide between other variables; we define ¸ · T ¸ · T ¸ · [ℵ X ] ℵ ℵ ℵTX ℵ Gℵ Z ∈ RN +M ×N +M (894) , = XT X T ℵ X TX Z T GX whose rank is 3 by assumption. So our problem’s variables are the two Gram matrices GX and Gℵ and matrix Z = ℵTX of cross products. Then measurements of distance-square d can be expressed as linear constraints on GX as in (892), dihedral angle ϕ measurements can be expressed as linear constraints on Gℵ by (865), and normal-vector η conditions can be expressed by vanishing linear constraints on cross-product matrix Z : Consider three points x labelled 1 , 2 , 3 assumed to lie in the ℓ th plane whose normal is ηℓ . There might occur, for example, the independent constraints ηℓT(x1 − x2 ) = 0 ηℓT(x2 − x3 ) = 0

(895)

which are expressible in terms of constant matrices A ∈ RM ×N ; hZ , Aℓ12 i = 0 hZ , Aℓ23 i = 0

(896)

Although normals η can be constrained exactly to unit length, δ(Gℵ ) = 1

(897)

NMR data is noisy; so measurements are given as upper and lower bounds. Given bounds on dihedral angles respecting 0 ≤ ϕj ≤ π and bounds on distances di and given constant matrices Ak (896) and symmetric matrices Φi (835) and Bj per (865), then a molecular conformation problem can be expressed:

408

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX find

GX

subject to

di

Gℵ ∈ SM , GX ∈ SN , Z∈ RM ×N

≤

cos ϕj ≤

tr(GX Φi ) ≤ di ,

∀ i ∈ I1

tr(Gℵ Bj ) ≤ cos ϕj , ∀ j ∈ I2

hZ , Ak i GX 1 δ(Gℵ ) ¸ · Gℵ Z Z T GX ¸ · Gℵ Z rank Z T GX

∀ k ∈ I3

=0, =0 =1

(898)

º0 =3

§

where GX 1 = 0 provides a geometrically centered list X ( 5.4.2.2). Ignoring the rank constraint would tend to force cross-product matrix Z to zero. What binds these three variables is the rank constraint; we show how to satisfy it in 4.4. 2

§

5.4.3

Inner-product form EDM definition

(p.22) We might, for example, realize a constellation given only interstellar distance (or, equivalently, distance from Earth and relative angular measurement; the Earth as vertex to two stars).

§

§

Equivalent to (833) is [355, 1-7] [309, 3.2] p dij = dik + dkj − 2 dik dkj cos θikj · ¸ "p # dik p ¤ £p 1 −e ıθikj = dik dkj p −ıθikj −e 1 dkj

(899)

√ called law of cosines where ı , −1 , i , j , k are positive integers, and θikj is the angle at vertex xk formed by vectors xi − xk and xj − xk ; cos θikj =

1 (d 2 ik

+ dkj − dij ) (xi − xk )T (xj − xk ) p = kxi − xk k kxj − xk k dik dkj

(900)

5.4. EDM DEFINITION

409

where the numerator forms an inner product of vectors. Distance-square ¸¶ µ· p d is a convex quadratic function5.23 on R2+ whereas dij (θikj ) is dij p ik dkj ⋆ quasiconvex ( 3.9) minimized over domain {−π ≤ θikj ≤ π} by θikj = 0, we get the Pythagorean theorem when θikj = ±π/2, and dij (θikj ) is maximized ⋆ when θikj = ±π ; p ¢2 ¡p dik + dkj , θikj = ±π dij = (901) dij = dik + dkj , θikj = ± π2 p ¢2 ¡p dik − dkj , θikj = 0 dij = so p p p p p | dik − dkj | ≤ dij ≤ dik + dkj (902)

§

Hence the triangle inequality, Euclidean metric property 4, holds for any EDM D . We may construct an inner-product form of the EDM definition for matrices by evaluating (899) for k = 1 : By defining p p p   d12 d13 cos θ213 d12 d14 cos θ214 · · · d12 d1N cos θ21N d12 p p p  d13 d13 d14 cos θ314 · · · d13 d1N cos θ31N   d12 d13 cos θ213 p  p N −1 ... p ΘT Θ ,  d13 d14 cos θ314 d14 d14 d1N cos θ41N   d12 d14 cos θ214 ∈ S   .. .. .. ... ...   . . . p p p d12 d1N cos θ21N d13 d1N cos θ31N d14 d1N cos θ41N · · · d1N (903) then any EDM may be expressed ¸ · · ¸ £ ¤ 0 0T 0 T T T 1 + 1 0 δ(Θ Θ) − 2 D(Θ) , ∈ EDMN δ(ΘT Θ) 0 ΘT Θ · ¸ 0 δ(ΘT Θ)T = δ(ΘT Θ) δ(ΘT Θ)1T + 1δ(ΘT Θ)T − 2ΘT Θ (904) EDMN =

©

D(Θ) | Θ ∈ RN −1×N −1

ª

(905) T

for which all Euclidean metric properties hold. Entries of Θ Θ result from vector inner-products as in (900); id est, 5.23

·

·p pdik dkj

¸ −e ıθikj º 0, having eigenvalues {0, 2}. Minimum is attained for −e−ıθikj 1 ¸ ½ µ1 , µ ≥ 0 , θikj = 0 = . (§D.2.1, [57, exmp.4.5]) 0 , −π ≤ θikj ≤ π , θikj 6= 0 1

410

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX Θ = [ x 2 − x1

x 3 − x1

√ xN − x1 ] = X 2VN ∈ Rn×N −1

···

Inner product ΘT Θ is obviously related to a Gram matrix (846), · ¸ 0 0T G= , x1 = 0 0 ΘT Θ

(906)

(907)

For D = D(Θ) and no condition on the list X (confer (854) (858)) ΘT Θ = −VNTDVN ∈ RN −1×N −1 5.4.3.1

(908)

Relative-angle form

The inner-product form EDM definition is not a unique definition of Euclidean distance matrix; there are approximately five flavors distinguished by their argument to operator D . Here is another one: Like D(X) (837), D(Θ) will make an EDM given any Θ ∈ Rn×N −1 , it is neither a convex function of Θ ( 5.4.3.2), and it is homogeneous in the sense (840). Scrutinizing ΘT Θ (903) we find that because of the arbitrary choice k = 1, distances therein are all with respect to point x1 . Similarly, relative angles in ΘT Θ are between all vector pairs having vertex x1 . Yet picking arbitrary θi1j to fill ΘT Θ will not necessarily make an EDM; inner product (903) must be positive semidefinite.

§

ΘT Θ = √    

d12

0

√

δ(d) Ω

p δ(d) ,

 √   1 cos θ213 · · · cos θ21N d12 0 √    ...  d13   cos θ cos θ 1 31N 213     ... .  . . . . .    .. .. .. .. .   p p 0 d1N d1N cos θ21N cos θ31N · · · 1 (909) 0

d13

p

Expression D(Θ) defines an EDM for any positive semidefinite relative-angle matrix Ω = [cos θi1j , i, j = 2 . . . N ] ∈ SN −1 (910) and any nonnegative distance vector d = [d1j , j = 2 . . . N ] = δ(ΘT Θ) ∈ RN −1

(911)

5.4. EDM DEFINITION

411

§

because ( A.3.1.0.5) Ω º 0 ⇒ ΘT Θ º 0

(912)

Decomposition (909) and the relative-angle matrix inequality Ω º 0 lead to a different expression of an inner-product form EDM definition (904)

D(Ω , d) ,

·

0 d

=

·

0 d

¸

s µ· ¸¶ · ¸ s µ· ¸¶ T 0 0 0 0 δ 1T + 1 0 d − 2 δ d d 0 Ω ¸ T dp p ∈ EDMN d1T + 1d T − 2 δ(d) Ω δ(d) (913) £

¤ T

and another expression of the EDM cone:

n o p EDMN = D(Ω , d) | Ω º 0 , δ(d) º 0

(914)

In the particular circumstance x1 = 0, we can relate interpoint angle matrix Ψ from the Gram decomposition in (846) to relative-angle matrix Ω in (909). Thus, ¸ · 1 0T , x1 = 0 (915) Ψ ≡ 0 Ω 5.4.3.2

Inner-product form −VNT D(Θ)VN convexity

On ·page p 409 ¸ we saw that each EDM entry dij is a convex quadratic function d of p ik and a quasiconvex function of θikj . Here the situation for dkj inner-product form EDM operator D(Θ) (904) is identical to that in 5.4.1 for list-form D(X) ; −D(Θ) is not a quasiconvex function of Θ by the same reasoning, and from (908)

§

−VNT D(Θ)VN = ΘT Θ

(916)

is a convex quadratic function of Θ on domain Rn×N −1 achieving its minimum at Θ = 0.

412 5.4.3.3

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX Inner-product form, discussion

We deduce that knowledge of interpoint distance is equivalent to knowledge of distance and angle from the perspective of one point, x1 in our chosen case. The total amount of information N (N − 1)/2 in ΘT Θ is unchanged5.24 with respect to EDM D .

5.5

Invariance

When D is an EDM, there exist an infinite number of corresponding N -point lists X (71) in Euclidean space. All those lists are related by isometric transformation: rotation, reflection, and translation (offset or shift).

5.5.1

Translation

Any translation common among all the points xℓ in a list will be cancelled in the formation of each dij . Proof follows directly from (833). Knowing that translation α in advance, we may remove it from the list constituting the columns of X by subtracting α1T . Then it stands to reason by list-form definition (837) of an EDM, for any translation α ∈ Rn D(X − α1T ) = D(X)

(917)

In words, interpoint distances are unaffected by offset; EDM D is translation invariant. When α = x1 in particular, √ [ x2 − x1 x3 − x1 · · · xN − x1 ] = X 2VN ∈ Rn×N −1 (906) and so i´ ³ h √ = D(X) D(X − x1 1 ) = D(X − Xe1 1 ) = D X 0 2VN T

T

(918)

The reason for amount O(N 2 ) information is because of the relative measurements. Use of a fixed reference in measurement of angles and distances would reduce required information but is antithetical. In the particular case n = 2, for example, ordering all points xℓ (in a length-N list) by increasing angle of vector xℓ − x1 with respect to j−1 P x2 − x1 , θi1j becomes equivalent to θk,1,k+1 ≤ 2π and the amount of information is 5.24

k=i

reduced to 2N − 3 ; rather, O(N ).

5.5. INVARIANCE

413

5.5.1.0.1 Example. Translating geometric center to origin. We might choose to shift the geometric center αc of an N -point list {xℓ } (arranged columnar in X) to the origin; [327] [149] α = αc , X bc , X1 N1 ∈ P ⊆ A

(919)

where A represents the list’s affine hull. If we were to associate a point-mass mℓ with each of the P points xℓ Pin the list, then their center of mass (or gravity) would be ( xℓ mℓ ) / mℓ . The geometric center is the same as the center of mass under the assumption of uniform mass density across points. [204] The geometric center always lies in the convex hull P of the list; id est, αc ∈ P because bcT 1 = 1 and bc º 0 .5.25 Subtracting the geometric center from every list member, 1 1 (920) X − αc 1T = X − X11T = X(I − 11T ) = XV ∈ Rn×N N N where V is the geometric centering matrix (859). So we have (confer (837)) D(X) = D(XV ) = δ(V TX TXV )1T + 1δ(V TX TXV )T − 2V TX TXV ∈ EDMN (921) 2 5.5.1.1

Gram-form invariance

Following from (921) and the linear Gram-form EDM operator (849):

§

D(G) = D(V G V ) = δ(V G V )1T + 1δ(V G V )T − 2V G V ∈ EDMN

(922)

The Gram-form consequently exhibits invariance to translation by a doublet ( B.2) u1T + 1uT ; D(G) = D(G − (u1T + 1uT ))

(923)

because, for any u ∈ RN , D(u1T + 1uT ) = 0. The collection of all such doublets forms the nullspace (939) to the operator; the translation-invariant ⊥ subspace SN (1923) of the symmetric matrices SN . This means matrix G c can belong to an expanse more broad than a positive semidefinite cone; N⊥ id est, G ∈ SN So explains Gram matrix sufficiency in EDM + − Sc . 5.26 definition (849). Any b from α = Xb chosen such that bT 1 = 1, more generally, makes an auxiliary V -matrix. (§B.4.5) 5.26 A constraint G1 = 0 would prevent excursion into the translation-invariant subspace (numerical unboundedness). 5.25

414

5.5.2

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Rotation/Reflection

Rotation of the list X ∈ Rn×N about some arbitrary point α ∈ Rn , or reflection through some affine subset containing α , can be accomplished via Q(X −α1T ) where Q is an orthogonal matrix ( B.5). We rightfully expect

§

¡ ¢ D Q(X − α1T ) = D(QX − β1T ) = D(QX) = D(X)

(924)

T X T QT p Qp X = X X

(925)

D(Qp X) = D(X)

(926)

Because list-form D(X) is translation invariant, we may safely ignore offset and consider only the impact of matrices that premultiply X . Interpoint distances are unaffected by rotation or reflection; we say, EDM D is rotation/reflection invariant. Proof follows from the fact, QT = Q−1 ⇒ X TQTQX = X TX . So (924) follows directly from (837). The class of premultiplying matrices for which interpoint distances are unaffected is a little more broad than orthogonal matrices. Looking at EDM definition (837), it appears that any matrix Qp such that

will have the property An example is skinny Qp ∈ Rm×n (m > n) having orthonormal columns. We call such a matrix orthonormal. 5.5.2.0.1 Example. Reflection prevention and quadrature rotation. Consider the EDM graph in Figure 120(b) representing known distance between vertices (Figure 120(a)) of a tilted-square diamond in R2 . Suppose some geometrical optimization problem were posed where isometric transformation is feasible excepting reflection, and where rotation must be quantized so that only quadrature rotations are feasible; only multiples of π/2. In two dimensions, a counterclockwise rotation of any vector about the origin by angle θ is prescribed by the orthogonal matrix Q =

·

cos θ − sin θ sin θ cos θ

¸

(927)

5.5. INVARIANCE

415

x2

x3

x2

x1

x3

x1

x4

x4

(a)

(b)

(c)

Figure 120: (a) Four points in quadrature in two dimensions about their geometric center. (b) Complete EDM graph of diamond-shaped vertices. (c) Quadrature rotation of a Euclidean body in R2 first requires a shroud that is the smallest Cartesian square containing it.

whereas reflection of any point through a hyperplane containing the origin ¯· ) ( ¸T ¯ 2 ¯ cos θ x=0 (928) ∂H = x ∈ R ¯ ¯ sin θ

§

is accomplished via multiplication with symmetric orthogonal matrix ( B.5.2) ¸ · sin(θ)2 − cos(θ)2 −2 sin(θ) cos(θ) (929) R = −2 sin(θ) cos(θ) cos(θ)2 − sin(θ)2 Rotation matrix Q is characterized by identical diagonal entries and by antidiagonal entries equal but opposite in sign, whereas reflection matrix R is characterized in the reverse sense. © ª Assign the diamond vertices xℓ ∈ R2 , ℓ = 1 . . . 4 to columns of a matrix X = [ x1 x2 x3 x4 ] ∈ R2×4

(71)

Our scheme to prevent reflection enforces a rotation matrix characteristic upon the coordinates of adjacent points themselves: First shift the geometric center of X to the origin; for geometric centering matrix V ∈ S4 ( 5.5.1.0.1), define Y , XV ∈ R2×4 (930)

§

416

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

To maintain relative quadrature between points (Figure 120(a)) and to prevent reflection, it is sufficient that all interpoint distances be specified and that adjacencies Y (: , 1 : 2) , Y (: , 2 : 3) , and Y (: , 3 : 4) be proportional to 2×2 rotation matrices; any clockwise rotation would ascribe a reflection matrix characteristic. Counterclockwise rotation is thereby enforced by constraining equality among diagonal and antidiagonal entries as prescribed by (927); ¸ · 0 1 Y (: , 2 : 4) (931) Y (: , 1 : 3) = −1 0

Quadrature quantization of rotation can be regarded as a constraint on tilt of the smallest Cartesian square containing the diamond as in Figure 120(c). Our scheme to quantize rotation requires that all square vertices be described by vectors whose entries are nonnegative when the square is translated anywhere interior to the nonnegative orthant. We capture the four square vertices as columns of a product Y C where   1 0 0 1  1 1 0 0   C = (932)  0 1 1 0  0 0 1 1

Then, assuming a unit-square shroud, the affine constraint ¸ · 1/2 T 1 ≥0 YC + 1/2 quantizes rotation, as desired. 5.5.2.1

(933) 2

Inner-product form invariance

Likewise, D(Θ) (904) is rotation/reflection invariant; D(Qp Θ) = D(QΘ) = D(Θ)

(934)

so (925) and (926) similarly apply.

5.5.3

Invariance conclusion

§

In the making of an EDM, absolute rotation, reflection, and translation information is lost. Given an EDM, reconstruction of point position ( 5.12, the list X) can be guaranteed correct only in affine dimension r and relative position. Given a noiseless complete EDM, this isometric reconstruction is unique in so far as every realization of a corresponding list X is congruent:

5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION

5.6

417

Injectivity of D & unique reconstruction

Injectivity implies uniqueness of isometric reconstruction; hence, we endeavor to demonstrate it. EDM operators list-form D(X) (837), Gram-form D(G) (849), and inner-product form D(Θ) (904) are many-to-one surjections ( 5.5) onto the same range; the EDM cone ( 6): (confer (850) (941))

§

§

© ª N −1×N EDMN = D(X) : RN −1×N → SN h | X∈ R © ª N N⊥ = D(G) : SN → SN h | G ∈ S+ − Sc © ª N −1×N −1 = D(Θ) : RN −1×N −1 → SN h | Θ ∈ R

(935)

§

where ( 5.5.1.1)

⊥ SN = {u1T + 1uT | u ∈ RN } ⊆ SN c

5.6.1

(1923)

Gram-form bijectivity

Because linear Gram-form EDM operator D(G) = δ(G)1T + 1δ(G)T − 2G

(849)

§

§

has no nullspace [81, A.1] on the geometric center subspace5.27 ( E.7.2.0.2) N SN c , {G ∈ S | G1 = 0} N

= {G ∈ S

(1921)

| N (G) ⊇ 1} = {G ∈ SN | R(G) ⊆ N (1T )}

= {V Y V | Y ∈ SN } ⊂ SN

(1922)

(936)

≡ {VN AVNT | A ∈ SN −1 }

then D(G) on that subspace is injective. The equivalence ≡ in (936) follows from the fact: Given B = V Y V = VN AVNT ∈ SN c with only matrix A ∈ SN −1 unknown, then VN† BVN†T = A or VN† Y VN†T = A . 5.27

418

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

⊥ ∈ basis SN h

N dim SN c = dim Sh =

N (N −1) 2

in RN (N +1)/2

⊥ ⊥ N (N +1)/2 dim SN = dim SN c h = N in R

basis SN c = V {Eij }V (confer (54))

SN c ⊥ basis SN c

SN h

⊥ ∈ basis SN h

Figure 121: Orthogonal complements in SN abstractly oriented in isometrically isomorphic RN (N +1)/2 . Case N = 2 accurately illustrated in R3 . ⊥ ⊥ ⊥ Orthogonal projection of basis for SN on SN yields another basis for SN h c c . ⊥ (Basis vectors for SN are illustrated lying in a plane orthogonal to SN c c in this dimension. Basis vectors for each ⊥ space outnumber those for its respective orthogonal complement; such is not the case in higher dimension.)

5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION

419

To prove injectivity of D(G) on SN Any matrix Y ∈ SN can be c : decomposed into orthogonal components in SN ; Y = V Y V + (Y − V Y V )

(937)

N⊥ where V Y V ∈ SN (1923). Because of translation c and Y − V Y V ∈ Sc ⊥ invariance ( 5.5.1.1) and linearity, D(Y −V Y V ) = 0 hence N (D) ⊇ SN c . It remains only to show

§

D(V Y V ) = 0 ⇔ V Y V = 0 (938) ¢ ⇔ Y = u1T + 1uT for some u ∈ RN . D(V Y V ) will vanish whenever 2V Y V = δ(V Y V )1T + 1δ(V Y V )T . But this implies R(1) ( B.2) were a subset of R(V Y V ) , which is contradictory. Thus we have ¡

§

⊥ N (D) = {Y | D(Y ) = 0} = {Y | V Y V = 0} = SN c

(939) ¨

Since G1 = 0 ⇔ X1 = 0 (857) simply means list X is geometrically centered at the origin, and because the Gram-form EDM operator D is ⊥ translation invariant and N (D) is the translation-invariant subspace SN c , then EDM definition D(G) (935) on5.28 (confer 6.6.1, 6.7.1, A.7.4.0.1)

§

§

§

N N N −1 T SN } ⊂ SN c ∩ S+ = {V Y V º 0 | Y ∈ S } ≡ {VN AVN | A ∈ S+

must be surjective onto EDMN ; (confer (850)) © ª N EDMN = D(G) | G ∈ SN c ∩ S+ 5.6.1.1

(940)

(941)

Gram-form operator D inversion

Define the linear geometric centering operator V ; (confer (858))

§

V(D) : SN → SN , −V D V

1 2

(942)

[84, 4.3]5.29 This orthogonal projector V has no nullspace on N SN h = aff EDM

(1195)

Equivalence ≡ in (940) follows from the fact: Given B = V Y V = VN AVNT ∈ SN + with −1 only matrix A unknown, then VN† BVN†T = A and A ∈ SN must be positive semidefinite + by positive semidefiniteness of B and Corollary A.3.1.0.5. 5.29 Critchley cites Torgerson (1958) [324, ch.11, §2] for a history and derivation of (942). 5.28

420

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

because the projection of −D/2 on SN c (1921) can be 0 if and only if N⊥ N⊥ N D ∈ Sc ; but Sc ∩ Sh = 0 (Figure 121). Projector V on SN h is therefore N injective hence uniquely invertible. Further, −V Sh V /2 is equivalent to the geometric center subspace SN c in the ambient space of symmetric matrices; a surjection, ¡ N ¢ ¡ ¢ N N⊥ SN = V SN (943) c = V(S ) = V Sh ⊕ Sh h because (67)

¡ ¢ ¡ ⊥¢ ¡ ¢ V SN ⊇ V SN = V δ 2(SN ) (944) h h ¡ ¢ ¡ ¢ N N Because D(G) on SN c is injective, and aff D V(EDM ) = D V(aff EDM ) by property (121) of the affine hull, we find for D ∈ SN h D(−V D V 21 ) = δ(−V D V 21 )1T + 1δ(−V D V 12 )T − 2(−V D V 12 )

(945)

id est, ¡ ¢ D = D V(D) ¡ ¢ −V D V = V D(−V D V )

or

(946) (947)

¡ ¢ N SN h = D V(Sh ) ¢ ¡ N −V SN h V = V D(−V Sh V )

(948) (949)

These operators V and¡ D ¢are mutual inverses. (849) is equivalent to SN The Gram-form D SN c h ; ¡ ¢ ¡ ¢ ¡ ¢ N⊥ N N⊥ N D SN = D V(SN (950) c h ⊕ Sh ) = Sh + D V(Sh ) = Sh ¡ ¢ N⊥ because SN h ⊇ D V(Sh ) . In summary, for the Gram-form we have the isomorphisms [85, 2] [84, p.76, p.107] [7, 2.1]5.30 [6, 2] [8, 18.2.1] [2, 2.1]

§

§

§

§

N SN h = D(Sc ) N SN c = V(Sh )

§

§

(951) (952)

and from the bijectivity results in 5.6.1, N EDMN = D(SN c ∩ S+ ) N N SN c ∩ S+ = V(EDM ) 5.30

In [7, p.6, line 20], delete sentence: Since G is also . . . not a singleton set. [7, p.10, line 11] x3 = 2 (not 1).

(953) (954)

5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION

5.6.2

421

Inner-product form bijectivity

The Gram-form EDM operator D(G) = δ(G)1T + 1δ(G)T − 2G (849) is an injective map, for example, on the domain that is the subspace of symmetric matrices having all zeros in the first row and column N SN 1 = {G ∈ S | Ge1 = 0} ¸ ¾ ¸ · ½· 0 0T 0 0T N |Y∈S Y = 0 I 0 I

(1925)

because it obviously has no nullspace there. Since Ge1 = 0 ⇔ Xe1 = 0 (851) N means the first point in the list X resides at the origin, then D(G) on SN 1 ∩ S+ must be surjective onto EDMN . Substituting ΘT Θ ← −VNTDVN (916) into inner-product form EDM definition D(Θ) (904), it may be further decomposed: ¸ ¸ · · h T ¡ ¢T i 0 0 0 T T ¢ 1 + 1 0 δ −VN DVN ¡ − 2 D(D) = 0 −VNTDVN δ −VNTDVN (955) This linear operator D is another flavor of inner-product form and an injective map of the EDM cone onto itself. Yet when its domain is instead the entire N symmetric hollow subspace SN h = aff EDM , D(D) becomes an injective map onto that same subspace. Proof follows directly from the fact: linear D N N has no nullspace [81, A.1] on SN h = aff D(EDM ) = D(aff EDM ) (121).

§

5.6.2.1

¡ ¢ Inversion of D −VNTDVN

Injectivity of D(D) suggests inversion of (confer (854)) VN (D) : SN → SN −1 , −VNTDVN

(956)

⊥ a linear surjective5.31 mapping onto SN −1 having nullspace5.32 SN ; c N −1 VN (SN h ) = S

(957)

5.31

Surjectivity of VN (D) is demonstrated via the Gram-form EDM operator D(G) : N N −1 Since SN , −VNT D(VN†T Y VN† /2)VN = Y . h = D(Sc ) (950), then for any Y ∈ S N⊥ 5.32 N (VN ) ⊇ Sc is apparent. There exists a linear mapping T (VN (D)) , VN†T VN (D)VN† = −V D V

1 2

= V(D)

422

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

N⊥ injective on domain SN ∩ SN h because Sc h = 0. Revising the argument of this inner-product form (955), we get another flavor · ¸ · ¸ h T ¢T i ¡ ¢ ¡ 0 0 0 T T T ¡ ¢ 1 + 1 0 δ −VN DVN −2 D −VN DVN = 0 −VNTDVN δ −VNTDVN (958) N and we obtain mutual inversion of operators VN and D , for D ∈ Sh ¡ ¢ D = D VN (D) (959) ¡ ¢ −VNTDVN = VN D(−VNTDVN ) (960)

or

¡ ¢ N SN = D V (S ) N h h ¡ ¢ T N −VN Sh VN = VN D(−VNT SN h VN )

(961) (962)

Substituting ΘT Θ ← Φ into inner-product form EDM definition (904), any EDM may be expressed by the new flavor · ¸ · ¸ £ ¤ 0 0 0T T T D(Φ) , 1 + 1 0 δ(Φ) − 2 ∈ EDMN δ(Φ) 0 Φ (963) ⇔ Φº0 where this D is a linear surjective operator onto EDMN by definition, −1 injective because it has no nullspace on domain SN . More broadly, + N −1 N −1 aff D(S+ ) = D(aff S+ ) (121), N −1 SN ) h = D(S

SN −1 = VN (SN h )

(964)

demonstrably isomorphisms, and by bijectivity of this inner-product form: −1 EDMN = D(SN ) + N −1 N S+ = VN (EDM )

(965) (966)

such that ⊥ N (T (VN )) = N (V) ⊇ N (VN ) ⊇ SN = N (V) c ⊥ where the equality SN = N (V) is known (§E.7.2.0.2). c

¨

5.7. EMBEDDING IN AFFINE HULL

5.7

423

Embedding in affine hull

The affine hull A (73) of a point list {xℓ } (arranged columnar in X ∈ Rn×N (71)) is identical to the affine hull of that polyhedron P (81) formed from all convex combinations of the xℓ ; [57, 2] [285, 17]

§

§

A = aff X = aff P

(967)

Comparing hull definitions (73) and (81), it becomes obvious that the xℓ and their convex hull P are embedded in their unique affine hull A ; A ⊇ P ⊇ {xℓ }

(968)

Recall: affine dimension r is a lower bound on embedding, equal to dimension of the subspace parallel to that nonempty affine set A in which the points are embedded. ( 2.3.1) We define dimension of the convex hull P to be the same as dimension r of the affine hull A [285, 2], but r is not necessarily equal to rank of X (987). For the particular example illustrated in Figure 107, P is the triangle in union with its relative interior while its three vertices constitute the entire list X . Affine hull A is the unique plane that contains the triangle, so affine dimension r = 2 in that example while rank of X is 3. Were there only two points in Figure 107, then the affine hull would instead be the unique line passing through them; r would become 1 while rank would then be 2.

§

5.7.1

§

Determining affine dimension

Knowledge of affine dimension r becomes important because we lose any absolute offset common to all the generating xℓ in Rn when reconstructing convex polyhedra given only distance information. ( 5.5.1) To calculate r , we first remove any offset that serves to increase dimensionality of the subspace required to contain polyhedron P ; subtracting any α ∈ A in the affine hull from every list member will work,

§

X − α1T

(969)

translating A to the origin:5.33

A − α = aff(X − α1T ) = aff(X) − α P − α = conv(X − α1T ) = conv(X) − α

5.33

(970) (971)

The manipulation of hull functions aff and conv follows from their definitions.

424

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Because (967) and (968) translate, Rn ⊇ A − α = aff(X − α1T ) = aff(P − α) ⊇ P − α ⊇ {xℓ − α} (972) where from the previous relations it is easily shown aff(P − α) = aff(P) − α

(973)

Translating A neither changes its dimension or the dimension of the embedded polyhedron P ; (72) r , dim A = dim(A − α) , dim(P − α) = dim P

(974)

For any α ∈ Rn , (970)-(974) remain true. [285, p.4, p.12] Yet when α ∈ A , the affine set A − α becomes a unique subspace of Rn in which the {xℓ − α} and their convex hull P − α are embedded (972), and whose dimension is more easily calculated. 5.7.1.0.1 Example. Translating first list-member to origin. Subtracting the first member α , x1 from every list member will translate their affine hull A and their convex hull P and, in particular, x1 ∈ P ⊆ A to the origin in Rn ; videlicet, i h √ 2VN ∈ Rn×N (975) X − x1 1T = X − Xe1 1T = X(I − e1 1T ) = X 0

where VN is defined in (843), and e1 in (853). Applying (972) to (975),

Rn ⊇ R(XVN ) = A−x1 = aff(X − x1 1T ) = aff(P − x1 ) ⊇ P − x1 ∋ 0 (976) n×N −1 where XVN ∈ R . Hence r = dim R(XVN )

(977) 2

§

Since shifting the geometric center to the origin ( 5.5.1.0.1) translates the affine hull to the origin as well, then it must also be true r = dim R(XV )

(978)

5.7. EMBEDDING IN AFFINE HULL

425

For any matrix whose range is R(V ) = N (1T ) we get the same result; e.g., r = dim R(XVN†T )

(979)

R(XV ) = {Xz | z ∈ N (1T )}

(980)

because

§

§

and R(V ) = R(VN ) = R(VN†T ) ( E). These auxiliary matrices ( B.4.2) are more closely related; V = VN VN† 5.7.1.1

(1587)

Affine dimension r versus rank

Now, suppose D is an EDM as defined by D(X) = δ(X TX)1T + 1δ(X TX)T − 2X TX ∈ EDMN

(837)

and we premultiply by −VNT and postmultiply by VN . Then because VNT 1 = 0 (844), it is always true that −VNTDVN = 2VNTX TXVN = 2VNT GVN ∈ SN −1 where G is a Gram matrix. (confer (858))

(981)

Similarly pre- and postmultiplying by V

−V D V = 2V X TX V = 2V G V ∈ SN

(982)

always holds because V 1 = 0 (1577). Likewise, multiplying inner-product form EDM definition (904), it always holds: −VNTDVN = ΘT Θ ∈ SN −1

(908)

§

For any matrix A , rank ATA = rank A = rank AT . [188, 0.4]5.34 So, by (980), affine dimension r = rank XV = rank XVN = rank XVN†T = rank Θ = rank V D V = rank V G V = rank VNTDVN = rank VNT GVN 5.34

For A ∈ Rm×n , N (ATA) = N (A). [309, §3.3]

(983)

426

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

§

By conservation of dimension, ( A.7.3.0.1)

N

For D ∈ EDM

r + dim N (VNTDVN ) = N − 1

(984)

r + dim N (V D V ) = N

(985)

−VNTDVN ≻ 0 ⇔ r = N − 1

(986)

but −V D V ⊁ 0. The general fact5.35 (confer (869)) r ≤ min{n , N − 1}

(987)

is evident from (975) but can be visualized in the example illustrated in Figure 107. There we imagine a vector from the origin to each point in the list. Those three vectors are linearly independent in R3 , but affine dimension r is 2 because the three points lie in a plane. When that plane is translated to the origin, it becomes the only subspace of dimension r = 2 that can contain the translated triangular polyhedron.

5.7.2

Pr´ ecis

We collect expressions for affine dimension r : for list X ∈ Rn×N and Gram matrix G ∈ SN + r , = = = = = = = = =

dim(P − α) = dim P = dim conv X dim(A − α) = dim A = dim aff X rank(X − x1 1T ) = rank(X − αc 1T ) rank Θ (906) rank XVN = rank XV = rank XVN†T rank X , X e1 = 0 or X 1 = 0 (988) T rank VN GVN = rank V G V = rank VN† GVN rank G , Ge1 = 0 (854) or G1 = 0 (858)  T rank VN DVN = rank V D V = rank VN† DVN = rank VN (VNTDVN )VNT    rank Λ (1074)µ· ¸¶ · ¸ D ∈ EDMN 0 1T 0 1T    = N − 1 − dim N = rank − 2 (995) 1 −D 1 −D 5.35

rank X ≤ min{n , N }

5.7. EMBEDDING IN AFFINE HULL

5.7.3

427

Eigenvalues of −V D V versus −VN† DVN

Suppose for D ∈ EDMN we are given eigenvectors vi ∈ RN of −V D V and corresponding eigenvalues λ ∈ RN so that −V D V vi = λ i vi ,

i = 1...N

(989)

From these we can determine the eigenvectors and eigenvalues of −VN† DVN : Define νi , VN† vi , λ i 6= 0 (990) Then we have:

−V D VN VN† vi = λ i vi −VN† V D VN νi = λ i VN† vi −VN† DVN νi = λ i νi

(991) (992) (993)

the eigenvectors of −VN† DVN are given by (990) while its corresponding nonzero eigenvalues are identical to those of −V D V although −VN† DVN is not necessarily positive semidefinite. In contrast, −VNTDVN is positive semidefinite but its nonzero eigenvalues are generally different.

§

§

§

5.7.3.0.1 Theorem. EDM rank versus affine dimension r . [149, 3] [170, 3] [148, 3] For D ∈ EDMN (confer (1150))

1. r = rank(D) − 1 ⇔ 1TD† 1 6= 0 Points constituting a list X generating the polyhedron corresponding to D lie on the relative boundary of an r-dimensional circumhypersphere having √ ¡ ¢−1/2 diameter = 2 1TD† 1 (994) †1 circumcenter = 1XD TD † 1 2. r = rank(D) − 2 ⇔ 1TD† 1 = 0 There can be no circumhypersphere whose relative boundary contains a generating list for the corresponding polyhedron.

§

§

§ §

3. In Cayley-Menger form [103, 6.2] [83, 3.3] [46, 40] ( 5.11.2), µ· ¸¶ · ¸ 0 1T 0 1T r = N − 1 − dim N = rank −2 (995) 1 −D 1 −D

§

Circumhyperspheres exist for r < rank(D)−2. [323, 7]

⋄

428

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

For all practical purposes, (987) max{0 , rank(D) − 2} ≤ r ≤ min{n , N − 1}

5.8 5.8.1

(996)

Euclidean metric versus matrix criteria Nonnegativity property 1

When D = [dij ] is an EDM (837), then it is apparent from (981) 2VNTX TXVN = −VNTDVN º 0

(997)

because for any matrix A , ATA º 0 .5.36 We claim nonnegativity of the dij is enforced primarily by the matrix inequality (997); id est, ) −VNTDVN º 0 ⇒ dij ≥ 0 , i 6= j (998) D ∈ SN h (The matrix inequality to enforce strict positivity differs by a stroke of the pen. (1001)) We now support our claim: If any matrix A ∈ Rm×m is positive semidefinite, then its main diagonal δ(A) ∈ Rm must have all nonnegative entries. [146, 4.2] Given D ∈ SN h

§

−VNTDVN = 

d12

1  2 (d12 + d13 − d23 )  1  2 (d1,j+1 + d1,i+1 − dj+1,i+1 )  ..  . 1 2 (d12 + d1N − d2N )

1 2 (d12 + d13 − d23 )

d13

1 2 (d1,j+1 + d1,i+1 − dj+1,i+1 )

.. . 1 (d + d 13 1N − d3N ) 2

1 2 (d1,i+1 + d1,j+1 − di+1,j+1 ) 1 2 (d1,i+1 + d1,j+1 − di+1,j+1 )

d1,i+1 .. . 1 2 (d14 + d1N − d4N )

= 21 (1D1,2:N + D2:N, 1 1T − D2:N, 2:N ) ∈ SN −1

For A ∈ Rm×n , ATA º 0 ⇔ y TATAy = kAyk2 ≥ 0 for all kyk = 1. full-rank skinny-or-square, ATA ≻ 0. 5.36

···

··· .. . .. . ···

1 2 (d12 + d1N − d2N ) 1 2 (d13 + d1N − d3N )



   1  2 (d14 + d1N − d4N )  ..  . d1N

(999) When A is

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA

429

where row,column indices i,j ∈ {1 . . . N − 1}. [290] It follows:

−VNTDVN º 0 D ∈ SN h

¾

⇒

δ(−VNTDVN )



  = 

d12 d13 .. . d1N



  º0 

(1000)

Multiplication of VN by any permutation matrix Ξ has null effect on its range and nullspace. In other words, any permutation of the rows or columns of VN produces a basis for N (1T ); id est, R(Ξ r VN ) = R(VN Ξc ) = R(VN ) = N (1T ). T Hence, −VNTDVN º 0 ⇔ −VNT ΞTrDΞ r VN º 0 (⇔ −ΞT c VN DVN Ξc º 0). 5.37 will sift the remaining dij similarly Various permutation matrices to (1000) thereby proving their nonnegativity. Hence −VNTDVN º 0 is a sufficient test for the first property ( 5.2) of the Euclidean metric, nonnegativity. ¨

§

When affine dimension r equals 1, in particular, nonnegativity symmetry and hollowness become necessary and sufficient criteria satisfying matrix inequality (997). ( 6.6.0.0.1)

§

5.8.1.1

Strict positivity

Should we require the points in Rn to be distinct, then entries of D off the main diagonal must be strictly positive {dij > 0, i 6= j} and only those entries along the main diagonal of D are 0. By similar argument, the strict matrix inequality is a sufficient test for strict positivity of Euclidean distance-square; −VNTDVN ≻ 0 D ∈ SN h

¾

⇒ dij > 0 , i 6= j

(1001)

N −1 The rule of thumb is: If Ξ r (i , 1) = 1, then δ(−VNT ΞT is some r DΞ rVN ) ∈ R th permutation of the i row or column of D excepting the 0 entry from the main diagonal. 5.37

430

5.8.2

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Triangle inequality property 4

§

§

In light of Kreyszig’s observation [211, 1.1, prob.15] that properties 2 through 4 of the Euclidean metric ( 5.2) together imply nonnegativity property 1, p p p p 2 djk = djk + dkj ≥ djj = 0 ,

j 6= k

(1002)

nonnegativity criterion (998) suggests that matrix inequality −VNTDVN º 0 might somehow take on the role of triangle inequality; id est,  δ(D) = 0  p p p DT = D (1003) ⇒ dij ≤ dik + dkj , i 6= j 6= k  −VNTDVN º 0

We now show that is indeed the case: Let T be the leading principal submatrix in S2 of −VNTDVN (upper left 2×2 submatrix from (999)); T ,

§

·

d12 1 (d +d13 −d23 ) 2 12

1 (d +d13 −d23 ) 2 12

d13

¸

§

(1004)

Submatrix T must be positive (semi)definite whenever −VNTDVN ( A.3.1.0.4, 5.8.3) Now we have, −VNTDVN º 0 ⇒ T º 0 ⇔ λ1 ≥ λ2 ≥ 0 −VNTDVN ≻ 0 ⇒ T ≻ 0 ⇔ λ1 > λ2 > 0

is.

(1005)

where λ1 and λ2 are the eigenvalues of T , real due only to symmetry of T : ³ ´ p 2 2 2 λ1 = 12 d12 + d13 + d23 − 2(d12 + d13 )d23 + 2(d12 + d13 ) ∈R ´ ³ (1006) p 2 2 2 − 2(d12 + d13 )d23 + 2(d12 + d13 ) ∈R λ2 = 12 d12 + d13 − d23 Nonnegativity of eigenvalue λ1 is guaranteed by only nonnegativity of the dij which in turn is guaranteed by matrix inequality (998). Inequality between the eigenvalues in (1005) follows from only realness of the dij . Since λ1 always equals or exceeds λ2 , conditions for the positive (semi)definiteness of submatrix T can be completely determined by examining λ2 the smaller of its two eigenvalues. A triangle inequality is made apparent when we express

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA

431

T eigenvalue nonnegativity in terms of D matrix entries; videlicet, T º 0 ⇔ det T = λ1 λ2 ≥ 0 , d12 , d13 ≥ 0 ⇔ λ2 ≥ 0 ⇔ √ √ √ √ √ | d12 − d23 | ≤ d13 ≤ d12 + d23

(c) (b)

(1007)

(a)

Triangle inequality (1007a) (confer (902) (1019)), in terms of three rooted entries from D , is equivalent to metric property 4 p p p d ≤ d + 13 12 p p pd23 d ≤ d + p 23 p 12 pd13 d12 ≤ d13 + d23

(1008)

for the corresponding points x1 , x2 , x3 from some length-N list.5.38

5.8.2.1

Comment

Given D whose dimension N equals or exceeds 3, there are N !/(3!(N − 3)!) distinct triangle inequalities in total like (902) that must be satisfied, of which each dij is involved in N−2, and each point xi is in (N − 1)!/(2!(N −1 − 2)!). We have so far revealed only one of those triangle inequalities; namely, (1007a) that came from T (1004). Yet we claim if −VNTDVN º 0 then all triangle inequalities will be satisfied simultaneously; p p p p p | dik − dkj | ≤ dij ≤ dik + dkj ,

i
(1009)

(There are no more.) To verify our claim, we must prove the matrix inequality −VNTDVN º 0 to be a sufficient test of all the triangle inequalities; more efficient, we mention, for larger N : 5.38

Accounting for symmetry property 3, the fourth metric property demands three inequalities be satisfied per one of type (1007a). The first of those inequalities in (1008) is self evident from (1007a), while the two remaining follow from the left-hand side of (1007a) and the fact for scalars, |a| ≤ b ⇔ a ≤ b and −a ≤ b .

432

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.8.2.1.1 Shore. The columns of Ξ r VN Ξc hold a basis for N (1T ) when Ξ r and Ξc are permutation matrices. In other words, any permutation of the rows or columns of VN leaves its range and nullspace unchanged; id est, R(Ξ r VN Ξc ) = R(VN ) = N (1T ) (844). Hence, two distinct matrix inequalities can be equivalent tests of the positive semidefiniteness of D on R(VN ) ; id est, −VNTDVN º 0 ⇔ −(Ξ r VN Ξc )TD(Ξ r VN Ξc ) º 0. By properly choosing permutation matrices,5.39 the leading principal submatrix TΞ ∈ S2 of −(Ξ r VN Ξc )TD(Ξ r VN Ξc ) may be loaded with the entries of D needed to test any particular triangle inequality (similarly to (999)-(1007)). Because all the triangle inequalities can be individually tested using a test equivalent to the lone matrix inequality −VNTDVN º 0, it logically follows that the lone matrix inequality tests all those triangle inequalities simultaneously. We conclude that −VNTDVN º 0 is a sufficient test for the fourth property of the Euclidean metric, triangle inequality. ¨ 5.8.2.2

Strict triangle inequality

Without exception, all the inequalities in (1007) and (1008) can be made strict while their corresponding implications remain true. The then strict inequality (1007a) or (1008) may be interpreted as a strict triangle inequality under which collinear arrangement of points is not allowed. [208, 24/6, p.322] Hence by similar reasoning, −VNTDVN ≻ 0 is a sufficient test of all the strict triangle inequalities; id est,

§

 δ(D) = 0  p p p ⇒ dij < dik + dkj , DT = D  −VNTDVN ≻ 0

5.8.3

i 6= j 6= k

(1010)

−VNTDVN nesting

From (1004) observe that T = −VNTDVN |N ← 3 . In fact, for D ∈ EDMN , the leading principal submatrices of −VNTDVN form a nested sequence (by inclusion) whose members are individually positive semidefinite [146] [188] [309] and have the same form as T ; videlicet,5.40 p p p p p To individually test triangle inequality | dik − dkj | ≤ dij ≤ dik + dkj for particular i, k , j, set Ξ r (i, 1) = Ξ r (k , 2) = Ξ r (j , 3) = 1 and Ξc = I . 0 5.40 −V D V |N ← 1 = 0 ∈ S+ (§B.4.1) 5.39

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA

433

−VNTDVN |N ← 1 = [ ∅ ]

(o)

−VNTDVN |N ← 2 = [d12 ] ∈ S+

(a)

−VNTDVN |N ← 3

−VNTDVN |N ← 4

d12 1 (d +d13 −d23 ) 2 12

1 (d +d13 −d23 ) 2 12

d12  1 =  2 (d12 +d13 −d23 ) 1 (d +d14 −d24 ) 2 12

1 (d +d13 −d23 ) 2 12

=

· 

d13

¸

d13 1 (d +d14 −d34 ) 2 13

= T ∈ S2+ 1 (d +d14 −d24 ) 2 12 1 (d +d14 −d34 ) 2 13

d14

.. .



−VNTDVN |N ← i−1 ν(i)



−VNTDVN |N ← N −1 ν(N )

−VNTDVN |N ← i = 

ν(i)T

.. .

−VNTDVN

= 

T

ν(N )

d1i



i−1  ∈ S+

d1N

(b) 

  (c)

(d)



−1  ∈ SN +

(e) (1011)

where

ν(i) ,



1   2

d12 +d1i −d2i d13 +d1i −d3i .. . d1,i−1 +d1i −di−1,i



   ∈ Ri−2 , 

i>2

(1012)

Hence, the leading principal submatrices of EDM D must also be EDMs.5.41 Bordered symmetric matrices in the form (1011d) are known to have intertwined [309, 6.4] (or interlaced [188, 4.3] [306, IV.4.1]) eigenvalues; (confer 5.11.1) that means, for the particular submatrices (1011a) and (1011b),

§

5.41

§

§

§

In fact, each and every principal submatrix of an EDM D is another EDM. [218, §4.1]

434

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX λ2 ≤ d12 ≤ λ1

(1013)

where d12 is the eigenvalue of submatrix (1011a) and λ1 , λ2 are the eigenvalues of T (1011b) (1004). Intertwining in (1013) predicts that should d12 become 0, then λ2 must go to 0 .5.42 The eigenvalues are similarly intertwined for submatrices (1011b) and (1011c); γ 3 ≤ λ 2 ≤ γ2 ≤ λ 1 ≤ γ1

(1014)

§

where γ1 , γ2 , γ3 are the eigenvalues of submatrix (1011c). Intertwining likewise predicts that should λ2 become 0 (a possibility revealed in 5.8.3.1), then γ3 must go to 0. Combining results so far for N = 2, 3, 4: (1013) (1014) γ3 ≤ λ2 ≤ d12 ≤ λ1 ≤ γ1

(1015)

The preceding logic extends by induction through the remaining members of the sequence (1011). 5.8.3.1

Tightening the triangle inequality

§

Now we apply Schur complement from A.4 to tighten the triangle inequality from (1003) in case: cardinality N = 4. We find that the gains by doing so are modest. From (1011) we identify: ¸ · A B (1016) , −VNTDVN |N ← 4 BT C A , T = −VNTDVN |N ← 3

(1017)

§

both positive semidefinite by assumption, where B = ν(4) (1012), and C = d14 . Using nonstrict CC † -form (1451), C º 0 by assumption ( 5.8.1) and CC † = I . So by the positive semidefinite ordering of eigenvalues theorem ( A.3.1.0.1), ½ −1 λ1 ≥ d14 kν(4)k2 −1 T T −VN DVN |N ← 4 º 0 ⇔ T º d14 ν(4)ν(4) ⇒ (1018) λ2 ≥ 0

§

−1 −1 ν(4)ν(4)T while λ1 , λ2 are kν(4)k2 , 0} are the eigenvalues of d14 where {d14 the eigenvalues of T . 5.42

If d12 were 0, eigenvalue λ2 becomes 0 (1006) because d13 must then be equal to d23 ; id est, d12 = 0 ⇔ x1 = x2 . (§5.4)

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA

435

5.8.3.1.1 Example. Small completion problem, II. Applying the inequality for λ1 in (1018) to√the small completion problem on on d14 (1.236 in (830)) is tightened page 376 Figure 108, the lower bound √ to 1.289 . The correct value of d14 to three significant figures is 1.414 . 2

5.8.4

Affine dimension reduction in two dimensions

§

(confer 5.14.4) The leading principal 2×2 submatrix T of −VNTDVN has largest eigenvalue λ1 (1006) which is a convex function of D .5.43 λ1 can never be 0 unless d12 = d13 = d23 = 0. Eigenvalue λ1 can never be negative while the dij are nonnegative. The remaining eigenvalue λ2 is a concave function of D that becomes 0 only at the upper and lower bounds of inequality (1007a) and its equivalent forms: (confer (1009)) √ √ √ √ √ | d12 − d23 | ≤ d13 ≤ d12 + d23 √ √ √⇔ √ √ | d12 − d13 | ≤ d23 ≤ d12 + d13 √ √⇔ √ √ √ | d13 − d23 | ≤ d12 ≤ d13 + d23

(a) (b)

(1019)

(c)

In between those bounds, λ2 is strictly positive; otherwise, it would be negative but prevented by the condition T º 0.

When λ2 becomes 0, it means triangle △123 has collapsed to a line segment; a potential reduction in affine dimension r . The same logic is valid for any particular principal 2×2 submatrix of −VNTDVN , hence applicable to other triangles. 5.43

The largest eigenvalue of any symmetric matrix is always a convex function of its entries, while the eigenvalue is always concave. [57, exmp.3.10] In our particular  smallest  d12 case, say d , d13 ∈ R3 . Then the Hessian (1684) ∇ 2 λ1 (d) º 0 certifies convexity d23 whereas ∇ 2 λ2 (d) ¹ 0 certifies concavity. Each Hessian has rank equal to 1. The respective gradients ∇λ1 (d) and ∇λ2 (d) are nowhere 0.

436

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.9

Bridge: Convex polyhedra to EDMs

The criteria for the existence of an EDM include, by definition (837) (904), the properties imposed upon its entries dij by the Euclidean metric. From 5.8.1 and 5.8.2, we know there is a relationship of matrix criteria to those properties. Here is a snapshot of what we are sure: for i , j , k ∈ {1 . . . N } (confer 5.2)

§

§

§

p pdij pdij pdij dij

≥ 0 , i 6= j −VNTDVN º 0 =p 0, i = j δ(D) = 0 ⇐ = pdji p DT = D ≤ dik + dkj , i 6= j 6= k

(1020)

all implied by D ∈ EDMN . In words, these four Euclidean metric properties are necessary conditions for D to be a distance matrix. At the moment, we have no converse. As of concern in 5.3, we have yet to establish metric requirements beyond the four Euclidean metric properties that would allow D to be certified an EDM or might facilitate polyhedron or list reconstruction from an incomplete EDM. We deal with this problem in 5.14. Our present goal is to establish ab initio the necessary and sufficient matrix criteria that will subsume all the Euclidean metric properties and any further requirements5.44 for all N > 1 ( 5.8.3); id est,

§

§

§

−VNTDVN º 0 D ∈ SN h

¾

⇔ D ∈ EDMN

(856)

or for EDM definition (913), Ωº0

p δ(d) º 0

)

⇔ D = D(Ω , d) ∈ EDMN

(1021)

In 1935, Schoenberg [290, (1)] first extolled matrix product −VNTDVN (999) (predicated on symmetry and self-distance) specifically incorporating VN , albeit algebraically. He showed: nonnegativity −y T VNTDVN y ≥ 0, for all y ∈ RN −1 , is necessary and sufficient for D to be an EDM. Gower [148, §3] remarks how surprising it is that such a fundamental property of Euclidean geometry was obtained so late. 5.44

5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS

437

Figure 122: Elliptope E 3 in isometrically isomorphic R6 (projected on R3 ) is a convex body that appears to possess some kind of symmetry in this dimension; it resembles a malformed pillow in the shape of a bulging tetrahedron. Elliptope relative boundary is not smooth and comprises all set members (1022) having at least one 0 eigenvalue. [221, 2.1] This elliptope has an infinity of vertices, but there are only four vertices corresponding to a rank-1 matrix. Those yy T , evident in the illustration, have binary vector y ∈ R3 with entries in {±1}.

§

438

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

0

Figure 123: Elliptope E 2 in isometrically isomorphic R3 is a line segment illustrated interior to positive semidefinite cone S2+ (Figure 41).

5.9.1

Geometric arguments

§

§

5.9.1.0.1 Definition. Elliptope: [221] [218, 2.3] [103, 31.5] a unique bounded immutable convex Euclidean body in S n ; intersection of n positive semidefinite cone S+ with that set of n hyperplanes defined by unity main diagonal; n E n , S+ ∩ {Φ ∈ S n | δ(Φ) = 1} (1022) a.k.a, the set of all correlation matrices of dimension dim E n = n(n − 1)/2 in Rn(n+1)/2

(1023)

An elliptope E n is not a polyhedron, in general, but has some polyhedral faces and an infinity of vertices.5.45 Of those, 2n−1 vertices (some extreme points of the elliptope) are extreme directions yy T of the positive semidefinite cone, where the entries of vector y ∈ Rn belong to {±1} and exercise every combination. Each of the remaining vertices has rank belonging to the set {k > 0 | k(k + 1)/2 ≤ n}. Each and every face of an elliptope is exposed. △ 5.45

Laurent defines vertex distinctly from the sense herein (§2.6.1.0.1); she defines vertex as a point with full-dimensional (nonempty interior) normal cone (§E.10.3.2.1). Her definition excludes point C in Figure 31, for example.

5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS

439

In fact, any positive semidefinite matrix whose entries belong to {±1} is a correlation matrix whose rank must be one: (As there are few equivalent conditions for rank constraints, this statement is rather important as a device for relaxing an integer, combinatorial, or Boolean problem.)

§

§

5.9.1.0.2 Theorem. Some elliptope vertices. [115, 2.1.1] For Y ∈ S n , y ∈ Rn , and all i, j ∈ {1 . . . n} (confer 2.3.1.0.2) Y º 0,

Yij ∈ {±1}

⇔

Y = yy T ,

yi ∈ {±1}

(1024) ⋄

The elliptope for dimension n = 2 is a line segment in isometrically n . The elliptope isomorphic Rn(n+1)/2 (Figure 123). Obviously, cone(E n ) 6= S+ for dimension n = 3 is realized in Figure 122.

§

5.9.1.0.3 Lemma. Hypersphere. (confer bullet p.386) [17, 4] N Matrix Ψ = [Ψij ] ∈ S belongs to the elliptope in SN iff there exist N points p on the boundary of a hypersphere in Rrank Ψ having radius 1 such that kpi − pj k2 = 2(1 − Ψij ) ,

i, j = 1 . . . N

(1025) ⋄

There is a similar theorem for Euclidean distance matrices: We derive matrix criteria for D to be an EDM, validating (856) using simple geometry; distance to the polyhedron formed by the convex hull of a list of points (71) in Euclidean space Rn . 5.9.1.0.4 EDM assertion. D is a Euclidean distance matrix if and only if D ∈ SN h and distances-square from the origin {kp(y)k2 = −y T VNTDVN y | y ∈ S − β}

(1026)

correspond to points p in some bounded convex polyhedron P − α = {p(y) | y ∈ S − β}

(1027)

having N or fewer vertices embedded in an r-dimensional subspace A − α of Rn , where α ∈ A = aff P and where the domain of linear surjection p(y) −1 is the unit simplex S ⊂ RN shifted such that its vertex at the origin is + N −1 translated to −β in R . When β = 0, then α = x1 . ⋄

440

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

In terms of VN , the unit simplex (267) in RN −1 has an equivalent representation: √ (1028) S = {s ∈ RN −1 | 2VN s º −e1 } where e1 is as in (853). Incidental to the EDM assertion, shifting the unit-simplex domain in RN −1 translates the polyhedron P in Rn . Indeed, there is a map from vertices of the unit simplex to members of the list generating P ; p

:

RN −1

  −β      e1 − β  e2 − β p  ..   .    e N −1 − β

Rn

→                  

=

  x1 − α      x2 − α x3 − α  ..   .    x −α N

      

(1029)

     

5.9.1.0.5 Proof. EDM assertion. (⇒) We demonstrate that if D is an EDM, then each distance-square kp(y)k2 described by (1026) corresponds to a point p in some embedded polyhedron P − α . Assume D is indeed an EDM; id est, D can be made from some list X of N unknown points in Euclidean space Rn ; D = D(X) for X ∈ Rn×N as in (837). Since D is translation invariant ( 5.5.1), we may shift the affine hull A of those unknown points to the origin as in (969). Then take any point p in their convex hull (81);

§

P − α = {p = (X − Xb 1T )a | aT 1 = 1, a º 0} where α = Xb ∈ A ⇔ bT 1 = 1. Solutions to aT 1 = 1 are:5.46 n o √ a ∈ e1 + 2VN s | s ∈ RN −1

(1030)

(1031)

√ where e1 is as in (853). Similarly, b = e1 + 2VN β . √ √ √ T P − α = {p = X(I − (e + β)1 )(e + s) | 2V 2V 2VN s º −e1 } 1 N 1 N √ √ (1032) = {p = X 2VN (s − β) | 2VN s º −e1 }

Since R(VN ) = N (1T ) and N (1T ) ⊥ R(1) , then over all s ∈ RN −1 , VN s is a hyperplane through the origin orthogonal to 1. Thus the solutions {a} constitute a hyperplane orthogonal to the vector 1, and offset from the origin in RN by any particular solution; in this case, a = e1 . 5.46

5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS that describes the domain of p(s) as the unit simplex √ −1 S = {s | 2VN s º −e1 } ⊂ RN +

441

(1028)

Making the substitution s − β ← y

√ P − α = {p = X 2VN y | y ∈ S − β}

(1033)

Point p belongs to a convex polyhedron P −α embedded in an r-dimensional subspace of Rn because the convex hull of any list forms a polyhedron, and because the translated affine hull A − α contains the translated polyhedron P − α (972) and the origin (when α ∈ A), and because A has dimension r by definition (974). Now, any distance-square from the origin to the polyhedron P − α can be formulated {pTp = kpk2 = 2y T VNTX TXVN y | y ∈ S − β}

(1034)

Applying (981) to (1034) we get (1026). (⇐) To validate the EDM assertion in the reverse direction, we prove: If each distance-square kp(y)k2 (1026) on the shifted unit-simplex S − β ⊂ RN −1 corresponds to a point p(y) in some embedded polyhedron P − α , then D is an EDM. The r-dimensional subspace A − α ⊆ Rn is spanned by p(S − β) = P − α

(1035)

because A − α = aff(P − α) ⊇ P − α (972). So, outside the domain S − β of linear surjection p(y) , the simplex complement \S − β ⊂ RN −1 must contain the domain of the distance-square kp(y)k2 = p(y)T p(y) to remaining points in the subspace A − α ; id est, to the polyhedron’s relative exterior \P − α . For kp(y)k2 to be nonnegative on the entire subspace A − α , −VNTDVN must be positive semidefinite and is assumed symmetric;5.47 −VNTDVN , ΘT p Θp

(1036)

where5.48 Θp ∈ Rm×N −1 for some m ≥ r . Because p(S − β) is a convex polyhedron, it is necessarily a set of linear combinations of points from some ¡ ¢ The antisymmetric part −VNTDVN − (−VNTDVN )T /2 is annihilated by kp(y)k2 . By the same reasoning, any positive (semi)definite matrix A is generally assumed symmetric because only the symmetric part (A + AT )/2 survives the test y TA y ≥ 0. [188, §7.1] 5.48 T A = A º 0 ⇔ A = RTR for some real matrix R . [309, §6.3] 5.47

442

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

§

length-N list because every convex polyhedron having N or fewer vertices can be generated that way ( 2.12.2). Equivalent to (1026) are {pTp | p ∈ P − α} = {pTp = y T ΘT p Θp y | y ∈ S − β}

(1037)

Because p ∈ P − α may be found by factoring (1037), the list Θp is found by factoring (1036). A unique EDM can be made from that list using inner-product form definition D(Θ)|Θ=Θp (904). That EDM will be identical to D if δ(D) = 0, by injectivity of D (955). ¨

5.9.2

Necessity and sufficiency

§

From (997) we learned that matrix inequality −VNTDVN º 0 is a necessary test for D to be an EDM. In 5.9.1, the connection between convex polyhedra and EDMs was pronounced by the EDM assertion; the matrix inequality together with D ∈ SN h became a sufficient test when the EDM assertion demanded that every bounded convex polyhedron have a corresponding EDM. For all N > 1 ( 5.8.3), the matrix criteria for the existence of an EDM in (856), (1021), and (832) are therefore necessary and sufficient and subsume all the Euclidean metric properties and further requirements.

§

5.9.3

Example revisited

Now we apply the necessary and sufficient EDM criteria (856) to an earlier problem.

§

5.9.3.0.1 Example. Small completion problem, III. (confer 5.8.3.1.1) Continuing Example 5.3.0.0.2 pertaining to Figure 108 where N = 4, distance-square d14 is ascertainable from the matrix inequality −VNTDVN º 0. √ Because all distances in (829) are known except d14 , we may simply calculate the smallest eigenvalue of −VNTDVN over a range of d14 as in Figure 124. We observe a unique value of d14 satisfying (856) where the abscissa axis is tangent to the hypograph of the smallest eigenvalue. Since the smallest eigenvalue of a symmetric matrix is known to be a concave function ( 5.8.4), we calculate its second partial derivative with respect to d14 evaluated at 2 and find −1/3. We conclude there are no other satisfying values of d14 . Further, that value of d14 does not meet an upper or lower bound of a triangle inequality like (1009), so neither does it cause the collapse

§

5.10. EDM-ENTRY COMPOSITION

443

§

of any triangle. Because the smallest eigenvalue is 0, affine dimension r of 2 any point list corresponding to D cannot exceed N − 2. ( 5.7.1.1)

5.10

EDM-entry composition

§

Laurent [218, 2.3] applies results from Schoenberg (1938) [291] to show certain nonlinear compositions of individual EDM entries yield EDMs; in particular, D ∈ EDMN ⇔ [1 − e−α dij ] ∈ EDMN ∀ α > 0 ⇔ [e−α dij ] ∈ E N ∀α> 0

(1038)

where D = [dij ] and E N is the elliptope (1022). 5.10.0.0.1

Proof. (Laurent, 2003)

[291] (confer [211])

Lemma 2.1. from A Tour d’Horizon . . . on Completion Problems. [218] The following assertions are equivalent: for D = [dij , i, j = 1 . . . N ] ∈ SN h and N N E the elliptope in S ( 5.9.1.0.1),

§

(i) D ∈ EDMN (ii) e−αD , [e−αdij ] ∈ E N for all α > 0 (iii) 11T − e−αD , [1 − e−αdij ] ∈ EDMN for all α > 0

⋄

1) Equivalence of Lemma 2.1 (i) (ii) is stated in Schoenberg’s Theorem 1 [291, p.527]. 2) (ii) ⇒ (iii) can be seen from the statement in the beginning of section 3, saying that a distance space embeds in L2 iff some associated matrix is PSD. We reformulate it: Let d = (dij )i,j=0,1...N be a distance space on N + 1 points (i.e., symmetric hollow matrix of order N + 1) and let p = (pij )i,j=1...N be the symmetric matrix of order N related by: (A) 2pij = d0i + d0j − dij for i, j = 1 . . . N or equivalently (B) d0i = pii , dij = pii + pjj − 2pij for i, j = 1 . . . N

444

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX smallest eigenvalue 1

2

4

3

d14

-0.2

-0.4

-0.6

Figure 124: Smallest eigenvalue of −VNTDVN makes it a PSD matrix for only one value of d14 : 2. 1/α

dij

1.4

1/α

log2 (1+ dij )

1.2

1− e−α dij

1

(a)

0.8 0.6 0.4 0.2 0.5

1

1.5

2

4

dij

3

(b)

1/α

2

dij 1/α log2 (1+ dij ) 1− e−α dij

1

0.5

1

1.5

2

α

Figure 125: Some entrywise EDM compositions: (a) α = 2. Concave nondecreasing in dij . (b) Trajectory convergence in α for dij = 2.

5.10. EDM-ENTRY COMPOSITION

445

Then d embeds in L2 iff p is a positive semidefinite matrix iff d is of negative type (second half page 525/top of page 526 in [291]). For the implication from (ii) to (iii), set: p = e−αd and define d ′ from p using (B) above. Then d ′ is a distance space on N + 1 points that embeds in L2 . Thus its subspace of N points also embeds in L2 and is precisely 1 − e−αd . Note that (iii) ⇒ (ii) cannot be read immediately from this argument since (iii) involves the subdistance of d ′ on N points (and not the full d ′ on N + 1 points). 3) Show (iii) ⇒ (i) by using the series expansion of the function 1 − e−αd : the constant term cancels, α factors out; there remains a summation of d plus a multiple of α . Letting α go to 0 gives the result. This is not explicitly written in Schoenberg, but he also uses such an argument; expansion of the exponential function then α → 0 (first proof on [291, p.526]). ¨

§

Schoenberg’s results [291, 6, thm.5] (confer [211, p.108 -109]) also suggest certain finite positive roots of EDM entries produce EDMs; specifically, 1/α D ∈ EDMN ⇔ [dij ] ∈ EDMN ∀ α > 1 (1039) The special case α = 2 is of interest because it corresponds to absolute distance; e.g., √ ◦ (1040) D ∈ EDMN ⇒ D ∈ EDMN Assuming that points constituting a corresponding list X are distinct (1001), then it follows: for D ∈ SN h 1/α

lim [dij ] = lim [1 − e−α dij ] = −E , 11T − I

α→∞

§

α→∞

§

(1041)

Negative elementary matrix −E ( B.3) is relatively interior to the EDM cone ( 6.6) and terminal to the respective trajectories (1038) and (1039) as functions of α . Both trajectories are confined to the EDM cone; in engineering terms, the EDM cone is an invariant set [288] to either trajectory. Further, if D is not an EDM but for some particular αp it becomes an EDM, then for all greater values of α it remains an EDM.

446

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

These preliminary findings lead one to speculate whether any concave nondecreasing composition of individual EDM entries dij on R+ will produce another EDM; e.g., given EDM D , empirical evidence suggests that for 1/α each fixed α ≥ 1 [sic] the composition [log2 (1 + dij )] is also an EDM. Figure 125 illustrates that composition’s concavity in dij together with functions from (1038) and (1039). 5.10.0.0.2 Exercise. Taxicab distance matrix as EDM. Determine whether taxicab distance matrices (D1 (X) in Example 3.8.0.0.2) are all numerically equivalent to EDMs. Explain why or why not. H

5.10.1

EDM by elliptope

§

N (confer (863)) For some κ ∈ R+ and C ∈ SN ( 5.9.1.0.1), + in the elliptope E Alfakih asserts any given EDM D is expressible [9] [103, 31.5]

§

D = κ(11T − C ) ∈ EDMN

(1042)

This expression exhibits nonlinear combination of variables κ and C . We therefore propose a different expression requiring redefinition of the elliptope (1022) by scalar parametrization; n Etn , S+ ∩ {Φ ∈ S n | δ(Φ) = t1}

(1043)

D = t11T − E ∈ EDMN

(1044)

where, of course, E n = E1n . Then any given EDM D is expressible which is linear in variables t ∈ R+ and E ∈ EtN .

5.11

EDM indefiniteness

§

By the known result in A.7.2 regarding a 0-valued entry on the main diagonal of a symmetric positive semidefinite matrix, there can be no positive or negative semidefinite EDM except the 0 matrix because EDMN ⊆ SN h (836) and N (1045) SN h ∩ S+ = 0

the origin. So when D ∈ EDMN , there can be no factorization D = ATA or −D = ATA . [309, 6.3] Hence eigenvalues of an EDM are neither all nonnegative or all nonpositive; an EDM is indefinite and possibly invertible.

§

5.11. EDM INDEFINITENESS

5.11.1

447

EDM eigenvalues, congruence transformation

§

For any symmetric −D , we can characterize its eigenvalues by congruence transformation: [309, 6.3] "

−W TD W = −

VNT T

1

#

D [ VN

"

1] = −

VNTDVN VNTD1 T

1 DVN

T

1 D1

#

∈ SN

(1046)

Because 1 ] ∈ RN ×N

W , [ VN

(1047)

is full-rank, then (1456) ¡ ¢ inertia(−D) = inertia −W TDW

(1048)

the congruence (1046) has the same number of positive, zero, and negative eigenvalues as −D . Further, if we denote by {γi , i = 1 . . . N − 1} the eigenvalues of −VNTDVN and denote eigenvalues of the congruence −W TDW by {ζ i , i = 1 . . . N } and if we arrange each respective set of eigenvalues in nonincreasing order, then by theory of interlacing eigenvalues for bordered symmetric matrices [188, 4.3] [309, 6.4] [306, IV.4.1]

§

§

§

ζN ≤ γN −1 ≤ ζN −1 ≤ γN −2 ≤ · · · ≤ γ2 ≤ ζ2 ≤ γ1 ≤ ζ1

(1049)

When D ∈ EDMN , then γi ≥ 0 ∀ i (1394) because −VNTDVN º 0 as we know. That means the congruence must have N − 1 nonnegative eigenvalues; ζi ≥ 0, i = 1 . . . N − 1. The remaining eigenvalue ζN cannot be nonnegative because then −D would be positive semidefinite, an impossibility; so ζN < 0. By congruence, nontrivial −D must therefore have exactly one negative eigenvalue;5.49 [103, 2.4.5]

§

5.49

All the entries of the corresponding eigenvector must have the same sign with respect to each other [84, p.116] because that eigenvector is the Perron vector corresponding to the spectral radius; [188, §8.2.6] the predominant characteristic of negative [sic] matrices.

448

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX  λ(−D)i ≥ 0 , i = 1 . . . N − 1    ¶ µP N N D ∈ EDM ⇒ λ(−D)i = 0  i=1    N ×N D ∈ SN h ∩ R+

(1050)

§

where the λ(−D)i are nonincreasingly ordered eigenvalues of −D whose sum must be 0 only because tr D = 0 [309, 5.1]. The eigenvalue summation condition, therefore, can be considered redundant. Even so, all these conditions are insufficient to determine whether some given H ∈ SN h is an 5.50 EDM, as shown by counter-example. 5.11.1.0.1 Exercise. Spectral inequality. Prove whether it holds: for D = [dij ] ∈ EDMN λ(−D)1 ≥ dij ≥ λ(−D)N −1

∀ i 6= j

(1051) H

Terminology: a spectral cone is a convex cone containing all eigenspectra Any [211, p.365] [306, p.26] corresponding to some set of matrices. positive semidefinite matrix, for example, possesses a vector of nonnegative eigenvalues corresponding to an eigenspectrum contained in a spectral cone that is a nonnegative orthant.

5.11.2

Spectral cones

Denoting the eigenvalues of Cayley-Menger matrix

λ 5.50

µ·

0 1T 1 −D

¸¶

·

0 1T 1 −D

¸

∈ SN +1 by

∈ RN +1

When N = 3, for example, the symmetric hollow matrix   0 1 1 N ×N H =  1 0 5  ∈ SN h ∩ R+ 1 5 0

is not an EDM, although λ(−H) = [5 0.3723 −5.3723]T conforms to (1050).

(1052)

5.11. EDM INDEFINITENESS

449

§

§

§

§

we have the Cayley-Menger form ( 5.7.3.0.1) of necessary and sufficient conditions for D ∈ EDMN from the literature: [170, 3]5.51 [70, 3] [103, 6.2] (confer (856) (832)) ¸¶  µ· T 0 1   λ ≥ 0, 1 −D i D ∈ EDMN ⇔   D ∈ SN h

  i=1...N 

⇔

 

(

−VNTDVN º 0 D ∈ SN h

(1053)

These conditions say the Cayley-Menger form has only one negative ¸¶ µ· one and 0 1T belong to that eigenvalue. When D is an EDM, eigenvalues λ 1 −D particular orthant in RN +1 having the N + 1 th coordinate as sole negative coordinate5.52 : ·

5.11.2.1

RN + R−

¸

= cone {e1 , e2 , · · · eN , −eN +1 }

(1054)

Cayley-Menger versus Schoenberg

Connection to the Schoenberg criterion (856) is made when the Cayley-Menger form is further partitioned:

·

0 1T 1 −D

¸



 =

·

0 1 1 0

¸

[ 1 −D2:N, 1 ]

·

1T −D1, 2:N

−D2:N, 2:N

¸ 

(1055)

 

¸ 0 1 Matrix D ∈ is an EDM if and only if the Schur complement of 1 0 ( A.4) in this partition is positive semidefinite; [17, 1] [202, 3] id est,

§

SN h

§

§

·

T Recall: for D ∈ SN h , −VN DVN º 0 subsumes nonnegativity property 1 (§5.8.1). Empirically, all except one entry of the corresponding eigenvector have the same sign with respect to each other. 5.51

5.52

450

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

(confer (999))

−D2:N, 2:N

D ∈ EDMN ⇔ ¸ ¸· · 1T 0 1 = −2VNTDVN º 0 − [ 1 −D2:N, 1 ] −D1, 2:N 1 0

(1056)

and

D ∈ SN h

§

Positive semidefiniteness of that Schur complement insures nonnegativity ×N (D ∈ RN , 5.8.1), whereas complementary inertia (1458) insures existence + of that lone negative eigenvalue of the Cayley-Menger form. Now we apply results from chapter 2 with regard to polyhedral cones and their duals. 5.11.2.2

Ordered eigenspectra

Conditions (1053) specify eigenvalue membership to the smallest pointed · ¸ 0 1T polyhedral spectral cone for : 1 −EDMN Kλ , {ζ ∈ RN +1 | ζ1 ≥ ζ 2 ≥ · · · ≥ ζN ≥ 0 ≥ ζN +1 , 1T ζ = 0} · N¸ R+ = KM ∩ ∩ ∂H R− µ· ¸¶ 0 1T = λ 1 −EDMN

(1057)

where ∂H , {ζ ∈ RN +1 | 1T ζ = 0}

§

(1058)

is a hyperplane through the origin, and KM is the monotone cone ( 2.13.9.4.3, implying ordered eigenspectra) which has nonempty interior but is not pointed; KM = {ζ ∈ RN +1 | ζ1 ≥ ζ 2 ≥ · · · ≥ ζN +1 } ∗

KM = { [ e1 − e2

(408)

e2 − e3 · · · eN − eN +1 ] a | a º 0 } ⊂ RN +1

(409)

5.11. EDM INDEFINITENESS

451

So because of the hyperplane, dim aff Kλ = dim ∂H = N

(1059)

indicating Kλ has empty interior. Defining  T eT 1 − e2 T  eT  2 − e3  ∈ RN ×N +1 , A, . ..   T eT N − eN +1 



  B,  

eT 1 eT 2 .. . eT N −eT N +1



   ∈ RN +1×N +1  

(1060)

we have the halfspace-description: Kλ = {ζ ∈ RN +1 | Aζ º 0 , Bζ º 0 , 1T ζ = 0}

(1061)

From this and (416) we get a vertex-description for a pointed spectral cone having empty interior: ( µ· ) ¸ ¶† ˆ A Kλ = VN (1062) ˆ VN b | b º 0 B where VN ∈ RN +1×N , and where [sic] 1×N +1 ˆ = eT B N ∈ R

and

 T eT 1 − e2 T  eT  2 − e3  ∈ RN −1×N +1 Aˆ =  . ..   T T eN −1 − eN 

(1063)

(1064)

hold those· rows¸ of A and B corresponding to conically independent rows A VN . ( 2.10) in B Conditions (1053) can be equivalently restated in terms of a spectral cone for Euclidean distance matrices:  µ· ¸¶ · N¸ T 0 1 R+   λ ∈ KM ∩ ∩ ∂H N 1 −D R− (1065) D ∈ EDM ⇔   N D ∈ Sh

§

452

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Vertex-description of the dual spectral cone is, (287) ∗ Kλ

∗ KM

=

·

+

RN + R−

¸∗

+ ∂H∗ ⊆ RN +1

nh i o ¤ ª ˆ T 1 −1 a | a º 0 AT B T 1 −1 b | b º 0 = AˆT B (1066) From (1062) and (417) we get a halfspace-description: ©£

=

∗ ˆ T ])† VNT y º 0} Kλ = {y ∈ RN +1 | (VNT [ AˆT B

(1067)

∗

This polyhedral dual spectral cone Kλ is closed, convex, has nonempty interior because Kλ is pointed, but is not pointed because Kλ has empty interior. 5.11.2.3

Unordered eigenspectra

Spectral cones are not unique; eigenspectra ordering can be rendered benign within a cone by presorting a vector of eigenvalues into nonincreasing order.5.53 Then things simplify: Conditions (1053) now specify eigenvalue membership to the spectral cone λ

µ·

0 1T 1 −EDMN

¸¶

=

·

RN + R−

¸

= {ζ ∈ R

∩ ∂H

N +1

(1068) T

| Bζ º 0 , 1 ζ = 0}

where B is defined in (1060), and ∂H in (1058). From (416) we get a vertex-description for a pointed spectral cone having empty interior: λ

5.53

µ·

0 1T 1 −EDMN

¸¶

n o ˜ N )† b | b º 0 = VN (BV ½· ¸ ¾ I = b|bº0 −1T

(1069)

Eigenspectra ordering (represented by a cone having monotone description such as (1057)) becomes benign in (1278), for example, where projection of a given presorted vector on the nonnegative orthant in a subspace is equivalent to its projection on the monotone nonnegative cone in that same subspace; equivalence is a consequence of presorting.

5.11. EDM INDEFINITENESS

453

where VN ∈ RN +1×N and  eT 1  T  ˜ ,  e.2  ∈ RN ×N +1 B  ..  eT N 

(1070)

holds only those rows of B corresponding to conically independent rows in BVN . For presorted eigenvalues, (1053) can be equivalently restated  µ· ¸¶ · N ¸ 0 1T R+   λ ∩ ∂H ∈ 1 −D R− (1071) D ∈ EDMN ⇔   N D ∈ Sh

Vertex-description of the dual spectral cone is, (287) µ· · N¸ ¸¶∗ 0 1T ∗ R+ λ = + ∂H ⊆ RN +1 1 −EDMN R− i o nh ©£ T ¤ ª T ˜ B 1 −1 a | a º 0 = B 1 −1 b | b º 0 = (1072) From (417) we get a halfspace-description: µ· ¸¶∗ 0 1T ˜ T )† V T y º 0} = {y ∈ RN +1 | (VNT B λ N (1073) 1 −EDMN N +1 = {y ∈ R | [ I −1 ] y º 0} This polyhedral dual spectral cone is closed, convex, has nonempty interior but is not pointed. (Notice that any nonincreasingly ordered eigenspectrum belongs to this dual spectral cone.) 5.11.2.4

Dual cone versus dual spectral cone

An open question regards the relationship of convex cones and their duals to the corresponding spectral cones and their duals. A positive semidefinite cone, for example, is self-dual. Both the nonnegative orthant and the monotone nonnegative cone are spectral cones for it. When we consider the nonnegative orthant, then that spectral cone for the self-dual positive semidefinite cone is also self-dual.

454

5.12

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

List reconstruction

The traditional term metric multidimensional scaling 5.54 [239] [97] [328] [95] [242] [84] refers to any reconstruction of a list X ∈ Rn×N in Euclidean space from interpoint distance information, possibly incomplete ( 6.4), ordinal ( 5.13.2), or specified perhaps only by bounding-constraints ( 5.4.2.3.9) [326]. Techniques for reconstruction are essentially methods for optimally embedding an unknown list of points, corresponding to given Euclidean distance data, in an affine subset of desired or minimum dimension. The oldest known precursor is called principal component analysis [151] which analyzes the correlation matrix ( 5.9.1.0.1); [49, 22] a.k.a, Karhunen−Lo´eve transform in the digital signal processing literature. Isometric reconstruction ( 5.5.3) of point list X is best performed by eigen decomposition of a Gram matrix; for then, numerical errors of factorization are easily spotted in the eigenvalues: Now we consider how rotation/reflection and translation invariance factor into a reconstruction.

§

§

§

5.12.1

§

§

§

x1 at the origin. VN

At the stage of reconstruction, we have D ∈ EDMN and wish to find a generating list ( 2.3.2) for polyhedron P − α by factoring Gram matrix −VNTDVN (854) as in (1036). One way to factor −VNTDVN is via diagonalization of symmetric matrices; [309, 5.6] [188] ( A.5.1, A.3)

§

§

§

§

−VNTDVN , QΛQT

(1074)

QΛQT º 0 ⇔ Λ º 0

(1075)

√ √ T −VNTDVN = 2VNTX TXVN , Q Λ QT p Qp Λ Q

(1076)

where Q ∈ RN −1×N −1 is an orthogonal matrix containing eigenvectors while Λ ∈ SN −1 is a diagonal matrix containing corresponding nonnegative eigenvalues ordered by nonincreasing value. From the diagonalization, identify the list using (981);

5.54

Scaling [324] means making a scale, i.e., a numerical representation of qualitative data. If the scale is multidimensional, it’s multidimensional scaling. −Jan de Leeuw When data comprises measurable distances, then reconstruction is termed metric multidimensional scaling. In one dimension, N coordinates in X define the scale.

5.12. LIST RECONSTRUCTION

455

√ √ √ √ n×N −1 is unknown as is where Λ QT p Qp Λ , Λ = Λ Λ and where Qp ∈ R its dimension n . Rotation/reflection is accounted for by Qp yet only its to first r columns are necessarily orthonormal.5.55√Assuming membership √ the unit simplex y ∈ S (1033), then point p = X 2VN y = Qp Λ QT y in Rn belongs to the translated polyhedron P − x1 whose generating list constitutes the columns of (975) h i √ √ ¤ £ 0 X 2VN = 0 Qp Λ QT ∈ Rn×N = [0

x 2 − x1

(1077)

(1078)

x3 − x1 · · · xN − x1 ]

The scaled auxiliary matrix VN represents that translation. A simple choice for Qp has n set to N − 1; id est, Qp = I . Ideally, each member of the generating list has at most r nonzero entries; r being, affine dimension √ (1079) rank VNTDVN = rank Qp Λ QT = rank Λ = r Each member then has at least N −1 − r zeros in its higher-dimensional coordinates because r ≤ N − 1. (987) To truncate those zeros, choose n equal to affine dimension which is the smallest n possible because XVN has rank r ≤ n (983).5.56 In that case, the simplest choice for Qp is [ I 0 ] having dimensions r × N − 1. We may wish to verify the list (1078) found from the diagonalization of −VNTDVN . Because of rotation/reflection and translation invariance ( 5.5), EDM D can be uniquely made from that list by calculating: (837)

§

5.55

Recall r signifies affine dimension. Qp is not necessarily an orthogonal matrix. Qp is constrained such that only its first r columns are necessarily orthonormal because there are only r nonzero eigenvalues in Λ when −VNTDVN has rank r (§5.7.1.1). Remaining columns of Qp are arbitrary.   λ1 0   T . q1 ..   .   5.56 λr If we write QT =  ..  as rowwise eigenvectors, Λ =   0   . T . qN −1 . 0 0 £ ¤ in terms of eigenvalues, and Qp = qp1 · · · qpN −1 as column vectors, then r p √ P Qp Λ QT = λ i qpi qiT is a sum of r linearly independent rank-one matrices (§B.1.1). i=1

Hence the summation has rank r .

456

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

D(X) = D(X[ 0

√

2VN ]) = D(Qp [ 0

√ T √ T Λ Q ]) = D([ 0 Λ Q ]) (1080)

This suggests a way to find EDM D given −VNTDVN (confer (959)) D=

·

¸ ¸ · h T ¡ ¢T i 0 0 0 T T ¡ ¢ 1 + 1 0 δ −VN DVN −2 0 −VNTDVN δ −VNTDVN

5.12.2

(955)

0 geometric center. V

§

Alternatively we may perform reconstruction using auxiliary matrix V ( B.4.1) and Gram matrix −V D V 21 (858) instead; to find a generating list for polyhedron P − αc

(1081)

whose geometric center has been translated to the origin. Redimensioning diagonalization factors Q , Λ ∈ RN ×N and unknown Qp ∈ Rn×N , (982) √ √ T T −V D V = 2V X TX V , Q Λ QT p Qp Λ Q , QΛQ

(1082)

where the geometrically centered generating list constitutes (confer (1078)) XV =

√1 2

√ Qp Λ QT ∈ Rn×N

= [ x1 − N1 X1

§

x2 − N1 X1

x3 − N1 X1 · · · xN − N1 X1 ]

(1083)

where αc = N1 X1. ( 5.5.1.0.1) The simplest choice for Qp is [ I 0 ] ∈ Rr×N . Now EDM D can be uniquely made from the list found: (837) √ √ 1 1 D(X) = D(XV ) = D( √ Qp Λ QT ) = D( Λ QT ) 2 2

(1084)

This EDM is, of course, identical to (1080). Similarly to (955), from −V D V we can find EDM D (confer (946)) D = δ(−V D V 12 )1T + 1δ(−V D V 21 )T − 2(−V D V 12 )

(945)

5.12. LIST RECONSTRUCTION

457

(a)

(c)

(b)

(d)

(f)

(e)

Figure 126: Map of United States of America showing some state boundaries and the Great Lakes. All plots made by connecting 5020 points. Any difference in scale in (a) through (d) is artifact of plotting routine. (a) Shows original map made from decimated (latitude, longitude) data. (b) Original map data rotated (freehand) to highlight curvature of Earth. (c) Map isometrically reconstructed from an EDM (from distance only). (d) Same reconstructed map illustrating curvature. (e)(f ) Two views of one isotonic reconstruction (from comparative distance); problem (1093) with no sort constraint Π d (and no hidden line removal).

458

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.13

Reconstruction examples

5.13.1

Isometric reconstruction

5.13.1.0.1 Example. Map of the USA. The most fundamental application of EDMs is to reconstruct relative point position given only interpoint distance information. Drawing a map of the United States is a good illustration of isometric reconstruction from complete distance data. We obtained latitude and longitude information for the coast, border, states, and Great Lakes from the usalo atlas data file within Matlab Mapping Toolbox; conversion to Cartesian coordinates (x, y , z) via: φ , π/2 − latitude θ , longitude x = sin(φ) cos(θ) y = sin(φ) sin(θ) z = cos(φ)

(1085)

We used 64% of the available map data to calculate EDM D from N = 5020 points. The original (decimated) data and its isometric reconstruction via (1076) are shown in Figure 126(a)-(d). Matlab code is on Wıκımization. The eigenvalues computed for (1074) are λ(−VNTDVN ) = [199.8 152.3 2.465 0 0 0 · · · ]T

(1086)

The 0 eigenvalues have absolute numerical error on the order of 2E-13 ; meaning, the EDM data indicates three dimensions (r = 3) are required for reconstruction to nearly machine precision. 2

5.13.2

Isotonic reconstruction

Sometimes only comparative information about distance is known (Earth is closer to the Moon than it is to the Sun). Suppose, for example, EDM D for three points is unknown:   0 d12 d13 (826) D = [dij ] =  d12 0 d23  ∈ S3h d13 d23 0

but comparative distance data is available:

d13 ≥ d23 ≥ d12

(1087)

5.13. RECONSTRUCTION EXAMPLES

459

With vectorization d = [d12 d13 d23 ]T ∈ R3 , we express the comparative data as the nonincreasing sorting      d13 0 1 0 d12 (1088) Π d =  0 0 1  d13  =  d23  ∈ KM+ d12 d23 1 0 0

where Π is a given permutation matrix expressing known sorting action on the entries of unknown EDM D , and KM+ is the monotone nonnegative cone ( 2.13.9.4.2)

§

N (N −1)/2

KM+ = {z | z1 ≥ z2 ≥ · · · ≥ zN (N −1)/2 ≥ 0} ⊆ R+

(401)

where N (N − 1)/2 = 3 for the present example. From sorted vectorization (1088) we create the sort-index matrix   0 12 32 O =  12 0 22  ∈ S3h ∩ R3×3 (1089) + 2 2 3 2 0

generally defined

¢ ¡ Oij , k 2 | dij = Ξ Π d k ,

j 6= i

(1090)

where Ξ is a permutation matrix (1654) completely reversing order of vector entries. Replacing EDM data with indices-square of a nonincreasing sorting like this is, of course, a heuristic we invented and may be regarded as a nonlinear introduction of much noise into the Euclidean distance matrix. For large data sets, this heuristic makes an otherwise intense problem computationally tractable; we see an example in relaxed problem (1094). Any process of reconstruction that leaves comparative distance information intact is called ordinal multidimensional scaling or isotonic reconstruction. Beyond rotation, reflection, and translation error, ( 5.5) list reconstruction by isotonic reconstruction is subject to error in absolute scale (dilation) and distance ratio. Yet Borg & Groenen argue: [49, 2.2] reconstruction from complete comparative distance information for a large number of points is as highly constrained as reconstruction from an EDM; the larger the number, the better.

§ §

460

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX λ(−VNT OVN )j

900

800

700

600

500

400

300

200

100

0

1

2

3

4

5

6

7

8

9

10

j

Figure 127: Largest ten eigenvalues, of −VNT OVN for map of USA, sorted by nonincreasing value.

5.13.2.1

Isotonic map of the USA

To test Borg & Groenen’s conjecture, suppose we make a complete sort-index N ×N matrix O ∈ SN for the map of the USA and then substitute O in place h ∩ R+ of EDM D in the reconstruction process of 5.12. Whereas EDM D returned only three significant eigenvalues (1086), the sort-index matrix O is generally not an EDM (certainly not an EDM with corresponding affine dimension 3) so returns many more. The eigenvalues, calculated with absolute numerical error approximately 5E-7 , are plotted in Figure 127: (In the code on Wıκımization, matrix O is normalized by (N (N − 1)/2)2 .)

§

λ(−VNT OVN ) = [880.1 463.9 186.1 46.20 17.12 9.625 8.257 1.701 0.7128 0.6460 · · · ]T (1091) The extra eigenvalues indicate that affine dimension corresponding to an EDM near O is likely to exceed 3. To realize the map, we must simultaneously reduce that dimensionality and find an EDM D closest to O in some sense5.57 while maintaining the known comparative distance relationship. For example: given permutation matrix Π expressing the known sorting action like (1088) on entries (68) 5.57

a problem explored more in §7.

5.13. RECONSTRUCTION EXAMPLES 

    1  d , √ dvec D =   2   

d12 d13 d23 d14 d24 d34 .. .

dN −1,N

461 

      ∈ RN (N −1)/2    

(1092)

of unknown D ∈ SN h , we can make sort-index matrix O input to the optimization problem minimize D

k−VNT (D − O)VN kF

subject to rank VNTDVN ≤ 3 Π d ∈ KM+ D ∈ EDMN

(1093)

that finds the EDM D (corresponding to affine dimension not exceeding 3 in isomorphic dvec EDMN ∩ ΠT KM+ ) closest to O in the sense of Schoenberg (856). Analytical solution to this problem, ignoring the sort constraint Π d ∈ KM+ , is known [328]: we get the convex optimization [sic] ( 7.1)

§

minimize D

k−VNT (D − O)VN kF

subject to rank VNTDVN ≤ 3 D ∈ EDMN

(1094)

Only the three largest nonnegative eigenvalues in (1091) need be retained to make list (1078); the rest are discarded. The reconstruction from EDM D found in this manner is plotted in Figure 126(e)(f). Matlab code is on Wıκımization. From these plots it becomes obvious that inclusion of the sort constraint is necessary for isotonic reconstruction. That sort constraint demands: any optimal solution D⋆ must possess the known comparative distance relationship that produces the original ordinal distance data O (1090). Ignoring the sort constraint, apparently, violates it. Yet even more remarkable is how much the map reconstructed using only ordinal data still resembles the original map of the USA after suffering the many violations produced by solving relaxed problem (1094). This suggests the simple reconstruction techniques of 5.12 are robust to a significant amount of noise.

§

462

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.13.2.2

Isotonic solution with sort constraint

Because problems involving rank are generally difficult, we will partition (1093) into two problems we know how to solve and then alternate their solution until convergence: minimize D

k−VNT (D − O)VN kF

subject to rank VNTDVN ≤ 3 D ∈ EDMN minimize σ

(a) (1095)

kσ − Π dk

(b)

subject to σ ∈ KM+

where sort-index matrix O (a given constant in (a)) becomes an implicit vector variable o i solving the i th instance of (1095b) 1 √ dvec Oi = o i , ΠTσ ⋆ ∈ RN (N −1)/2 , 2

i ∈ {1, 2, 3 . . .}

(1096)

As mentioned in discussion of relaxed problem (1094), a closed-form solution to problem (1095a) exists. Only the first iteration of (1095a) sees the original sort-index matrix O whose entries are nonnegative whole N ×N numbers; id est, O0 = O ∈ SN (1090). Subsequent iterations i take h ∩ R+ the previous solution of (1095b) as input √ Oi = dvec−1 ( 2 o i ) ∈ SN (1097) real successors, estimating distance-square not order, to the sort-index matrix O . New convex problem (1095b) finds the unique minimum-distance projection of Π d on the monotone nonnegative cone KM+ . By defining Y †T = [e1 − e2

e2 − e3

e3 − e4 · · · em ] ∈ Rm×m

(402)

where m , N (N − 1)/2, we may rewrite (1095b) as an equivalent quadratic program; a convex problem in terms of the halfspace-description of KM+ : minimize σ

(σ − Π d)T (σ − Π d)

subject to Y † σ º 0

(1098)

5.13. RECONSTRUCTION EXAMPLES

463

§

This quadratic program can be converted to a semidefinite program via Schur-form ( 3.6.2); we get the equivalent problem minimize t∈ R , σ

subject to

t ·

tI σ − Πd (σ − Π d)T 1

Y †σ º 0 5.13.2.3

§

¸

º0

(1099)

Convergence

In E.10 we discuss convergence of alternating projection on intersecting convex sets in a Euclidean vector space; convergence to a point in their intersection. Here the situation is different for two reasons: Firstly, sets of positive semidefinite matrices having an upper bound on rank are generally not convex. Yet in 7.1.4.0.1 we prove (1095a) is equivalent to a projection of nonincreasingly ordered eigenvalues on a subset of the nonnegative orthant:

§

minimize D

subject to

k−VNT (D − O)VN kF rank VNTDVN N

D ∈ EDM

≤3

kΥ − ΛkF · 3 ¸ R+ subject to δ(Υ) ∈ 0

minimize ≡

Υ

(1100)

where −VNTDVN , U ΥU T ∈ SN −1 and −VNT OVN , QΛQT ∈ SN −1 are ordered diagonalizations ( A.5). It so happens: optimal orthogonal U ⋆ always equals Q given. Linear operator T (A) = U ⋆TAU ⋆ , acting on square matrix A , is a bijective isometry because the Frobenius norm is orthogonally invariant (43). This isometric isomorphism T thus maps a nonconvex problem to a convex one that preserves distance. Secondly, the second half (1095b) of the alternation takes place in a N −1 different vector space; SN ). From 5.6 we know these two h (versus S vector spaces are related by an isomorphism, SN −1 = VN (SN h ) (964), but not by an isometry. We have, therefore, no guarantee from theory of alternating projection that the alternation (1095) converges to a point, in the set of all EDMs corresponding to affine dimension not in excess of 3, belonging to dvec EDMN ∩ ΠT KM+ .

§

§

464

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.13.2.4

Interlude

Map reconstruction from comparative distance data, isotonic reconstruction, would also prove invaluable to stellar cartography where absolute interstellar distance is difficult to acquire. But we have not yet implemented the second half (1098) of alternation (1095) for USA map data because memory-demands exceed capability of our computer.

§

5.13.2.4.1 Exercise. Convergence of isotonic solution by alternation. Empirically demonstrate convergence, discussed in 5.13.2.3, on a smaller data set. H It would be remiss not to mention another method of solution to this isotonic reconstruction problem: Once again we assume only comparative distance data like (1087) is available. Given known set of indices I minimize

rank V D V

D

subject to dij ≤ dkl ≤ dmn D ∈ EDMN

∀(i, j, k , l , m , n) ∈ I

(1101)

this problem minimizes affine dimension while finding an EDM whose entries satisfy known comparative relationships. Suitable rank heuristics are discussed in 4.4.1 and 7.2.2 that will transform this to a convex optimization problem. Using contemporary computers, even with a rank heuristic in place of the objective function, this problem formulation is more difficult to compute than the relaxed counterpart problem (1094). That is because there exist efficient algorithms to compute a selected few eigenvalues and eigenvectors from a very large matrix. Regardless, it is important to recognize: the optimal solution set for this problem (1101) is practically always different from the optimal solution set for its counterpart, problem (1093).

§

5.14

§

Fifth property of Euclidean metric

§

We continue now with the question raised in 5.3 regarding the necessity for at least one requirement more than the four properties of the Euclidean metric ( 5.2) to certify realizability of a bounded convex polyhedron or to

§

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC

465

reconstruct a generating list for it from incomplete distance information. There we saw that the four Euclidean metric properties are necessary for D ∈ EDMN in the case N = 3, but become insufficient when cardinality N exceeds 3 (regardless of affine dimension).

5.14.1

Recapitulate

In the particular case N = 3, −VNTDVN º 0 (1005) and D ∈ S3h are necessary and sufficient conditions for D to be an EDM. By (1007), triangle inequality is then the only Euclidean condition bounding the necessarily nonnegative dij ; and those bounds are tight. That means the first four properties of the Euclidean metric are necessary and sufficient conditions for D to be an EDM in the case N = 3 ; for i, j ∈ {1, 2, 3} p pdij ≥ 0 , i 6= j 0, i = j −VNTDVN º 0 pdij = p ⇔ ⇔ D ∈ EDM3 3 D ∈ S d = d ij ji h p p p dij ≤ dik + dkj , i 6= j 6= k (1102)

Yet those four properties become insufficient when N > 3.

5.14.2

Derivation of the fifth

§

Correspondence between the triangle inequality and the EDM was developed in 5.8.2 where a triangle inequality (1007a) was revealed within the leading principal 2 × 2 submatrix of −VNTDVN when positive semidefinite. Our choice of the leading principal submatrix was arbitrary; actually, a unique triangle inequality like (902) corresponds to any one of the (N − 1)!/(2!(N −1 − 2)!) principal 2 × 2 submatrices.5.58 Assuming D ∈ S4h and −VNTDVN ∈ S3 , then by the positive (semi)definite principal submatrices theorem ( A.3.1.0.4) it is sufficient to prove all dij are nonnegative, all triangle inequalities are satisfied, and det(−VNTDVN ) is nonnegative. When N = 4, in other words, that nonnegative determinant becomes the fifth and last Euclidean metric requirement for D ∈ EDMN . We now endeavor to ascribe geometric meaning to it.

§

There are fewer principal 2×2 submatrices in −VNTDVN than there are triangles made by four or more points because there are N !/(3!(N − 3)!) triangles made by point triples. The triangles corresponding to those submatrices all have vertex x1 . (confer §5.8.2.1) 5.58

466 5.14.2.1

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX Nonnegative determinant

By (908) when D ∈ EDM4 , −VNTDVN is equal to inner product (903), p p  d12 d12 d13 cos θ213 pd12 d14 cos θ214 p ΘT Θ =  pd12 d13 cos θ213 p d13 d13 d14 cos θ314  d12 d14 cos θ214 d13 d14 cos θ314 d14 

(1103)

Because Euclidean space is an inner-product space, the more concise inner-product form of the determinant is admitted; ¡ ¢ det(ΘT Θ) = −d12 d13 d14 cos(θ213 )2 + cos(θ214 )2 + cos(θ314 )2 − 2 cos θ213 cos θ214 cos θ314 − 1 (1104) The determinant is nonnegative if and only if p p sin(θ214 )2 sin(θ314 )2 ≤ cos θ213 ≤ cos θ214 cos θ314 + sin(θ214 )2 sin(θ314 )2 ⇔ p p 2 2 cos θ213 cos θ314 − sin(θ213 ) sin(θ314 ) ≤ cos θ214 ≤ cos θ213 cos θ314 + sin(θ213 )2 sin(θ314 )2 ⇔ p p 2 2 cos θ213 cos θ214 − sin(θ213 ) sin(θ214 ) ≤ cos θ314 ≤ cos θ213 cos θ214 + sin(θ213 )2 sin(θ214 )2 (1105) cos θ214 cos θ314 −

which simplifies, for 0 ≤ θi1ℓ , θℓ1j , θi1j ≤ π and all i 6= j 6= ℓ ∈ {2, 3, 4} , to cos(θi1ℓ + θℓ1j ) ≤ cos θi1j ≤ cos(θi1ℓ − θℓ1j )

(1106)

Analogously to triangle inequality (1019), the determinant is 0 upon equality on either side of (1106) which is tight. Inequality (1106) can be equivalently written linearly as a triangle inequality between relative angles [368, 1.4];

§

|θi1ℓ − θℓ1j | ≤ θi1j ≤ θi1ℓ + θℓ1j θi1ℓ + θℓ1j + θi1j ≤ 2π 0 ≤ θi1ℓ , θℓ1j , θi1j ≤ π

(1107)

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC

467

θ213

-θ214 -θ314

π Figure 128: The relative-angle inequality tetrahedron (1108) bounding EDM4 is regular; drawn in entirety. Each angle θ (900) must belong to this solid to be realizable.

Generalizing this:

§

§

5.14.2.1.1 Fifth property of Euclidean metric - restatement. Relative-angle inequality. (confer 5.3.1.0.1) [45] [46, p.17, p.107] [218, 3.1] Augmenting the four fundamental Euclidean metric properties in Rn , for all i, j, ℓ 6= k ∈ {1 . . . N } , i < j < ℓ , and for N ≥ 4 distinct points {xk } , the inequalities |θikℓ − θℓkj | ≤ θikj ≤ θikℓ + θℓkj θikℓ + θℓkj + θikj ≤ 2π 0 ≤ θikℓ , θℓkj , θikj ≤ π

(a) (b) (c)

(1108)

where θikj = θjki is the angle between vectors at vertex xk (as defined in (900) and illustrated in Figure 109), must be satisfied at each point xk regardless of affine dimension. ⋄

468

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Because point labelling is arbitrary, this fifth Euclidean metric requirement must apply to each of the N points as though each were in turn labelled x1 ; hence the new index k in (1108). Just as the triangle inequality is the ultimate test for realizability of only three points, the relative-angle inequality is the ultimate test for only four. For four distinct points, the triangle inequality remains a necessary although penultimate test; ( 5.4.3)

§

§

−VNTDVN º 0 Four Euclidean metric properties ( 5.2). ⇔ D = D(Θ) ∈ EDM4 ⇔ 4 Angle θ inequality (831) or (1108). D ∈ Sh (1109) The relative-angle inequality, for this case, is illustrated in Figure 128. 5.14.2.2

Beyond the fifth metric property

When cardinality N exceeds 4, the first four properties of the Euclidean metric and the relative-angle inequality together become insufficient conditions for realizability. In other words, the four Euclidean metric properties and relative-angle inequality remain necessary but become a sufficient test only for positive semidefiniteness of all the principal 3 × 3 submatrices [sic] in −VNTDVN . Relative-angle inequality can be considered the ultimate test only for realizability at each vertex xk of each and every purported tetrahedron constituting a hyperdimensional body. When N = 5 in particular, relative-angle inequality becomes the penultimate Euclidean metric requirement while nonnegativity of then unwieldy det(ΘT Θ) corresponds (by the positive (semi)definite principal submatrices theorem in A.3.1.0.4) to the sixth and last Euclidean metric requirement. Together these six tests become necessary and sufficient, and so on. Yet for all values of N , only assuming nonnegative dij , relative-angle matrix inequality in (1021) is necessary and sufficient to certify realizability; ( 5.4.3.1)

§

§

§

−VNTDVN º 0 Euclidean metric property 1 ( 5.2). ⇔ D = D(Ω , d) ∈ EDMN ⇔ Angle matrix inequality Ω º 0 (909). D ∈ SN h (1110) Like matrix criteria (832), (856), and (1021), the relative-angle matrix inequality and nonnegativity property subsume all the Euclidean metric properties and further requirements.

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC

5.14.3

469

Path not followed

As a means to test for realizability of four or more points, an intuitively appealing way to augment the four Euclidean metric properties is to recognize generalizations of the triangle inequality: In the case of cardinality N = 4, the three-dimensional analogue to triangle & distance is tetrahedron & facet-area, while in case N = 5 the four-dimensional analogue is polychoron & facet-volume, ad infinitum. For N points, N + 1 metric properties are required.

5.14.3.1

N= 4

Each of the four facets of a general tetrahedron is a triangle and its relative interior. Suppose we identify each facet of the tetrahedron by its area-square: c1 , c2 , c3 , c4 . Then analogous to metric property 4, we may write a tight5.59 area inequality for the facets √

ci ≤

√

cj +

√

ck +

√

cℓ ,

i 6= j 6= k 6= ℓ ∈ {1, 2, 3, 4}

(1111)

§

which is a generalized “triangle” inequality [211, 1.1] that follows from √

ci =

√

cj cos ϕij +

√ √ ck cos ϕik + cℓ cos ϕiℓ

(1112)

[225] [348, Law of Cosines] where ϕij is the dihedral angle at the common edge between triangular facets i and j . If D is the EDM corresponding to the whole tetrahedron, then area-square of the i th triangular facet has a convenient formula in terms of Di ∈ EDMN −1 the EDM corresponding to that particular facet: From the Cayley-Menger determinant 5.60 for simplices, [348] [121] [148, 4] [83, 3.3] the i th facet area-square for i ∈ {1 . . . N } is ( A.4.1)

§

5.59

§

§

The upper bound is met when all angles in (1112) are simultaneously 0 ; that occurs, for example, if one point is relatively interior to the convex hull of the three remaining. 5.60 whose foremost characteristic is: the determinant vanishes if and · ¸ only if affine 0 1T dimension does not equal penultimate cardinality; id est, det = 0 ⇔ r < N−1 1 −D where D is any EDM (§5.7.3.0.1). Otherwise, the determinant is negative.

470

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX ¸ · −1 0 1T det c i = N −2 1 −Di 2 (N − 2)! 2 N ¡ T −1 ¢ (−1) = N −2 2 det Di 1 Di 1 2 (N − 2)! (−1)N T T = N −2 2 1 cof(Di ) 1 2 (N − 2)!

(1113) (1114) (1115)

where Di is the i th principal N − 1 × N − 1 submatrix5.61 of D ∈ EDMN , and cof(Di ) is the N − 1 × N − 1 matrix of cofactors [309, 4] corresponding to Di . The number of principal 3 × 3 submatrices in D is, of course, equal to the number of triangular facets in the tetrahedron; four (N !/(3!(N −3)!)) when N = 4.

§

5.14.3.1.1 Exercise. Sufficiency conditions for an EDM of four points. Triangle inequality (property 4) and area inequality (1111) are conditions necessary for D to be an EDM. Prove their sufficiency in conjunction with the remaining three Euclidean metric properties. H 5.14.3.2

N= 5

Moving to the next level, we might encounter a Euclidean body called polychoron: a bounded polyhedron in four dimensions.5.62 Our polychoron has five (N !/(4!(N −4)!)) facets, each of them a general tetrahedron whose volume-square c i is calculated using the same formula; (1113) where D is the EDM corresponding to the polychoron, and Di is the EDM corresponding to the i th facet (the principal 4 × 4 submatrix of D ∈ EDMN corresponding to the i th tetrahedron). The analogue to triangle & distance is now polychoron & facet-volume. We could then write another generalized “triangle” inequality like (1111) but in terms of facet volume; [354, IV]

§

√

5.61

ci ≤

√

cj +

√

ck +

√

cℓ +

√

cm ,

i 6= j 6= k 6= ℓ 6= m ∈ {1 . . . 5}

(1116)

Every principal submatrix of an EDM remains an EDM. [218, §4.1] The simplest polychoron is called a pentatope [348]; a regular simplex hence convex. (A pentahedron is a three-dimensional body having five vertices.) 5.62

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC

471

.

Figure 129: Length of one-dimensional face a equals height h = a = 1 of this convex nonsimplicial pyramid in R3 with square base inscribed in a circle of radius R centered at the origin. [348, Pyramid ]

§

5.14.3.2.1 Exercise. Sufficiency for an EDM of five points. For N = 5, triangle (distance) inequality ( 5.2), area inequality (1111), and volume inequality (1116) are conditions necessary for D to be an EDM. Prove their sufficiency. H 5.14.3.3

Volume of simplices

§

There is no known formula for the volume of a bounded general convex polyhedron expressed either by halfspace or vertex-description. [366, 2.1] [261, p.173] [216] [159] [160] Volume is a concept germane to R3 ; in higher dimensions it is called content. Applying the EDM assertion ( 5.9.1.0.4) and a result from [57, p.407], a general nonempty simplex ( 2.12.3) in RN −1 corresponding to an EDM D ∈ SN h has content

§

√

c = content(S)

q det(−VNTDVN )

§

(1117)

where content-square of the unit simplex S ⊂ RN −1 is proportional to its Cayley-Menger determinant; ¸ · −1 0 1T content(S) = N −1 det 1 −D([ 0 e1 e2 · · · eN −1 ]) 2 (N − 1)! 2 2

where ei ∈ RN −1 and the EDM operator used is D(X) (837).

(1118)

472

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.14.3.3.1 Example. Pyramid. A formula for volume of a pyramid is known;5.63 it is 31 the product of its base area with its height. [208] The pyramid in Figure 129 has volume 31 . To find its volume using EDMs, we must first decompose the pyramid into simplicial parts. Slicing it in half along the plane containing the line segments corresponding to radius R and height h we find the vertices of one simplex, 

1/2 1/2  X = 1/2 −1/2 0 0

−1/2 −1/2 0

 0 0  ∈ Rn×N 1

(1119)

where N = n + 1 for any nonempty simplex in Rn . √The volume of this simplex is half that of the entire pyramid; id est, c = 61 found by 2 evaluating (1117). With that, we conclude digression of path.

5.14.4 Affine dimension reduction in three dimensions

§

§

(confer 5.8.4) The determinant of any M ×M matrix is equal to the product of its M eigenvalues. [309, 5.1] When N = 4 and det(ΘT Θ) is 0, that means one or more eigenvalues of ΘT Θ ∈ R3×3 are 0. The determinant will go to 0 whenever equality is attained on either side of (831), (1108a), or (1108b), meaning that a tetrahedron has collapsed to a lower affine dimension; id est, r = rank ΘT Θ = rank Θ is reduced below N − 1 exactly by the number of 0 eigenvalues ( 5.7.1.1).

§

In solving completion problems of any size N where one or more entries of an EDM are unknown, therefore, dimension r of the affine hull required to contain the unknown points is potentially reduced by selecting distances to attain equality in (831) or (1108a) or (1108b). 5.63

Pyramid volume is independent of the paramount vertex position as long as its height remains constant.

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 5.14.4.1

473

Exemplum redux

We now apply the fifth Euclidean metric property to an earlier problem:

§

5.14.4.1.1 Example. Small completion problem, IV. (confer 5.9.3.0.1) Returning again to Example 5.3.0.0.2 that pertains to Figure 108 where N = 4, distance-square d14 is ascertainable from the fifth Euclidean metric √ property. Because all distances in (829) are known except d14 , then cos θ123 = 0 and θ324 = 0 result from identity (900). Applying (831), cos(θ123 + θ324 ) ≤ cos θ124 ≤ cos(θ123 − θ324 ) 0 ≤ cos θ124 ≤ 0

(1120)

It follows again from (900) that d14 can only be 2. As explained in this subsection, affine dimension r cannot exceed N − 2 because equality is attained in (1120). 2

474

CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Chapter 6 Cone of distance matrices For N > 3, the cone of EDMs is no longer a circular cone and the geometry becomes complicated. . .

§

−Hayden, Wells, Liu, & Tarazaga (1991) [171, 3]

In the subspace of symmetric matrices SN , we know that the convex cone of Euclidean distance matrices EDMN (the EDM cone) does not intersect the positive semidefinite cone SN + (PSD cone) except at the origin, their only vertex; there can be no positive or negative semidefinite EDM. (1045) [218] EDMN ∩ SN + = 0

§

(1121)

Even so, the two convex cones can be related. In 6.8.1 we prove the equality ¡ N⊥ ¢ EDMN = SN − SN (1214) h ∩ Sc +

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

475

476

CHAPTER 6. CONE OF DISTANCE MATRICES

a resemblance to EDM definition (837) where SN h =

©

ª A ∈ SN | δ(A) = 0

(61)

§

is the symmetric hollow subspace ( 2.2.3) and where ⊥ SN = {u1T + 1uT | u ∈ RN } c

(1923)

§

is the orthogonal complement of the geometric center subspace ( E.7.2.0.2) N SN c = {Y ∈ S | Y 1 = 0}

6.0.1

(1921)

gravity

Equality (1214) is equally important as the known isomorphisms (953) (954) −1 (965) (966) relating the EDM cone EDMN to positive semidefinite cone SN + 6.1 ( 5.6.2.1) or to an N (N − 1)/2-dimensional face of SN But + ( 5.6.1.1). those isomorphisms have never led to this equality relating whole cones EDMN and SN + . Equality (1214) is not obvious from the various EDM definitions such as (837) or (1140) because inclusion must be proved algebraically in order to N⊥ establish equality; EDMN ⊇ SN − SN h ∩ (Sc + ). We will instead prove (1214) using purely geometric methods.

§

§

6.0.2

§

highlight

In 6.8.1.7 we show: the Schoenberg criterion for discriminating Euclidean distance matrices ( −1 −VNTDVN ∈ SN + N D ∈ EDM ⇔ (856) D ∈ SN h

§

is a discretized membership relation ( 2.13.4, dual generalized inequalities) between the EDM cone and its ordinary dual. 6.1

Because both positive semidefinite cones are frequently in play, dimension is explicit.

6.1. DEFINING EDM CONE

6.1

477

Defining EDM cone

We invoke a popular matrix criterion to illustrate correspondence between the EDM and PSD cones belonging to the ambient space of symmetric matrices: ( −V D V ∈ SN + D ∈ EDMN ⇔ (860) D ∈ SN h

§

where V ∈ SN is the geometric centering matrix ( B.4). The set of all EDMs of dimension N ×N forms a closed convex cone EDMN because any pair of EDMs satisfies the definition of a convex cone (160); videlicet, for each and every ζ1 , ζ2 ≥ 0 ( A.3.1.0.2)

§

V D1 V º 0 , ζ1 V D1 V + ζ2 V D2 V º 0 ⇐ N D1 ∈ SN ζ1 D1 + ζ2 D2 ∈ Sh h ,

V D2 V º 0 D2 ∈ SN h

(1122)

and convex cones are invariant to inverse linear transformation [285, p.22]. 6.1.0.0.1 Definition. Cone of Euclidean distance matrices. In the subspace of symmetric matrices, the set of all Euclidean distance matrices forms a unique immutable pointed closed convex cone called the EDM cone: for N > 0 © ª N EDMN , D ∈ SN | −V D V ∈ S h + ª T © (1123) N T = D ∈ S | hzz , −Di ≥ 0 , δ(D) = 0 z∈N (1T )

The EDM cone in isomorphic RN (N +1)/2 [sic] is the intersection of an infinite number (when N > 2) of halfspaces about the origin and a finite number of hyperplanes through the origin in vectorized variable D = [dij ] . Hence EDMN has empty interior with respect to SN because it is confined to the symmetric hollow subspace SN h . The EDM cone relative interior comprises © ª T rel int EDMN = D ∈ SN | hzz T , −Di > 0 , δ(D) = 0 (1124) z∈N (1T ) © ª = D ∈ EDMN | rank(V D V ) = N − 1 while its relative boundary comprises © ª rel ∂ EDMN = D ∈ EDMN | hzz T , −Di = 0 for some z ∈ N (1T ) (1125) © ª = D ∈ EDMN | rank(V D V ) < N − 1 △

478 d13

0 0.2

CHAPTER 6. CONE OF DISTANCE MATRICES dvec rel ∂ EDM3 d12 0.2 0.4

0.4

0.6

0.6

0.8

0.8

1

1 1

d23

0.8 0.6

d23

0.4 0.2

(a)

(d)

0

d13 √

d13 0.6

0.8

0.8

d23

0.4

0.2 0.4

√

0

d12

0.2 0.4

0.6 0.8

0.4 0.8

0.8

1

1 1 0.8

0.6

0.6

0.4

0.4

0.2

(b)

0.2 0.6

0.6 1

1 1

√

0 0.2

d12

0.2

0

0

(c)

Figure 130: Relative boundary (tiled) of EDM cone EDM3 drawn truncated in isometrically isomorphic subspace R3 . (a) EDM cone drawn in usual distance-square coordinates dij . View is from interior toward origin. Unlike positive semidefinite cone, EDM cone is not self-dual; neither is it proper in ambient symmetric subspace (dual EDM cone for this example belongs p to isomorphic R6 ). (b) Drawn in its natural coordinates dij (absolute distance), cone remains convex (confer 5.10); intersection of three halfspaces (1008) whose partial boundaries each contain origin. Cone geometry becomes nonconvex (nonpolyhedral) in higher dimension. ( 6.3) (c) Two coordinate systems artificially superimposed. Coordinate transformation from dij to p dij appears a topological contraction. (d) Sitting on its vertex 0, pointed EDM3 is a circular cone having axis of revolution dvec(−E ) = dvec(11T − I ) (1041) (68). (Rounded vertex is plot artifact.)

§

§

6.2. POLYHEDRAL BOUNDS

479

This cone is more easily visualized in the isomorphic vector subspace R corresponding to SN h : In the case N = 1 point, the EDM cone is the origin in R0 . In the case N = 2, the EDM cone is the nonnegative real line in R ; a halfline in a subspace of the realization in Figure 138. The EDM cone in the case N = 3 is a circular cone in R3 illustrated in Figure 130(a)(d); rather, the set of all matrices N (N −1)/2

 0 d12 d13 D =  d12 0 d23  ∈ EDM3 d13 d23 0 

(1126)

makes a circular cone in this dimension. In this case, the first four Euclidean metric properties are necessary and sufficient tests to certify realizability of triangles; (1102). Thus triangle inequality property 4 describes three halfspacesp(1008) whose intersection makes a polyhedral cone in R3 of realizable dij (absolute distance); an isomorphic subspace representation of the set of all EDMs D in the natural coordinates   √ √ 0 d d 12 13 √ √ √ ◦ d23  D ,  √d12 √0 (1127) d13 d23 0 illustrated in Figure 130(b).

6.2

Polyhedral bounds

The convex cone of EDMs is nonpolyhedral in dij for N > 2 ; e.g., Figure 130(a). Still we found necessary and sufficient bounding polyhedral relations consistent with EDM cones for cardinality N = 1, 2, 3, 4: N = 3. Transforming distance-square coordinates dij by taking their positive square root provides the polyhedral cone in Figure 130(b); polyhedral because an intersection of three halfspaces in natural coordinates p dij is provided by triangle inequalities (1008). This polyhedral cone implicitly encompasses necessary and sufficient metric properties: nonnegativity, self-distance, symmetry, and triangle inequality.

480

CHAPTER 6. CONE OF DISTANCE MATRICES

(b)

(c)

dvec rel ∂ EDM3

(a)

∂H 0

Figure 131: (a) In isometrically isomorphic subspace R3 , intersection of EDM3 with hyperplane ∂H representing one fixed symmetric entry d23 = κ (both drawn truncated, rounded vertex is artifact of plot). EDMs in this dimension corresponding to affine dimension 1 comprise relative boundary of EDM cone ( 6.6). Since intersection illustrated includes a nontrivial subset of cone’s relative boundary, then it is apparent there exist infinitely many EDM completions corresponding to affine dimension 1. In this dimension it is impossible to represent a unique nonzero completion corresponding to affine dimension 1, for example, using a single hyperplane because any hyperplane supporting relative boundary at a particular point Γ contains an entire ray {ζ Γ | ζ ≥ 0} belonging to rel ∂ EDM3 by Lemma 2.8.0.0.1. (b) d13 = κ . (c) d12 = κ .

§

6.3.

√

EDM CONE IS NOT CONVEX

481

N = 4. Relative-angle inequality (1108) together with four Euclidean metric properties are necessary and sufficient tests for realizability of tetrahedra. (1109) Albeit relative angles θikj (900) are nonlinear functions of the dij , relative-angle inequality provides a regular tetrahedron in R3 [sic] (Figure 128) bounding angles θikj at vertex xk consistently with EDM4 .6.2

§

Yet were we to employ the procedure outlined in 5.14.3 for making generalized triangle inequalities, then we would find all the necessary and sufficient dij -transformations for generating bounding polyhedra consistent with EDMs of any higher dimension (N > 3).

6.3

√

EDM cone is not convex

For some applications, like a molecular conformation problem (Figure p 3, Figure 119) or multidimensional scaling [94] [329], absolute distance dij is the preferred variable. Taking square root of the entries in all EDMs D of dimension N , we get another cone but not a convex cone when N > 3 (Figure 130(b)): [84, 4.5.2] p √ ◦ (1128) EDMN , { D | D ∈ EDMN } √ where ◦ D is defined like (1127). It is a cone simply because any cone is completely constituted by rays emanating from the origin: ( 2.7) Any given Γ ∈ RN (N −1)/2 | ζ ≥ 0} remains a ray under entrywise square root: p ray {ζ { ζ Γ ∈ RN (N −1)/2 | ζ ≥ 0}. It is already established that √ ◦ D ∈ EDMN ⇒ D ∈ EDMN (1040) p But because of how EDMN is defined, it is obvious that (confer 5.10) p √ ◦ D ∈ EDMN ⇔ D ∈ EDMN (1129) p p √ √ EDMN convex, then given ◦ D1 , ◦ D2 ∈ EDMN we Were p would √ √ ◦ ◦ expect their conic combination D1 + D2 to be a member of EDMN .

§

§

§

6.2

Still, property-4 triangle inequalities (1008) corresponding to each principal 3 × 3 p submatrix of −VNTDVN demand that the corresponding dij belong to a polyhedral cone like that in Figure 130(b).

482

CHAPTER 6. CONE OF DISTANCE MATRICES

That is √easily proven false by counter-example via (1129), for then √ √ √ ◦ ◦ ◦ ( D1 + D2 ) ◦ ( D1 + ◦ D2 ) would need to be a member of EDMN . Notwithstanding, p EDMN ⊆ EDMN (1130)

by (1040) (Figure 130), and we learn how p to transform a nonconvex proximity problem in the natural coordinates dij to a convex optimization in 7.2.1.

§

6.4

A geometry of completion It is not known how to proceed if one wishes to restrict the dimension of the Euclidean space in which the configuration of points may be constructed.

§

−Michael W. Trosset (2000) [327, 1] Given an incomplete noiseless EDM, intriguing is the question of whether a list in X ∈ Rn×N (71) may be reconstructed and under what circumstances reconstruction is unique. [2] [4] [5] [6] [8] [17] [63] [191] [202] [217] [218] [219] If one or more entries of a particular EDM are fixed, then geometric interpretation of the feasible set of completions is the intersection of the EDM cone EDMN in isomorphic subspace RN (N −1)/2 with as many hyperplanes as there are fixed symmetric entries.6.3 Assuming a nonempty intersection, then the number of completions is generally infinite, and those corresponding to particular affine dimension r < N − 1 belong to some generally nonconvex subset of that intersection (confer 2.9.2.6.2) that can be as small as a point.

§

§

6.4.0.0.1 Example. Maximum variance unfolding. [347] A process minimizing affine dimension ( 2.1.5) of certain kinds of Euclidean manifold by topological transformation can be posed as a completion problem (confer E.10.2.1.2). Weinberger & Saul, who originated the technique, specify an applicable manifold in three dimensions by analogy to an ordinary sheet of paper (confer 2.1.6); imagine, we find it deformed from flatness in some way introducing neither holes, tears, or self-intersections. [347, 2.2] The physical process is intuitively described as unfurling,

§

§

§

Depicted in Figure 131a is an intersection of the EDM cone EDM3 with a single hyperplane representing the set of all EDMs having one fixed symmetric entry. 6.3

6.4. A GEOMETRY OF COMPLETION

483

(a)

2 nearest neighbors

(b)

3 nearest neighbors

Figure 132: Neighborhood graph (dashed) with one dimensionless EDM subgraph completion (solid) superimposed (but not obscuring dashed). Local view of a few dense samples # from relative interior of some arbitrary Euclidean manifold whose affine dimension appears two-dimensional in this neighborhood. All line segments measure absolute distance. Dashed line segments help visually locate nearest neighbors; suggesting, best number of nearest neighbors can be greater than value of embedding dimension after topological transformation. (confer [197, 2]) Solid line segments represent completion of EDM subgraph from available distance data for an arbitrarily chosen sample and its nearest neighbors. Each distance from EDM subgraph becomes distance-square in corresponding EDM submatrix.

§

484

CHAPTER 6. CONE OF DISTANCE MATRICES

unfolding, diffusing, decompacting, or unraveling. In particular instances, the process is a sort of flattening by stretching until taut (but not by crushing); e.g., unfurling a three-dimensional Euclidean body resembling a billowy national flag reduces that manifold’s affine dimension to r = 2. Data input to the proposed process originates from distances between neighboring relatively dense samples of a given manifold. Figure 132 realizes a densely sampled neighborhood; called, neighborhood graph. Essentially, the algorithmic process preserves local isometry between nearest neighbors allowing distant neighbors to excurse expansively by “maximizing variance” (Figure 5). The common number of nearest neighbors to each sample is a data-dependent algorithmic parameter whose minimum value connects the graph. The dimensionless EDM subgraph between each sample and its nearest neighbors is completed from available data and included as input; one such EDM subgraph completion is drawn superimposed upon the neighborhood graph in Figure 132 .6.4 The consequent dimensionless EDM graph comprising all the subgraphs is incomplete, in general, because the neighbor number is relatively small; incomplete even though it is a superset of the neighborhood graph. Remaining distances (those not graphed at all) are squared then made variables within the algorithm; it is this variability that admits unfurling. To demonstrate, consider untying the trefoil knot drawn in Figure 133(a). A corresponding Euclidean distance matrix D = [dij , i, j = 1 . . . N ] employing only 2 nearest neighbors is banded having the incomplete form   0 dˇ12 dˇ13 ? · · · ? dˇ1,N −1 dˇ1N   ˇ2N  ˇ23 dˇ24 . . .  dˇ12 0 d ? ? d     . . ˇ ˇ ˇ   d13 . ? ? ? d23 0 d34     . . .. .. ˇ24 dˇ34 0   ? d ? ?  (1131) D =   .. ... ... ... ... ... ...   . ?     . . . . ˇN −2,N −1 dˇN −2,N   . . ? ? ? 0 d   ... ˇ   ˇ ? ? dN −2,N −1 0 dˇN −1,N   d1,N −1 ? dˇ1N

dˇ2N

?

?

?

dˇN −2,N

dˇN −1,N

0

where dˇij denotes a given fixed distance-square. The unfurling algorithm can 6.4

Local reconstruction of point position, from the EDM submatrix corresponding to a complete dimensionless EDM subgraph, is unique to within an isometry (§5.6, §5.12).

6.4. A GEOMETRY OF COMPLETION

485

(a)

(b)

Figure 133: (a) Trefoil knot in R3 from Weinberger & Saul [347]. (b) Topological transformation algorithm employing 4 nearest neighbors and N = 539 samples reduces affine dimension of knot to r = 2. Choosing instead 2 nearest neighbors would make this embedding more circular.

486

CHAPTER 6. CONE OF DISTANCE MATRICES

be expressed as an optimization problem; constrained total distance-square maximization: maximize h−V , Di D

T 1 ˇ subject to hD , ei eT j + ej ei i 2 = dij

rank(V D V ) = 2

∀(i, j) ∈ I

(1132)

D ∈ EDMN where ei ∈ RN is the i th member of the standard basis, where set I indexes the given distance-square data like that in (1131), where V ∈ RN ×N is the geometric centering matrix ( B.4.1), and where

§

h−V , Di = tr(−V D V ) = 2 tr G =

1 X dij N i,j

(861)

where G is the Gram matrix producing D assuming G1 = 0. If the (rank) constraint on affine dimension is ignored, then problem (1132) becomes convex, a corresponding solution D⋆ can be found, and a nearest rank-2 solution is then had by ordered eigen of · decomposition ¸ 2 R + −V D⋆ V followed by spectral projection ( 7.1.3) on ⊂ RN . This 0 two-step process is necessarily suboptimal. Yet because the decomposition for the trefoil knot reveals only two dominant eigenvalues, the spectral projection is nearly benign. Such a reconstruction of point position ( 5.12) utilizing 4 nearest neighbors is drawn in Figure 133(b); a low-dimensional embedding of the trefoil knot. This problem (1132) can, of course, be written equivalently in terms of Gram matrix G , facilitated by (867); videlicet, for Φij as in (835)

§

§

maximize hI , Gi G∈SN c

subject to hG , Φij i = dˇij rank G = 2 Gº0

∀(i, j) ∈ I

(1133)

The advantage to converting EDM to Gram is: Gram matrix G is a bridge between point list X and EDM D ; constraints on any or all of these three variables may now be introduced. (Example 5.4.2.3.5) Confinement to the geometric center subspace SN c (implicit constraint G1 = 0) keeps G

6.4. A GEOMETRY OF COMPLETION

487

Figure 134: Trefoil ribbon; courtesy, Kilian Weinberger. Same topological transformation algorithm with 5 nearest neighbors and N = 1617 samples.

488

CHAPTER 6. CONE OF DISTANCE MATRICES

§

⊥ independent of its translation-invariant subspace SN ( 5.5.1.1, Figure 140) c so as not to become numerically unbounded. A problem dual to maximum variance unfolding problem (1133) (less the Gram rank constraint) has been called the fastest mixing Markov process. That dual has simple interpretations in graph and circuit theory and in mechanical and thermal systems, explored in [317], and has direct application to quick calculation of PageRank by search engines [214, 4]. Optimal Gram rank turns out to be tightly bounded above by minimum multiplicity of the second smallest eigenvalue of a dual optimal variable. 2

§

6.5

EDM definition in 11T

Any EDM D corresponding to affine dimension r has representation D(VX ) , δ(VX VXT )1T + 1δ(VX VXT )T − 2VX VXT ∈ EDMN

(1134)

where R(VX ∈ RN ×r ) ⊆ N (1T ) = 1⊥ , VXT VX = δ 2(VXT VX ) and VX is full-rank with orthogonal columns. (1135) Equation (1134) is simply the standard EDM definition (837) with a centered list X as in (921); Gram matrix X TX has been replaced with the subcompact singular value decomposition ( A.6.2)6.5

§

N VX VXT ≡ V TX TX V ∈ SN c ∩ S+

(1136)

This means: inner product VXT VX is an r×r diagonal matrix Σ of nonzero singular values. Vector δ(VX VXT ) may me decomposed into complementary parts by projecting it on orthogonal subspaces 1⊥ and R(1) : namely, ¡ ¢ P1⊥ δ(VX VXT ) = V δ(VX VXT ) (1137) ¡ ¢ 1 P1 δ(VX VXT ) = 11T δ(VX VXT ) (1138) N Of course

δ(VX VXT ) = V δ(VX VXT ) +

1 T 11 δ(VX VXT ) N

(1139)

√ √ Subcompact SVD: VX VXT , Q Σ Σ QT ≡ V TX TX V . So VXT is not necessarily XV (§5.5.1.0.1), although affine dimension r = rank(VXT ) = rank(XV ). (978) 6.5

6.5. EDM DEFINITION IN 11T

489

§

by (859). Substituting this into EDM definition (1134), we get the Hayden, Wells, Liu, & Tarazaga EDM formula [171, 2] D(VX , y) , y1T + 1y T +

λ T 11 − 2VX VXT ∈ EDMN N

(1140)

where λ , 2kVX k2F = 1T δ(VX VXT )2

and

y , δ(VX VXT ) −

λ 1 = V δ(VX VXT ) 2N (1141)

and y = 0 if and only if 1 is an eigenvector of EDM D . Scalar λ becomes an eigenvalue when corresponding eigenvector 1 exists.6.6 Then the particular dyad sum from (1140) λ T ⊥ 11 ∈ SN (1142) c N must belong to the orthogonal complement of the geometric center subspace N (p.703), whereas VX VXT ∈ SN c ∩ S+ (1136) belongs to the positive semidefinite cone in the geometric center subspace. y1T + 1y T +

Proof. We validate eigenvector 1 and eigenvalue λ . (⇒) Suppose 1 is an eigenvector of EDM D . Then because VXT 1 = 0

(1143)

it follows D1 = δ(VX VXT )1T 1 + 1δ(VX VXT )T 1 = N δ(VX VXT ) + kVX k2F 1 ⇒ δ(VX VXT ) ∝ 1

For some κ ∈ R+

δ(VX VXT )T 1 = N κ = tr(VXT VX ) = kVX k2F ⇒ δ(VX VXT ) = so y = 0. (⇐) Now suppose δ(VX VXT ) =

1 kVX k2F 1 (1145) N

λ 1 ; id est, y = 0. Then 2N

λ T 11 − 2VX VXT ∈ EDMN N 1 is an eigenvector with corresponding eigenvalue λ . D=

6.6

(1144)

e.g., when X = I in EDM definition (837).

(1146) ¨

490

CHAPTER 6. CONE OF DISTANCE MATRICES

δ(VX VXT )

N (1T )

1

VX

Figure 135: Example of VX selection to make an EDM corresponding to cardinality N = 3 and affine dimension r = 1 ; VX is a vector in nullspace N (1T ) ⊂ R3 . Nullspace of 1T is hyperplane in R3 (drawn truncated) having normal 1. Vector δ(VX VXT ) may or may not be in plane spanned by {1 , VX } , but belongs to nonnegative orthant which is strictly supported by N (1T ).

6.5. EDM DEFINITION IN 11T

6.5.1

§

491

Range of EDM D

From B.1.1 pertaining to linear independence of dyad sums: If the transpose halves of all the dyads in the sum (1134)6.7 make a linearly independent set, then the nontranspose halves constitute a basis for the range of EDM D . Saying this mathematically: For D ∈ EDMN R(D) = R([ δ(VX VXT ) 1 VX ]) ⇐ rank([ δ(VX VXT ) 1 VX ]) = 2 + r R(D) = R([ 1 VX ])

⇐ otherwise

(1147)

To prove this, we need that condition under which the rank equality is satisfied: We know R(VX ) ⊥ 1, but what is the relative geometric orientation of δ(VX VXT ) ? δ(VX VXT ) º 0 because VX VXT º 0, and δ(VX VXT ) ∝ 1 remains possible (1144); this means δ(VX VXT ) ∈ / N (1T ) simply because it has no negative entries. (Figure 135) If the projection of δ(VX VXT ) on N (1T ) does not belong to R(VX ) , then that is a necessary and sufficient condition for linear independence (l.i.) of δ(VX VXT ) with respect to R([ 1 VX ]) ; id est, V δ(VX VXT ) 6= VX a for any a ∈ Rr

1 11T )δ(VX VXT ) N δ(VX VXT ) − N1 kVX k2F 1

(I −

δ(VX VXT ) −

λ 1 2N

6= VX a

(1148)

6= VX a

= y 6= VX a ⇔ {1 , δ(VX VXT ) , VX } is l.i.

On the other hand when this condition is violated (when (1141) y = VX ap for some particular a ∈ Rr ), then from (1140) we have ¢ ¡ ¢ ¡ T R D = y1T + 1y T + Nλ 11T − 2VX VXT = R (VX ap + Nλ 1)1T + (1aT p − 2VX )VX = R([ VX ap +

λ 1 N

= R([ 1 VX ])

1aT p − 2VX ])

(1149)

An example of such a violation is (1146) where, in particular, ap = 0. ¨

6.7

Then a statement parallel to (1147) is, for D ∈ EDMN (Theorem 5.7.3.0.1) ¡ ¢ rank(D) = r + 2 ⇔ y ∈ / R(VX ) ⇔ 1TD† 1 = 0 (1150) ¡ ¢ rank(D) = r + 1 ⇔ y ∈ R(VX ) ⇔ 1TD† 1 6= 0 Identifying columns VX , [ v1 · · · vr ] , then VX VXT =

P i

vi viT is also a sum of dyads.

492

CHAPTER 6. CONE OF DISTANCE MATRICES 1 0 -1

(a) 1

0

-1

VX ∈ R3×1

տ

-1 0 1

←−

dvec D(VX ) ⊂ dvec rel ∂ EDM3

(b)

Figure 136: (a) Vector VX from Figure 135 spirals in N (1T ) ⊂ R3 decaying toward origin. (Spiral is two-dimensional in vector space R3 .) (b) Corresponding trajectory D(VX ) on EDM cone relative boundary creates a vortex also decaying toward origin. There are two complete orbits on EDM cone boundary about axis of revolution for every single revolution of VX about origin. (Vortex is three-dimensional in isometrically isomorphic R3 .)

6.5. EDM DEFINITION IN 11T

6.5.2

493

Boundary constituents of EDM cone

Expression (1134) has utility in forming the set of all EDMs corresponding to affine dimension r : ª D ∈ EDMN | rank(V D V ) = r © ª = D(VX ) | VX ∈ RN ×r , rank VX = r , VXT VX = δ 2(VXT VX ) , R(VX ) ⊆ N (1T ) (1151)

©

whereas {D ∈ EDMN | rank(V D V ) ≤ r} is the closure of this same set; ©

ª © ª D ∈ EDMN | rank(V D V ) ≤ r = D ∈ EDMN | rank(V D V ) = r

(1152)

For example,

rel ∂ EDMN =

ª D ∈ EDMN | rank(V D V ) < N − 1 NS −2© ª = D ∈ EDMN | rank(V D V ) = r ©

(1153)

r=0

None of these are necessarily convex sets, although EDMN =

NS −1© r=0

ª

ª D ∈ EDMN | rank(V D V ) = N − 1 © ª = D ∈ EDMN | rank(V D V ) = N − 1 =

rel int EDMN

D ∈ EDMN | rank(V D V ) = r

©

(1154)

are pointed convex cones. When cardinality N = 3 and affine dimension r = 2, for example, the relative interior rel int EDM3 is realized via (1151). ( 6.6) When N = 3 and r = 1, the relative boundary of the EDM cone dvec rel ∂ EDM3 is realized in isomorphic R3 as in Figure 130(d). This figure could be constructed via (1152) by spiraling vector VX tightly about the origin in N (1T ) ; as can be imagined with aid of Figure 135. Vectors close to the origin in N (1T ) are correspondingly close to the origin in EDMN . As vector VX orbits the origin in N (1T ) , the corresponding EDM orbits the axis of revolution while remaining on the boundary of the circular cone dvec rel ∂ EDM3 . (Figure 136)

§

494

6.5.3

CHAPTER 6. CONE OF DISTANCE MATRICES

Faces of EDM cone

Like the positive semidefinite cone, EDM cone faces are EDM cones. 6.5.3.0.1 Exercise. Isomorphic faces. Prove that in high cardinality N , any set of EDMs made via (1151) or (1152) with particular affine dimension r is isomorphic with any set admitting the same affine dimension but made in lower cardinality. H 6.5.3.1

Smallest face containing an EDM

Now suppose we are given a particular EDM D(VXp ) ∈ EDMN corresponding to affine dimension r and parametrized by VXp in (1134). The EDM cone’s smallest face that contains D(VXp ) is ¡ ¢ F EDMN ∋ D(VXp ) © ª = D(VX ) | VX ∈ RN ×r , rank VX = r , VXT VX = δ 2(VXT VX ) , R(VX ) ⊆ R(VXp ) ≃ EDM r+1

(1155)

which is isomorphic6.8 with the convex cone EDM r+1 , hence of dimension ¡ ¢ dim F EDMN ∋ D(VXp ) = (r + 1)r/2 (1156) in isomorphic RN (N −1)/2 . Not all dimensions are represented; e.g., the EDM cone has no two-dimensional faces. When cardinality N = 4 and affine dimension r = 2 so that R(VXp ) is any two-dimensional subspace of three-dimensional N (1T ) in R4 , for example, then the corresponding face of EDM4 is isometrically isomorphic with: (1152)

EDM3 = {D ∈ EDM3 | rank(V D V ) ≤ 2} ≃ F(EDM4 ∋ D(VXp )) (1157) Each two-dimensional subspace of N (1T ) corresponds to another three-dimensional face. Because each and every principal submatrix of an EDM in EDMN ( 5.14.3) is another EDM [218, 4.1], for example, then each principal submatrix belongs to a particular face of EDMN .

§

6.8

§

The fact that the smallest face is isomorphic with another EDM cone (perhaps smaller than EDMN ) is implicit in [171, §2].

6.5. EDM DEFINITION IN 11T 6.5.3.2

495

Extreme directions of EDM cone

§

In particular, extreme directions ( 2.8.1) of EDMN correspond to affine dimension r = 1 and are simply represented: for any particular cardinality N ≥ 2 ( 2.8.2) and each and every nonzero vector z in N (1T )

§

Γ , (z ◦ z)1T + 1(z ◦ z)T − 2zz T ∈ EDMN

(1158)

= δ(zz T )1T + 1δ(zz T )T − 2zz T

is an extreme direction corresponding to a one-dimensional face of the EDM cone EDMN that is a ray in isomorphic subspace RN (N −1)/2 . Proving this would exercise the fundamental definition (171) of extreme direction. Here is a sketch: Any EDM may be represented D(VX ) = δ(VX VXT )1T + 1δ(VX VXT )T − 2VX VXT ∈ EDMN

(1134)

§

where matrix VX (1135) has orthogonal columns. For the same reason (1411) that zz T is an extreme direction of the positive semidefinite cone ( 2.9.2.4) for any particular nonzero vector z , there is no conic combination of distinct EDMs (each conically independent of Γ ( 2.10)) equal to Γ . ¥

§

§

6.5.3.2.1 Example. Biorthogonal expansion of an EDM. (confer 2.13.7.1.1) When matrix D belongs to the EDM cone, nonnegative coordinates for biorthogonal expansion are the eigenvalues λ ∈ RN of −V D V 21 : For any D ∈ SN h it holds ¡ ¢ ¡ ¢T ¢ ¡ D = δ −V D V 12 1T + 1δ −V D V 12 − 2 −V D V 21

§

(945)

By diagonalization −V D V 21 , QΛQT ∈ SN c ( A.5.1) we may write D=δ =

µN P

N P

i=1

i=1

¶

λ i qi qiT

T

1 + 1δ

λ i δ(qi qiT )1T + ¡

µN P

i=1

¶T

λ i qi qiT

1δ(qi qiT )T −

−2

2qi qiT

¢

N P

i=1

λ i qi qiT

(1159)

where qi is the i th eigenvector of −V D V 21 arranged columnar in orthogonal matrix Q = [ q1 q2 · · · qN ] ∈ RN ×N (372)

496

CHAPTER 6. CONE OF DISTANCE MATRICES

and where {δ(qi qiT )1T + 1δ(qi qiT )T − 2qi qiT , i = 1 . . . N } are extreme directions of some pointed polyhedral cone K ⊂ SN h and extreme directions N of EDM . Invertibility of (1159) −V D V

1 2

= −V =

N P

i=1

N P

i=1

¡ ¢ λ i δ(qi qiT )1T + 1δ(qi qiT )T − 2qi qiT V

1 2

implies linear independence of those extreme directions. biorthogonal expansion is expressed

where

(1160)

λ i qi qiT

¢ ¡ dvec D = Y Y † dvec D = Y λ −V D V 21

Then the

(1161)

£ ¡ ¢ ¤ Y , dvec δ(qi qiT )1T + 1δ(qi qiT )T − 2qi qiT , i = 1 . . . N ∈ RN (N −1)/2×N (1162)

When D belongs to the EDM cone in the subspace of symmetric hollow matrices, unique coordinates Y † dvec D for this biorthogonal expansion must be the nonnegative eigenvalues λ of −V D V 12 . This means D simultaneously belongs to the EDM cone and to the pointed polyhedral cone dvec K = cone(Y ). 2

6.5.3.3

Open question

Result (1156) is analogous to that for the positive semidefinite cone (207), although the question remains open whether all faces of EDMN (whose dimension is less than dimension of the cone) are exposed like they are for the positive semidefinite cone.6.9 ( 2.9.2.3) [323]

§

6.6

N −1 Correspondence to PSD cone S+

§

Hayden, Wells, Liu, & Tarazaga [171, 2] assert one-to-one correspondence of EDMs with positive semidefinite matrices in the symmetric subspace. Because rank(V D V ) ≤ N − 1 ( 5.7.1.1), that PSD cone corresponding to

§

6.9

Elementary example of a face not exposed is given in Figure 31 and Figure 40.

−1 6.6. CORRESPONDENCE TO PSD CONE SN +

497

§

−1 the EDM cone can only be SN . [8, 18.2.1] To clearly demonstrate this + correspondence, we invoke inner-product form EDM definition

D(Φ) =

·

0 δ(Φ)

¸

T

£

T

1 + 1 0 δ(Φ) ⇔ Φº0

¤

− 2

·

0 0

0T Φ

¸

∈ EDMN

(963)

Then the EDM cone may be expressed © ª −1 EDMN = D(Φ) | Φ ∈ SN +

(1163)

Hayden & Wells’ assertion can therefore be equivalently stated in terms of an inner-product form EDM operator −1 D(SN ) = EDMN +

(965)

−1 VN (EDMN ) = SN +

(966)

§

identity (966) holding because R(VN ) = N (1T ) (844), linear functions D(Φ) and VN (D) = −VNTDVN ( 5.6.2.1) being mutually inverse. In terms of affine dimension r , Hayden & Wells claim particular correspondence between PSD and EDM cones: r = N − 1: Symmetric hollow matrices −D positive definite on N (1T ) correspond to points relatively interior to the EDM cone. r < N − 1: Symmetric hollow matrices −D positive semidefinite on N (1T ) , where −VNTDVN has at least one 0 eigenvalue, correspond to points on the relative boundary of the EDM cone. r = 1: Symmetric hollow nonnegative matrices rank-one on N (1T ) correspond to extreme directions (1158) of the EDM cone; id est, for some nonzero vector u ( A.3.1.0.7)

§

rank VNTDVN = 1 N ×N D ∈ SN h ∩ R+

)

D ∈ EDMN ⇔ ⇔ D is an extreme direction

(

−VNTDVN ≡ uuT D ∈ SN h (1164)

498

CHAPTER 6. CONE OF DISTANCE MATRICES

§

6.6.0.0.1 Proof. Case r = 1 is easily proved: From the nonnegativity development in 5.8.1, extreme direction (1158), and Schoenberg criterion (856), we need show only sufficiency; id est, prove rank VNTDVN = 1 N ×N D ∈ SN h ∩ R+

)

D ∈ EDMN ⇒ D is an extreme direction

Any symmetric matrix D satisfying the rank condition must have the form, for z,q ∈ RN and nonzero z ∈ N (1T ) , D = ±(1q T + q 1T − 2zz T )

§

(1165)

§

because ( 5.6.2.1, confer E.7.2.0.2) N (VN (D)) = {1q T + q 1T | q ∈ RN } ⊆ SN

(1166)

Hollowness demands q = δ(zz T ) while nonnegativity demands choice of positive sign in (1165). Matrix D thus takes the form of an extreme direction (1158) of the EDM cone. ¨ The foregoing proof is not extensible in rank: An EDM with corresponding affine dimension r has the general form, for {zi ∈ N (1T ) , i = 1 . . . r} an independent set, D = 1δ

µ

r P

i=1

zi ziT

¶T

+δ

µ

r P

i=1

zi ziT

¶

1T − 2

r P

i=1

zi ziT ∈ EDMN

(1167)

P T The EDM so defined relies principally on the sum zi zi having positive summand coefficients (⇔ −VNTDVN º 0)6.10 . Then it is easy to find a sum incorporating negative coefficients while meeting rank, nonnegativity, and symmetric hollowness conditions but not positive semidefiniteness on subspace R(VN ) ; e.g., from page 448, 

6.10

 0 1 1 1 −V  1 0 5 V = z1 z1T − z2 z2T 2 1 5 0

(⇐) For ai ∈ RN −1 , let zi = VN†Tai .

(1168)

−1 6.6. CORRESPONDENCE TO PSD CONE SN +

499

6.6.0.0.2

Example. Extreme rays versus rays on the boundary.   0 1 4 The EDM D =  1 0 1  is an extreme direction of EDM3 where 4 1 0 · ¸ 1 in (1164). Because −VNTDVN has eigenvalues {0, 5} , the ray u= 2 whose direction is D also lies on· the relative boundary of EDM3 . ¸ 0 1 In exception, EDM D = κ , for any particular κ > 0, is an 1 0 extreme direction of EDM2 but −VNTDVN has only one eigenvalue: {κ}. Because EDM2 is a ray whose relative boundary ( 2.6.1.3.1) is the origin, this conventional boundary does not include D which belongs to the relative 2 interior in this dimension. ( 2.7.0.0.1)

§

§

6.6.1

−1 Gram-form correspondence to SN +

§

§

With respect to D(G) = δ(G)1T + 1δ(G)T − 2G (849) the linear Gram-form EDM operator, results in 5.6.1 provide [2, 2.6] ¡ ¢ ¡ ¢ −1 T EDMN = D V(EDMN ) ≡ D VN SN VN +

(1169)

¡ ¡ ¢¢ N −1 T −1 T VN SN VN ≡ V D VN SN VN = V(EDMN ) , −V EDMN V 21 = SN c ∩ S+ + + (1170) N N −1 a one-to-one correspondence between EDM and S+ .

6.6.2

EDM cone by elliptope

Having defined the elliptope parametrized by scalar t > 0 N EtN = SN + ∩ {Φ ∈ S | δ(Φ) = t1}

(1043)

then following Alfakih [9] we have EDMN = cone{11T − E1N } = {t(11T − E1N ) | t ≥ 0} Identification E N = E1N equates the standard Figure 122) to our parametrized elliptope.

elliptope

(1171)

§

( 5.9.1.0.1,

500

CHAPTER 6. CONE OF DISTANCE MATRICES

dvec rel ∂ EDM3

dvec(11T − E 3 )

EDMN = cone{11T − E N } = {t(11T − E N ) | t ≥ 0}

(1171)

Figure 137: Three views of translated negated elliptope 11T − E13 (confer Figure 122) shrouded by truncated EDM cone. Fractal on EDM cone relative boundary is numerical artifact belonging to intersection with elliptope relative boundary. The fractal is trying to convey existence of a neighborhood about the origin where the translated elliptope boundary and EDM cone boundary intersect.

−1 6.6. CORRESPONDENCE TO PSD CONE SN +

501

6.6.2.0.1 Expository. Normal cone, tangent cone, elliptope. Define T E (11T ) to be the tangent cone to the elliptope E at point 11T ; id est, (1172) T E (11T ) , {t(E − 11T ) | t ≥ 0} The normal cone KE⊥ (11T ) to the elliptope at 11T is a closed convex cone defined ( E.10.3.2.1, Figure 166)

§

KE⊥ (11T ) , {B | hB , Φ − 11T i ≤ 0 , Φ ∈ E }

(1173)

The polar cone of any set K is the closed convex cone (confer (271)) ◦

K , {B | hB , Ai ≤ 0 , for all A ∈ K}

(1174)

The normal cone is well known to be the polar of the tangent cone,

§

◦

KE⊥ (11T ) = T E (11T )

(1175)

and vice versa; [185, A.5.2.4] ◦

KE⊥ (11T ) = TE (11T )

(1176)

From Deza & Laurent [103, p.535] we have the EDM cone EDM = −TE (11T )

(1177)

The polar EDM cone is also expressible in terms of the elliptope. From (1175) we have ◦ EDM = −KE⊥ (11T ) (1178) ⋆

§

In 5.10.1 we proposed the expression for EDM D D = t11T − E ∈ EDMN

(1044)

where t ∈ R+ and E belongs to the parametrized elliptope EtN . We further propose, for any particular t > 0 EDMN = cone{t11T − EtN } Proof. Pending.

(1179) ¨

Relationship of the translated negated elliptope with the EDM cone is illustrated in Figure 137. We speculate EDMN = lim t11T − EtN t→∞

(1180)

502

CHAPTER 6. CONE OF DISTANCE MATRICES

6.7

Vectorization & projection interpretation

§ §

In E.7.2.0.2 we learn: −V D V can be interpreted as orthogonal projection [6, 2] of vectorized −D ∈ SN h on the subspace of geometrically centered symmetric matrices N SN c = {G ∈ S | G1 = 0}

(1921)

= {G ∈ SN | N (G) ⊇ 1} = {G ∈ SN | R(G) ⊆ N (1T )}

= {V Y V | Y ∈ SN } ⊂ SN

(1922)

(936)

≡ {VN AVNT | A ∈ SN −1 }

§

because elementary auxiliary matrix V is an orthogonal projector ( B.4.1). Yet there is another useful projection interpretation: Revising the fundamental matrix criterion for membership to the EDM cone (832),6.11 ) hzz T , −Di ≥ 0 ∀ zz T | 11Tzz T = 0 ⇔ D ∈ EDMN (1181) N D ∈ Sh this is equivalent, of course, to the Schoenberg criterion ) −VNTDVN º 0 ⇔ D ∈ EDMN D ∈ SN h

(856)

because N (11T ) = R(VN ). When D ∈ EDMN , correspondence (1181) means −z TDz is proportional to a nonnegative coefficient of orthogonal projection ( E.6.4.2, Figure 139) of −D in isometrically isomorphic RN (N +1)/2 on the range of each and every vectorized ( 2.2.2.1) symmetric dyad ( B.1) in the nullspace of 11T ; id est, on each and every member of © ª T , svec(zz T ) | z ∈ N (11T ) = R(VN ) ⊂ svec ∂ SN + (1182) © ª N −1 T T = svec(VN υυ VN ) | υ ∈ R

§

§

§

whose dimension is

dim T = N (N − 1)/2 6.11

N (11T ) = N (1T ) and R(zz T ) = R(z)

(1183)

6.7. VECTORIZATION & PROJECTION INTERPRETATION

503

The set of all symmetric dyads {zz T | z ∈ RN } constitute the extreme directions of the positive semidefinite cone ( 2.8.1, 2.9) SN + , hence lie on its boundary. Yet only those dyads in R(VN ) are included in the test (1181), thus only a subset T of all vectorized extreme directions of SN + is observed. 2 2 In the particularly simple case D ∈ EDM = {D ∈ Sh | d12 ≥ 0} , for example, only one extreme direction of the PSD cone is involved: ¸ · 1 −1 T (1184) zz = −1 1

§

§

Any nonnegative scaling of vectorized zz T belongs to the set T illustrated in Figure 138 and Figure 139.

6.7.1

Face of PSD cone SN + containing V

In any case, set T (1182) constitutes the vectorized extreme directions of an N (N − 1)/2-dimensional face of the PSD cone SN + containing auxiliary N −1 rank V matrix V ; a face isomorphic with S+ = S+ ( 2.9.2.3).

§

To show this, we must first find the smallest face that contains auxiliary matrix V and then determine its extreme directions. From (206), ¡ ¢ N F SN = {W ∈ SN + ∋V + | N (W ) ⊇ N (V )} = {W ∈ S+ | N (W ) ⊇ 1} −1 = {V Y V º 0 | Y ∈ SN } ≡ {VN B VNT | B ∈ SN } +

V ≃ Srank = −VNT EDMN VN +

§

(1185)

where the equivalence ≡ is from 5.6.1 while isomorphic equality¡ ≃ with¢ transformed EDM cone is from (966). Projector V belongs to F SN + ∋V † †T T because VN VN VN VN = V . ( B.4.3) Each and every rank-one matrix belonging to this face is therefore of the form:

§

VN υυ T VNT | υ ∈ RN −1

(1186)

¡ ¢ −1 Because F SN is isomorphic with a positive semidefinite cone SN , + ∋V + then T constitutes the vectorized extreme directions of F , the origin constitutes the extreme points of F , and auxiliary matrix V is some convex combination of those extreme points and directions by the extremes theorem ( 2.8.1.1.1). ¨

§

504

CHAPTER 6. CONE OF DISTANCE MATRICES ·

d11 d12 d12 d22

¸

svec ∂ S2+

svec EDM2

d22

0 −T √

d11

T ,

½

· svec(zz ) | z ∈ N (11 ) = κ T

T

2d12

¸ ¾ 1 , κ ∈ R ⊂ svec ∂ S2+ −1

Figure 138: Truncated boundary of positive semidefinite cone S2+ in isometrically isomorphic R3 (via svec (51)) is, in this dimension, constituted solely by its extreme directions. Truncated cone of Euclidean distance matrices EDM2 in isometrically isomorphic subspace R . Relative boundary of √EDM cone is constituted solely by matrix 0. Halfline T = {κ2 [ 1 − 2 1 ]T | κ ∈ R} on PSD cone boundary depicts that lone extreme ray (1184) on which orthogonal projection of −D must be positive semidefinite if D is to belong to EDM2 . aff cone T = svec S2c . (1189) Dual EDM cone is halfspace in R3 whose bounding hyperplane has inward-normal svec EDM2 .

6.7. VECTORIZATION & PROJECTION INTERPRETATION ·

d11 d12 d12 d22

505 ¸

svec ∂ S2+

svec S2h

d22 D 0 −T

−D

√

d11

2d12

Projection of vectorized −D on range of vectorized zz T : Psvec zzT (svec(−D)) =

N

D ∈ EDM

⇔

(

hzz T , −Di T zz hzz T , zz T i hzz T , −Di ≥ 0 ∀ zz T | 11Tzz T = 0 D ∈ SN h

(1181)

Figure 139: Given-matrix D is assumed to belong to symmetric hollow subspace S2h ; a line in this dimension. Negating D , we find its polar along S2h . Set T (1182) has only one ray member in this dimension; not orthogonal to S2h . Orthogonal projection of −D on T (indicated by half dot) has nonnegative projection coefficient. Matrix D must therefore be an EDM.

506

CHAPTER 6. CONE OF DISTANCE MATRICES

In fact, the smallest face that contains auxiliary matrix V of the PSD cone SN + is the intersection with the geometric center subspace (1921) (1922); ¡ ¢ © ª N −1 T T F SN ∋ V = cone V υυ V | υ ∈ R + N N N = SN c ∩ S+

(1187)

≡ {X º 0 | hX , 11T i = 0}

(1525)

In isometrically isomorphic RN (N +1)/2 ¡ ¢ svec F SN + ∋ V = cone T related to SN c by

6.7.2

§

(1188)

aff cone T = svec SN c

(1189)

EDM criteria in 11T

§

(confer 6.5, (863)) Laurent specifies an elliptope trajectory condition for EDM cone membership: [218, 2.3] D ∈ EDMN ⇔ [1 − e−α dij ] ∈ EDMN ∀ α > 0

§

§

(1038)

From the parametrized elliptope EtN in 6.6.2 and 5.10.1 we propose ) t ∈ R + D ∈ EDMN ⇔ ∃ Ä D = t11T − E (1190) E ∈ EtN

§

Chabrillac & Crouzeix [70, 4] prove a different criterion they attribute to Finsler (1937) [132]. We apply it to EDMs: for D ∈ SN h (986) −VNTDVN ≻ 0 ⇔ ∃ κ > 0 Ä −D + κ11T ≻ 0 ⇔ N D ∈ EDM with corresponding affine dimension r = N − 1

(1191)

This Finsler criterion has geometric interpretation in terms of the vectorization & projection already discussed in connection with (1181). With reference to Figure 138, the offset 11T is simply a direction orthogonal to T in isomorphic R3 . Intuitively, translation of −D in direction 11T is like orthogonal projection on T in so far as similar information can be obtained.

6.8. DUAL EDM CONE

507

When the Finsler criterion (1191) is applied despite lower affine dimension, the constant κ can go to infinity making the test −D + κ11T º 0 impractical for numerical computation. Chabrillac & Crouzeix invent a criterion for the semidefinite case, but is no more practical: for D ∈ SN h D ∈ EDMN (1192) ⇔ ∃ κp > 0 Ä ∀ κ ≥ κp , −D − κ11T [sic] has exactly one negative eigenvalue

6.8

Dual EDM cone Ambient SN

6.8.1

We consider finding the ordinary dual EDM cone in ambient space SN where EDMN is pointed, closed, convex, but has empty interior. The set of all EDMs in SN is a closed convex cone because it is the intersection of halfspaces about the origin in vectorized variable D ( 2.4.1.1.1, 2.7.2): ª T © T T EDMN = D ∈ SN | hei eT i , Di ≥ 0 , hei ei , Di ≤ 0 , hzz , −Di ≥ 0 z∈ N (1T ) (1193) i=1...N

§

§

∗

By definition (271), dual cone K comprises each and every vector inward-normal to a hyperplane supporting convex cone K . The dual EDM cone in the ambient space of symmetric matrices is therefore expressible as the aggregate of every conic combination of inward-normals from (1193): ∗

T T T T EDMN = cone{ei eT i , −ej ej | i , j = 1 . . . N } − cone{zz | 11 zz = 0}

={

N P

i=1

ζ i ei eT i −

N P

j=1

T T T ξj ej eT j | ζ i , ξ j ≥ 0} − cone{zz | 11 zz = 0}

© ª = {δ(u) | u ∈ R } − cone VN υυ T VNT | υ ∈ RN −1 , (kvk = 1) ⊂ SN N

−1 = {δ 2(Y ) − VN ΨVNT | Y ∈ SN , Ψ ∈ SN } +

(1194)

The EDM cone is not self-dual in ambient SN because its affine hull belongs to a proper subspace aff EDMN = SN (1195) h

§

The ordinary dual EDM cone cannot, therefore, be pointed. ( 2.13.1.1)

508

CHAPTER 6. CONE OF DISTANCE MATRICES

When N = 1, the EDM cone is the point at the origin in R . Auxiliary matrix VN is empty [ ∅ ] , and dual cone EDM∗ is the real line. When N = 2, the EDM cone is a nonnegative real line in isometrically isomorphic R3 ; there S2h is a real line containing the EDM cone. Dual cone ∗ EDM2 is the particular halfspace in R3 whose boundary has inward-normal EDM2 . Diagonal matrices {δ(u)} in (1194) are represented by a hyperplane through the origin {d | [ 0 1 0 ]d = 0} while the term cone{VN υυ T VNT } is represented by the halfline T in Figure 138 belonging to the positive semidefinite cone boundary. The dual EDM cone is formed by translating the hyperplane along the negative semidefinite halfline −T ; the union of each and every translation. (confer 2.10.2.0.1) When cardinality N exceeds 2, the dual EDM cone can no longer be polyhedral simply because the EDM cone cannot. ( 2.13.1.1)

§

6.8.1.1

§

EDM cone and its dual in ambient SN

Consider the two convex cones K1 , SN h ª T © K2 , A ∈ SN | hyy T , −Ai ≥ 0 y∈N (1T )

(1196)

ª A ∈ SN | −z T V A V z ≥ 0 ∀ zz T(º 0) © ª = A ∈ SN | −V A V º 0 =

so

©

K1 ∩ K2 = EDMN

(1197)

© ª ∗ K2 = − cone VN υυ T VNT | υ ∈ RN −1 ⊂ SN

(1198)

∗ K1 =

⊥ N SN (67) is the subspace of diagonal matrices. Dual cone h ⊆S From (1194) via (287),

§

Gaffke & Mathar [135, 5.3] observe that projection on K1 and K2 have simple closed forms: Projection on subspace K1 is easily performed by symmetrization and zeroing the main diagonal or vice versa, while projection of H ∈ SN on K2 is PK2 H = H − PSN+ (V H V ) (1199)

6.8. DUAL EDM CONE

509

Proof. First, we observe membership of H −PSN+ (V H V ) to K2 because ³ ´ PSN+ (V H V ) − H = PSN+ (V H V ) − V H V + (V H V − H) (1200)

The term PSN+ (V H V ) − V H V necessarily belongs to the (dual) positive semidefinite cone by Theorem E.9.2.0.1. V 2 = V , hence ³ ´ −V H − PSN+ (V H V ) V º 0 (1201)

by Corollary A.3.1.0.5. Next, we require

hPK2 H − H , PK2 H i = 0

Expanding,

(1202)

h−PSN+ (V H V ) , H − PSN+ (V H V )i = 0

(1203)

hPSN+ (V H V ) , (V H V − H)i = 0

(1205)

hPSN+ (V H V ) , (PSN+ (V H V ) − V H V ) + (V H V − H)i = 0

§

(1204)

Product V H V belongs to the geometric center subspace; ( E.7.2.0.2) N V H V ∈ SN c = {Y ∈ S | N (Y ) ⊇ 1}

§

(1206)

Diagonalize V H V , QΛQT ( A.5) whose nullspace is spanned by the eigenvectors corresponding to 0 eigenvalues by Theorem A.7.3.0.1. Projection of V H V on the PSD cone ( 7.1) simply zeroes negative eigenvalues in diagonal matrix Λ . Then

§

N (PSN+ (V H V )) ⊇ N (V H V ) ( ⊇ N (V ) )

(1207)

from which it follows: PSN+ (V H V ) ∈ SN c

(1208)

⊥ so PSN+ (V H V ) ⊥ (V H V −H) because V H V −H ∈ SN c . ∗ Finally, we must have PK2 H − H = −PSN+ (V H V ) ∈ K2 . From 6.7.1 ¡ ¢ ∗ is the negative of the positive we know dual cone K2 = −F SN + ∋V semidefinite cone’s face that contains auxiliary matrix V . Thus ¡ smallest ¢ PSN+ (V H V ) ∈ F SN ∋ V ⇔ N (PSN+ (V H V )) ⊇ N (V ) ( 2.9.2.3) which was + ¨ already established in (1207).

§

§

510

CHAPTER 6. CONE OF DISTANCE MATRICES

¡ ¢ 2 EDM2 = S2h ∩ S2⊥ c − S+ svec S2⊥ c

svec S2c svec ∂ S2+

svec S2h 0

§

Figure 140: Orthogonal complement S2⊥ c (1923) ( B.2) of geometric center subspace (a plane in isometrically isomorphic R3 ; drawn is a tiled fragment) apparently supporting positive semidefinite cone. (Rounded vertex is artifact of plot.) Line svec S2c = aff cone T (1189) intersects svec ∂ S2+ , also drawn in Figure 138; it runs along PSD cone boundary. (confer Figure 121)

6.8. DUAL EDM CONE

511

¡ ¢ 2 EDM2 = S2h ∩ S2⊥ c − S+ svec S2⊥ c

svec S2c

svec S2h 0

− svec ∂ S2+

Figure 141: EDM cone construction in isometrically R3 by adding ¡ 2⊥ isomorphic ¢ 2⊥ 2 polar PSD cone to svec Sc . Difference svec Sc − S+ is halfspace partially 2 bounded by svec S2⊥ c . EDM cone is nonnegative halfline along svec Sh in this dimension.

512

CHAPTER 6. CONE OF DISTANCE MATRICES

§

From the results in E.7.2.0.2, we know matrix product V H V is the orthogonal projection of H ∈ SN on the geometric center subspace SN c . Thus the projection product PK2 H = H − PSN+ PSNc H 6.8.1.1.1

(1209)

Lemma. Projection on PSD cone ∩ geometric center subspace. PSN+ ∩ SNc = PSN+ PSNc

(1210) ⋄

Proof. For each and every H ∈ SN , projection of PSNc H on the positive semidefinite cone remains in the geometric center subspace N SN c = {G ∈ S | G1 = 0} N

= {G ∈ S

(1921)

| N (G) ⊇ 1} = {G ∈ SN | R(G) ⊆ N (1T )}

= {V Y V | Y ∈ SN } ⊂ SN

(936)

(1922)

§

That is because: eigenvectors of PSNc H corresponding to 0 eigenvalues span its nullspace N (PSNc H ). ( A.7.3.0.1) To project PSNc H on the positive semidefinite cone, its negative eigenvalues are zeroed. ( 7.1.2) The nullspace is thereby expanded while eigenvectors originally spanning N (PSNc H ) remain intact. Because the geometric center subspace is invariant to projection on the PSD cone, then the rule for projection on a convex set in a subspace governs ( E.9.5, projectors do not commute) and statement (1210) follows directly. ¨

§

§

From the lemma it follows {PSN+ PSNc H | H ∈ SN } = {PSN+ ∩ SNc H | H ∈ SN }

(1211)

Then from (1947) ¡ ¢ N ∗ − SN = {H − PSN+ PSNc H | H ∈ SN } c ∩ S+

From (287) we get closure of a vector sum ¡ ¢ ⊥ N ∗ = SN − SN K2 = − SN c + c ∩ S+

(1212)

(1213)

therefore the equality

¡ N⊥ ¢ EDMN = K1 ∩ K2 = SN − SN h ∩ Sc +

(1214)

6.8. DUAL EDM CONE

513

whose veracity is intuitively evident, in hindsight, [84, p.109] from the most fundamental EDM definition (837). Formula (1214) is not a matrix criterion for membership to the EDM cone, it is not an EDM definition, and it is not an equivalence between EDM operators or an isomorphism. Rather, it is a recipe for constructing the EDM cone whole from large Euclidean bodies: the positive semidefinite cone, orthogonal complement of the geometric center subspace, and symmetric hollow subspace. A realization of this construction in low dimension is illustrated in Figure 140 and Figure 141. The dual EDM cone follows directly from (1214) by standard properties of cones ( 2.13.1.1):

§

∗

∗

∗

⊥ N EDMN = K1 + K2 = SN − SN h c ∩ S+

(1215)

which bears strong resemblance to (1194). 6.8.1.2

nonnegative orthant contains EDMN

That EDMN is a proper subset of the nonnegative orthant is not obvious from (1214). We wish to verify ¡ N⊥ ¢ N ×N EDMN = SN − SN (1216) h ∩ Sc + ⊂ R+

While there are many ways to prove this, it is sufficient to show that all entries of the extreme directions of EDMN must be nonnegative; id est, for any particular nonzero vector z = [zi , i = 1 . . . N ] ∈ N (1T ) ( 6.5.3.2),

§

δ(zz T )1T + 1δ(zz T )T − 2zz T ≥ 0

(1217)

where the inequality denotes entrywise comparison. The inequality holds because the i, j th entry of an extreme direction is squared: (z i − zj )2 . We observe that the dyad 2zz T ∈ SN + belongs to the positive semidefinite cone, the doublet ⊥ δ(zz T )1T + 1δ(zz T )T ∈ SN (1218) c to the orthogonal complement (1923) of the geometric center subspace, while their difference is a member of the symmetric hollow subspace SN h . Here is an algebraic method to prove nonnegativity: Suppose we are ⊥ N given A ∈ SN and B = [Bij ] ∈ SN Then we have, for c + and A − B ∈ Sh . T T some vector u , A = u1 + 1u = [Aij ] = [ui + uj ] and δ(B) = δ(A) = 2u . Positive semidefiniteness of B requires nonnegativity A − B ≥ 0 because

(ei −ej )TB(ei −ej ) = (Bii −Bij )−(Bji −Bjj ) = 2(ui +uj )−2Bij ≥ 0

(1219)

514 6.8.1.3

CHAPTER 6. CONE OF DISTANCE MATRICES Dual Euclidean distance matrix criterion

Conditions necessary for membership of a matrix D∗ ∈ SN to the dual ∗ ∗ EDM cone EDMN may be derived from (1194): D∗ ∈ EDMN ⇒ D∗ = δ(y) − VN AVNT for some vector y and positive semidefinite matrix −1 A ∈ SN . This in turn implies δ(D∗ 1) = δ(y) . Then, for D∗ ∈ SN + ∗

D∗ ∈ EDMN ⇔ δ(D∗ 1) − D∗ º 0

(1220)

δ(D∗ 1) − D∗ ∈ SN c

(1221)

where, for any symmetric matrix D∗

To show sufficiency of the matrix criterion in (1220), recall Gram-form EDM operator D(G) = δ(G)1T + 1δ(G)T − 2G

(849)

§

where Gram matrix G is positive semidefinite by definition, and recall the self-adjointness property of the main-diagonal linear operator δ ( A.1): ­ ® hD , D∗ i = δ(G)1T + 1δ(G)T − 2G , D∗ = hG , δ(D∗ 1) − D∗ i 2 (867)

Assuming hG , δ(D∗ 1) − D∗ i ≥ 0 (1426), then we have known membership relation ( 2.13.2.0.1)

§

D∗ ∈ EDMN

∗

⇔ hD , D∗ i ≥ 0 ∀ D ∈ EDMN

(1222) ¨

Elegance of this matrix criterion (1220) for membership to the dual EDM cone is the lack of any other assumptions except D∗ be symmetric. (Recall: Schoenberg criterion (856) for membership to the EDM cone requires membership to the symmetric hollow subspace.) Linear Gram-form EDM operator (849) has adjoint, for Y ∈ SN DT (Y ) , (δ(Y 1) − Y ) 2

(1223)

Then we have: [84, p.111] ∗

EDMN = {Y ∈ SN | δ(Y 1) − Y º 0}

(1224)

the dual EDM cone expressed in terms of the adjoint operator. A dual EDM cone determined this way is illustrated in Figure 143.

6.8. DUAL EDM CONE

515

§

6.8.1.3.1 Exercise. Dual EDM spectral cone. ∗ Find a spectral cone as in 5.11.2 corresponding to EDMN . 6.8.1.4

H

Nonorthogonal components of dual EDM

Now we tie construct (1215) for the dual EDM cone together with the matrix criterion (1220) for dual EDM cone membership. For any D∗ ∈ SN it is obvious: ⊥ δ(D∗ 1) ∈ SN (1225) h any diagonal matrix belongs to the subspace of diagonal matrices (62). We ∗ know when D∗ ∈ EDMN N δ(D∗ 1) − D∗ ∈ SN c ∩ S+

(1226)

this adjoint expression (1223) belongs to that face (1187) of the positive semidefinite cone SN Any nonzero + in the geometric center subspace. dual EDM ⊥ N N D∗ = δ(D∗ 1) − (δ(D∗ 1) − D∗ ) ∈ SN ⊖ SN h c ∩ S+ = EDM

∗

(1227)

can therefore be expressed as the difference of two linearly independent nonorthogonal components (Figure 121, Figure 142). 6.8.1.5

§

Affine dimension complementarity

−1 (confer (1226)) From 6.8.1.3 we have, for some A ∈ SN + N δ(D∗ 1) − D∗ = VN AVNT ∈ SN c ∩ S+

(1228)

if and only if D∗ belongs to the dual EDM cone. Call rank(VN AVNT ) dual affine dimension. Empirically, we find a complementary relationship in affine dimension between the projection of some arbitrary symmetric matrix H on ◦ ∗ the polar EDM cone, EDMN = −EDMN , and its projection on the EDM cone; id est, the optimal solution of 6.12 minimize D◦ ∈ SN

kD◦ − HkF

subject to D◦ − δ(D◦ 1) º 0 6.12

(1229)

This polar projection can be solved quickly (without semidefinite programming) via

516

CHAPTER 6. CONE OF DISTANCE MATRICES ⊥ N N D◦ = δ(D◦ 1) + (D◦ − δ(D◦ 1)) ∈ SN ⊕ SN h c ∩ S+ = EDM

◦

N SN c ∩ S+

EDMN

◦

D◦

δ(D◦ 1)

D◦ − δ(D◦ 1) 0

EDMN

◦

⊥ SN h

Figure 142: Hand-drawn abstraction of polar EDM cone (drawn truncated). Any member D◦ of polar EDM cone can be decomposed into two linearly independent nonorthogonal components: δ(D◦ 1) and D◦ − δ(D◦ 1).

6.8. DUAL EDM CONE

517

has dual affine dimension complementary to affine dimension corresponding to the optimal solution of minimize D∈SN h

kD − HkF

subject to −VNTDVN º 0

(1230)

Precisely, rank(D◦⋆ − δ(D◦⋆ 1)) + rank(VNTD⋆ VN ) = N − 1

(1231)

and rank(D◦⋆ − δ(D◦⋆ 1)) ≤ N − 1 because vector 1 is always in the nullspace of rank’s argument. This is similar to the known result for projection on the self-dual positive semidefinite cone and its polar: rank P−SN+ H + rank PSN+ H = N

(1232)

When low affine dimension is a desirable result of projection on the EDM cone, projection on the polar EDM cone should be performed instead. Convex polar problem (1229) can be solved for D◦⋆ by transforming to an equivalent Schur-form semidefinite program ( 3.6.2). Interior-point methods for numerically solving semidefinite programs tend to produce high-rank solutions. ( 4.1.1.1) Then D⋆ = H − D◦⋆ ∈ EDMN by Corollary E.9.2.2.1, and D⋆ will tend to have low affine dimension. This approach breaks when attempting projection on a cone subset discriminated by affine dimension or rank, because then we have no complementarity relation like (1231) or (1232) ( 7.1.4.1).

§

§

§

Lemma 6.8.1.1.1; rewriting, minimize D ◦ ∈ SN

subject to

k(D◦ − δ(D◦ 1)) − (H − δ(D◦ 1))kF

D◦ − δ(D◦ 1) º 0

N which is the projection of affinely transformed optimal solution H − δ(D◦⋆ 1) on SN c ∩ S+ ;

D◦⋆ − δ(D◦⋆ 1) = PSN PSN (H − δ(D◦⋆ 1)) c + Foreknowledge of an optimal solution D◦⋆ as argument to projection suggests recursion.

518

CHAPTER 6. CONE OF DISTANCE MATRICES

6.8.1.6

§

EDM cone is not self-dual

In 5.6.1.1, via Gram-form EDM operator D(G) = δ(G)1T + 1δ(G)T − 2G ∈ EDMN

⇐

Gº0

(849)

we established clear connection between the EDM cone and that face (1187) of positive semidefinite cone SN + in the geometric center subspace: N EDMN = D(SN c ∩ S+ )

N V(EDMN ) = SN c ∩ S+

(953) (954)

where V(D) = −V D V

§

1 2

(942)

In 5.6.1 we established N N −1 T SN VN c ∩ S+ = VN S+

(940)

Then from (1220), (1228), and (1194) we can deduce ∗

∗

−1 T N δ(EDMN 1) − EDMN = VN SN VN = SN c ∩ S+ +

(1233)

which, by (953) and (954), means the EDM cone can be related to the dual EDM cone by an equality: ³ ´ N∗ N∗ EDM = D δ(EDM 1) − EDM N

∗

V(EDMN ) = δ(EDMN 1) − EDMN

∗

(1234) (1235)

This means projection −V(EDMN ) of the EDM cone on the geometric center subspace SN c ( E.7.2.0.2) is a linear transformation of the dual EDM ∗ ∗ cone: EDMN − δ(EDMN 1). Secondarily, it means the EDM cone is not self-dual in SN .

§

6.8. DUAL EDM CONE 6.8.1.7

519

Schoenberg criterion is discretized membership relation

We show the Schoenberg criterion ) −1 −VNTDVN ∈ SN + D∈

SN h

⇔ D ∈ EDMN

(856)

§

to be a discretized membership relation ( 2.13.4) between a closed convex ∗ cone K and its dual K like ∗

hy , xi ≥ 0 for all y ∈ G(K ) ⇔ x ∈ K

(339)

∗

where G(K ) is any set of generators whose conic hull constructs closed ∗ convex dual cone K : The Schoenberg criterion is the same as ) hzz T , −Di ≥ 0 ∀ zz T | 11Tzz T = 0 ⇔ D ∈ EDMN (1181) N D ∈ Sh which, by (1182), is the same as © ª ) hzz T , −Di ≥ 0 ∀ zz T ∈ VN υυ T VNT | υ ∈ RN −1 D∈

SN h

⇔ D ∈ EDMN (1236)

§

where the zz T constitute ¡ Na set¢of generators G for the positive semidefinite cone’s smallest face F S+ ∋ V ( 6.7.1) that contains auxiliary matrix V . From the aggregate in (1194) we get the ordinary membership relation, assuming only D ∈ SN [185, p.58] ∗

hD∗ , Di ≥ 0 ∀ D∗ ∈ EDMN ⇔ D ∈ EDMN (1237) © ª hD∗ , Di ≥ 0 ∀ D∗ ∈ {δ(u) | u ∈ RN } − cone VN υυ T VNT | υ ∈ RN −1 ⇔ D ∈ EDMN Discretization (339) yields: N −1 T T T hD∗ , Di ≥ 0 ∀ D∗ ∈ {ei eT } ⇔ D ∈ EDMN i , −ej ej , −VN υυ VN | i , j = 1 . . . N , υ ∈ R (1238)

520

CHAPTER 6. CONE OF DISTANCE MATRICES

­ ® Because {δ(u) | u ∈ RN } , D ≥ 0 ⇔ D ∈ SN h , we can restrict observation to the symmetric hollow subspace without loss of generality. Then for D ∈ SN h © ª hD∗ , Di ≥ 0 ∀ D∗ ∈ −VN υυ T VNT | υ ∈ RN −1 ⇔ D ∈ EDMN

(1239)

this discretized membership relation becomes (1236); identical to the Schoenberg criterion. Hitherto a correspondence between the EDM cone and a face of a PSD cone, the Schoenberg criterion is now accurately interpreted as a discretized membership relation between the EDM cone and its ordinary dual.

Ambient SN h

6.8.2

When instead we consider the ambient space of symmetric hollow matrices (1195), then still we find the EDM cone is not self-dual for N > 2. The simplest way to prove this is as follows:

§

Given a set of generators G = {Γ} (1158) for the pointed closed convex EDM cone, the discretized membership theorem in 2.13.4.2.1 asserts that members of the dual EDM cone in the ambient space of symmetric hollow matrices can be discerned via discretized membership relation:

∗

N N ∗ ∗ EDMN ∩ SN h , {D ∈ Sh | hΓ , D i ≥ 0 ∀ Γ ∈ G(EDM )}

(1240)

T T T T T ∗ T = {D∗ ∈ SN h | hδ(zz )1 + 1δ(zz ) − 2zz , D i ≥ 0 ∀ z ∈ N (1 )}

T T T ∗ T = {D∗ ∈ SN h | h1δ(zz ) − zz , D i ≥ 0 ∀ z ∈ N (1 )}

By comparison T T EDMN = {D ∈ SN h | h−zz , Di ≥ 0 ∀ z ∈ N (1 )}

(1241)

the term δ(zz T )T D∗ 1 foils any hope of self-dualness in ambient SN h . ¨

6.9. THEOREM OF THE ALTERNATIVE

521

§

To find the dual EDM cone in ambient SN h per 2.13.9.4 we prune the aggregate in (1194) describing the ordinary dual EDM cone, removing any member having nonzero main diagonal: © 2 ª ∗ N −1 T T T T EDMN ∩ SN = cone δ (V υυ V ) − V υυ V | υ ∈ R h N N N N −1 = {δ 2(VN ΨVNT ) − VN ΨVNT | Ψ ∈ SN } +

(1242)

When N = 1, the EDM cone and its dual in ambient Sh each comprise the origin in isomorphic R0 ; thus, self-dual in this dimension. (confer (98)) When N = 2, the EDM cone is the nonnegative real line in isomorphic R . ∗ (Figure 138) EDM2 in S2h is identical, thus self-dual in this dimension. This ·result¸ is in agreement ·with¸ (1240), verified directly: for all κ ∈ R , 1 1 ∗ ⇒ d12 ≥ 0. and δ(zz T ) = κ2 z=κ 1 −1 The first case adverse to self-dualness N = 3 may be deduced from Figure 130; the EDM cone is a circular cone in isomorphic R3 corresponding to no rotation of Lorentz cone (163) (the self-dual circular cone). Figure 143 illustrates the EDM cone and its dual in ambient S3h ; no longer self-dual.

6.9

§

Theorem of the alternative

In 2.13.2.1.1 we showed how alternative systems of generalized inequality can be derived from closed convex cones and their duals. This section is, therefore, a fitting postscript to the discussion of the dual EDM cone. 6.9.0.0.1 Theorem. EDM alternative. Given D ∈ SN h D ∈ EDMN

or in the alternative ( 1Tz = 1 ∃ z such that Dz = 0

§

[149, 1]

(1243)

In words, either N (D) intersects hyperplane {z | 1Tz = 1} or D is an EDM; the alternatives are incompatible. ⋄

522

CHAPTER 6. CONE OF DISTANCE MATRICES

dvec rel ∂ EDM3

0

∗

dvec rel ∂(EDM3 ∩ S3h ) ∗

D∗ ∈ EDMN ⇔ δ(D∗ 1) − D∗ º 0

(1220)

Figure 143: Ordinary dual EDM cone projected on S3h shrouds EDM3 ; drawn tiled in isometrically isomorphic R3 . (It so happens: intersection ∗ N EDMN ∩ SN h ( 2.13.9.3) is identical to projection of dual EDM cone on Sh .)

§

6.10. POSTSCRIPT

523

§

When D is an EDM [241, 2]

§ §

N (D) ⊂ N (1T ) = {z | 1Tz = 0}

(1244)

Because [149, 2] ( E.0.1) DD† 1 = 1 1T D† D = 1T

(1245)

R(1) ⊂ R(D)

(1246)

then

6.10

Postscript

We provided an equality (1214) relating the convex cone of Euclidean distance matrices to the convex cone of positive semidefinite matrices. Projection on a positive semidefinite cone, constrained by an upper bound on rank, is easy and well known; [119] simply, a matter of truncating a list of eigenvalues. Projection on a positive semidefinite cone with such a rank constraint is, in fact, a convex optimization problem. ( 7.1.4) In the past, it was difficult to project on the EDM cone under a constraint on rank or affine dimension. A surrogate method was to invoke the Schoenberg criterion (856) and then project on a positive semidefinite cone under a rank constraint bounding affine dimension from above. But a solution acquired that way is necessarily suboptimal. In 7.3.3 we present a method for projecting directly on the EDM cone under a constraint on rank or affine dimension.

§

§

524

CHAPTER 6. CONE OF DISTANCE MATRICES

Chapter 7 Proximity problems In the “extremely large-scale case” (N of order of tens and hundreds of thousands), [iteration cost O(N 3 )] rules out all advanced convex optimization techniques, including all known polynomial time algorithms. −Arkadi Nemirovski (2004)

A problem common to various sciences is to find the Euclidean distance matrix (EDM) D ∈ EDMN closest in some sense to a given complete matrix of measurements H under a constraint on affine dimension 0 ≤ r ≤ N − 1 ( 2.3.1, 5.7.1.1); rather, r is bounded above by desired affine dimension ρ .

§

§

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

525

526

CHAPTER 7. PROXIMITY PROBLEMS

7.0.1

Measurement matrix H

Ideally, we want a given matrix of measurements H ∈ RN ×N to conform with the first three Euclidean metric properties ( 5.2); to belong to the ×N intersection of the orthant of nonnegative matrices RN with the symmetric + N hollow subspace Sh ( 2.2.3.0.1). Geometrically, we want H to belong to the polyhedral cone ( 2.12.1.0.1)

§

§

§

N ×N K , SN h ∩ R+

(1247)

Yet in practice, H can possess significant measurement uncertainty (noise). Sometimes realization of an optimization problem demands that its input, the given matrix H , possess some particular characteristics; perhaps symmetry and hollowness or nonnegativity. When that H given does not possess the desired properties, then we must impose them upon H prior to optimization:

ˆ When measurement matrix H is not symmetric or hollow, taking its symmetric hollow part is equivalent to orthogonal projection on the symmetric hollow subspace SN h .

ˆ When measurements of distance in H are negative, zeroing negative entries effects unique minimum-distance projection on the orthant of nonnegative matrices R in isomorphic R (§E.9.2.2.3). N ×N +

7.0.1.1

N2

Order of imposition

Since convex cone K (1247) is the intersection of an orthant with a subspace, we want to project on that subset of the orthant belonging to the subspace; on the nonnegative orthant in the symmetric hollow subspace that is, in fact, the intersection. For that reason alone, unique minimum-distance projection 2 of H on K (that member of K closest to H in isomorphic RN in the Euclidean sense) can be attained by first taking its symmetric hollow part, and only then clipping negative entries of the result to 0 ; id est, there is only one correct order of projection, in general, on an orthant intersecting a subspace:

527

ˆ project on the subspace, then project the result on the orthant in that subspace. (confer §E.9.5) In contrast, order of projection on an intersection of subspaces is arbitrary. That order-of-projection rule applies more generally, of course, to the intersection of any convex set C with any subspace. Consider the proximity problem 7.1 over convex feasible set SN h ∩ C given nonsymmetric N ×N nonhollow H ∈ R : minimize B∈ SN h

kB − Hk2F

(1248)

subject to B ∈ C

a convex optimization problem. Because the symmetric hollow subspace SN h ×N ⊥ 2.2.3), then is orthogonal to the antisymmetric antihollow subspace RN ( h for B ∈ SN h µ µ ¶¶ T 1 T 2 tr B (H − H ) + δ (H) = 0 (1249) 2

§

so the objective function is equivalent to kB −

Hk2F

° ° °2 µ ¶°2 °1 ° ° ° 1 T 2 T 2 ° + ° (H − H ) + δ (H)° B − ≡ ° (H + H ) − δ (H) °2 ° ° ° 2 F F (1250)

This means the antisymmetric antihollow part of given matrix H would be ignored by minimization with respect to symmetric hollow variable B under the Frobenius norm; id est, minimization proceeds as though given the symmetric hollow part of H . This action of the Frobenius norm (1250) is effectively a Euclidean projection (minimum-distance projection) of H on the symmetric hollow subspace SN h prior to minimization. Thus minimization proceeds inherently following the correct order for projection on SN Therefore we h ∩ C. may either assume H ∈ SN , or take its symmetric hollow part prior to h optimization. 7.1

There are two equivalent interpretations of projection (§E.9): one finds a set normal, the other, minimum distance between a point and a set. Here we realize the latter view.

528

CHAPTER 7. PROXIMITY PROBLEMS .............................................................................................................. ............................. ................... ................... ................ .............. ................ ............ .............. . . . . . . . . . . . ........... ........ . . . . .......... . . . . . .......... ...... . . . . . . . . . ......... ...... . . . ......... . . . . ........ ..... . . . . . . . ....... ..... . . ....... . . . . ....... .... . . . . . ...... .... . . ...... . . . ..... ..... . . . .... ... . .... . .. .... . . . .... ... . . ... .. . ... . ... ... . ... .. . ... .. . ... .. . ... .... ... ... .. N .. ... .. N .. ... .. .. .. .. h .. . ... .. . ... ... ... ... ... .. ... .. . . ... . ... ... ... ... .... .... .... N N ×N .... . .... . ... .... h + ... ..... ..... ...... ...... ...... N ....... ...... . . . . . ....... ....... ........ ........ ........ ....... ......... ......... ......... ......... . . .......... . . . . . . .. .......... .......... ............ ........... ............ ............ ............... .............. ................. ................ . . . . ..................... . . . . . . . . . . . . . . ................................ ....... N ×N ................................................................................................

"

"

"b

b

b b © b © " b © © " b © " b © " b © " b © " ©© b " h b hhhhh " 0© b c hhhhh " b hhhhh EDM " S b c h " b h h c " b c " b c " b " b c " b c " b c K=S ∩ R " b c " b " b c S " b c " b c " b " b b" "

"

R

Figure 144: Pseudo-Venn diagram: The EDM cone belongs to the intersection of the symmetric hollow subspace with the nonnegative orthant; N ×N EDMN ⊆ K (836). EDMN cannot exist outside SN does. h , but R+

7.0.1.2

Egregious input error under nonnegativity demand

More pertinent to the optimization problems presented herein where

N ×N C , EDMN ⊆ K = SN h ∩ R+

(1251)

then should some particular realization of a proximity problem demand input H be nonnegative, and were we only to zero negative entries of a nonsymmetric nonhollow input H prior to optimization, then the ensuing projection on EDMN would be guaranteed incorrect (out of order). Now comes a surprising fact: Even were we to correctly follow the order-of-projection rule and provide H ∈ K prior to optimization, then the ensuing projection on EDMN will be incorrect whenever input H has negative entries and some proximity problem demands nonnegative input H .

529

0 H

SN h

EDMN N ×N K = SN h ∩ R+

Figure 145: Pseudo-Venn diagram from Figure 144 showing elbow placed in path of projection of H on EDMN ⊂ SN h by an optimization problem demanding nonnegative input matrix H . The first two line segments leading away from H result from correct order-of-projection required to provide nonnegative H prior to optimization. Were H nonnegative, then its projection on SN h would instead belong to K ; making the elbow disappear. (confer Figure 157)

530

CHAPTER 7. PROXIMITY PROBLEMS

This is best understood referring to Figure 144: Suppose nonnegative input H is demanded, and then the problem realization correctly projects N its input first on SN That demand h and then directly on C = EDM . for nonnegativity effectively requires imposition of K on input H prior to optimization so as to obtain correct order of projection (on SN h first). Yet such an imposition prior to projection on EDMN generally introduces an elbow into the path of projection (illustrated in Figure 145) caused by the technique itself; that being, a particular proximity problem realization requiring nonnegative input. Any procedure for imposition of nonnegativity on input H can only be incorrect in this circumstance. There is no resolution unless input H is guaranteed nonnegative with no tinkering. Otherwise, we have no choice but to employ a different problem realization; one not demanding nonnegative input.

7.0.2

Lower bound

Most of the problems we encounter in this chapter have the general form: minimize B

kB − AkF

subject to B ∈ C

(1252)

where A ∈ Rm×n is given data. This particular objective denotes Euclidean projection ( E) of vectorized matrix A on the set C which may or may not be convex. When C is convex, then the projection is unique minimum-distance because the Frobenius norm when squared is a strictly convex function of variable B and because the optimal solution is the same regardless of the square (483). When C is a subspace, then the direction of projection is orthogonal to C . Denoting by A , UA ΣA QAT and B , UB ΣB QBT their full singular value decompositions (whose singular values are always nonincreasingly ordered ( A.6)), there exists a tight lower bound on the objective over the manifold of orthogonal matrices;

§

§

kΣB − ΣA kF ≤

§

inf

UA , UB , QA , QB

kB − AkF

(1253)

§

This least lower bound holds more generally for any orthogonally invariant norm on Rm×n ( 2.2.1) including the Frobenius and spectral norm [306, II.3]. [188, 7.4.51]

§

531

7.0.3

Problem approach

Problems traditionally posed in terms of point position {xi ∈ Rn , i = 1 . . . N } such as X minimize (kxi − xj k − hij )2 (1254) {xi }

or

minimize {xi }

i, j ∈ I

X

(kxi − xj k2 − hij )2

(1255)

i, j ∈ I

(where I is an abstract set of indices and hij is given data) are everywhere converted herein to the distance-square variable D or to Gram matrix G ; the Gram matrix acting as bridge between position and distance. (That conversion is performed regardless of whether known data is complete.) Then the techniques of chapter 5 or chapter 6 are applied to find relative or absolute position. This approach is taken because we prefer introduction of rank constraints into convex problems rather than searching a googol of local minima in nonconvex problems like (1254) [96] ( 3.7.4.0.2, 7.2.2.7.1) or (1255).

§

7.0.4

§

Three prevalent proximity problems

There are three statements of the closest-EDM problem prevalent in the literature, the multiplicity due primarily to choice of projection on the EDM versus positive semidefinite (PSD) cone and vacillation between the p distance-square variable dij versus absolute distance dij . In their most fundamental form, the three prevalent proximity problems are (1256.1), √ p ◦ (1256.2), and (1256.3): [319] for D , [dij ] and D , [ dij ] minimize

(1)

D

subject to rank V D V ≤ ρ D ∈ EDMN minimize

(3)

k−V (D − H)V k2F

D

kD − Hk2F

subject to rank V D V ≤ ρ D ∈ EDMN

minimize √ ◦ D

√ k ◦ D − Hk2F

subject to rank V D V ≤ ρ p √ ◦ D ∈ EDMN minimize √ ◦ D

√ ◦

(2) (1256)

k−V ( D − H)V k2F

subject to rank V D V ≤ ρ p √ ◦ D ∈ EDMN

(4)

532

CHAPTER 7. PROXIMITY PROBLEMS

where we have made explicit an imposed upper bound ρ on affine dimension r = rank VNTDVN = rank V D V

(988)

that is benign when ρ = N − 1 or H were realizable with r ≤ ρ . Problems (1256.2) and (1256.3) are Euclidean projections of a vectorized matrix H on an EDM cone ( 6.3), whereas problems (1256.1) and (1256.4) are Euclidean projections of a vectorized matrix −VHV on a PSD cone.7.2 Problem (1256.4) is not posed in the literature because it has limited theoretical foundation.7.3 Analytical solution to (1256.1) is known in closed form for any bound ρ although, as the problem is stated, it is a convex optimization only in the case ρ = N − 1. We show in 7.1.4 how (1256.1) becomes a convex optimization problem for any ρ when transformed to the spectral domain. When expressed as a function of point list in a matrix X as in (1254), problem (1256.2) becomes a variant of what is known in statistics literature as a stress problem. [49, p.34] [94] [329] Problems (1256.2) and (1256.3) are convex optimization problems in D for the case ρ = N − 1 wherein (1256.3) becomes equivalent to (1255). Even with the rank constraint removed from (1256.2), we will see that the convex problem remaining inherently minimizes affine dimension. Generally speaking, each problem in (1256) produces a different result because there is no isometry relating them. Of the various auxiliary V -matrices ( B.4), the geometric centering matrix V (859) appears in the literature most often although VN (843) is the auxiliary matrix naturally consequent to Schoenberg’s seminal exposition [290]. Substitution of any auxiliary matrix or its pseudoinverse into these problems produces another valid problem. Substitution of VNT for left-hand V in (1256.1), in particular, produces a different result because

§

§

§

minimize D

k−V (D − H)V k2F

subject to D ∈ EDMN

(1257)

Because −VHV is orthogonal projection of −H on the geometric center subspace SN c (§E.7.2.0.2), problems (1256.1) and (1256.4) may be interpreted as oblique (nonminimum distance) projections√of −H on a positive √ semidefinite cone. 7.3 D ∈ EDMN ⇒ ◦ D ∈ EDMN , −V ◦ D V ∈ SN + (§5.10) 7.2

7.1. FIRST PREVALENT PROBLEM:

533

finds D to attain Euclidean distance of vectorized −V H V to the positive semidefinite cone in ambient isometrically isomorphic RN (N +1)/2 , whereas minimize D

k−VNT (D − H)VN k2F

(1258)

subject to D ∈ EDMN

attains Euclidean distance of vectorized −VNTH VN to the positive semidefinite cone in isometrically isomorphic subspace RN (N −1)/2 ; quite different projections7.4 regardless of whether affine dimension is constrained. But substitution of auxiliary matrix VWT ( B.4.3) or VN† yields the same result as (1256.1) because V = VW VWT = VN VN† ; id est,

§

k−V (D − H)V k2F = k−VW VWT (D − H)VW VWT k2F = k−VWT (D − H)VW k2F = k−VN VN† (D − H)VN VN† k2F = k−VN† (D − H)VN k2F

(1259)

§

§

We see no compelling reason to prefer one particular auxiliary V -matrix over another. Each has its own coherent interpretations; e.g., 5.4.2, 6.7. Neither can we say any particular problem formulation produces generally better results than another.7.5

7.1

First prevalent problem: Projection on PSD cone

This first problem  k−VNT (D − H)VN k2F   D subject to rank VNTDVN ≤ ρ   D ∈ EDMN minimize

Problem 1

(1260)

N The isomorphism T (Y ) = VN†T Y VN† onto SN c = {V X V | X ∈ S } relates the map in (1258) to that in (1257), but is not an isometry. 7.5 All four problem formulations (1256) produce identical results when affine dimension r , implicit to a realizable measurement matrix H , does not exceed desired affine dimension ρ ; because, the optimal objective value will vanish (k · k = 0). 7.4

534

CHAPTER 7. PROXIMITY PROBLEMS

poses a Euclidean projection of −VNTH VN in subspace SN −1 on a generally nonconvex subset (when ρ < N − 1) of the positive semidefinite cone −1 whose elemental matrices have rank no greater than boundary ∂ SN + desired affine dimension ρ ( 5.7.1.1). Problem 1 finds the closest EDM D in the sense of Schoenberg. (856) [290] As it is stated, this optimization problem is convex only when desired affine dimension is largest ρ = N − 1 although its analytical solution is known [239, thm.14.4.2] for all nonnegative ρ ≤ N − 1 .7.6 We assume only that the given measurement matrix H is symmetric;7.7

§

H ∈ SN

(1261)

Arranging the eigenvalues λ i of −VNTH VN in nonincreasing order for all i , λ i ≥ λ i+1 with vi the corresponding i th eigenvector, then an optimal solution to Problem 1 is [328, 2]

§

−VNTD⋆ VN

=

ρ X i=1

max{0, λ i } vi viT

(1262)

where −VNTH VN

,

N −1 X i=1

λ i vi viT ∈ SN −1

(1263)

is an eigenvalue decomposition and D⋆ ∈ EDMN

(1264)

is an optimal Euclidean distance matrix. In 7.1.4 we show how to transform Problem 1 to a convex optimization problem for any ρ .

§

7.6

being first pronounced in the context of multidimensional scaling by Mardia [238] in 1978 who attributes the generic result (§7.1.2) to Eckart & Young, 1936 [119]. −1 7.7 Projection in Problem 1 is on a rank ρ subset of the positive semidefinite cone SN + N −1 (§2.9.2.1) in the subspace of symmetric matrices S . It is wrong here to zero the main diagonal of given H because first projecting H on the symmetric hollow subspace places an elbow in the path of projection in Problem 1. (Figure 145)

7.1. FIRST PREVALENT PROBLEM:

7.1.1

535

Closest-EDM Problem 1, convex case

7.1.1.0.1 Proof. Solution (1262), convex case. When desired affine dimension is unconstrained, ρ = N − 1, the rank function disappears from (1260) leaving a convex optimization problem; a simple −1 unique minimum-distance projection on the positive semidefinite cone SN : + videlicet minimize k−VNT (D − H)VN k2F D∈ SN (1265) h subject to −VNTDVN º 0 by (856). Because SN −1 = −VNT SN h VN

(957)

then the necessary and sufficient conditions for projection in isometrically −1 isomorphic RN (N −1)/2 on the self-dual (349) positive semidefinite cone SN + are:7.8 ( E.9.2.0.1) (1522) (confer (1954))

§

−VNTD⋆ VN −VNTD⋆ VN −VNTD⋆ VN

¢ ¡º 0 T ⋆ −VN D VN + VNTH VN = 0 + VNTH VN º 0

(1266)

Symmetric −VNTH VN is diagonalizable hence decomposable in terms of its eigenvectors v and eigenvalues λ as in (1263). Therefore (confer (1262)) −VNTD⋆ VN

=

N −1 X i=1

max{0, λ i }vi viT

satisfies (1266), optimally solving (1265). To see that, recall: eigenvectors constitute an orthogonal set and −VNTD⋆ VN

7.8

+

VNTH VN

= −

N −1 X i=1

min{0, λ i }vi viT

(1267) these

(1268) ¨

These conditions for projection on a convex cone are identical to the Karush-Kuhn-Tucker (KKT) optimality conditions [259, p.328] for problem (1265).

536

7.1.2

CHAPTER 7. PROXIMITY PROBLEMS

Generic problem

Prior to determination of D⋆ , analytical solution (1262) to Problem 1 is equivalent to solution of a generic rank-constrained projection problem: Given desired affine dimension ρ and A ,

−VNTH VN

=

N −1 X i=1

λ i vi viT ∈ SN −1

(1263)

Euclidean projection on a rank ρ subset of a PSD cone (on a generally −1 nonconvex subset of the PSD cone boundary ∂ SN when ρ < N − 1) +  kB − Ak2F   B∈ SN −1 subject to rank B ≤ ρ   Bº0 minimize

Generic 1

(1269)

has well known optimal solution (Eckart & Young) [119] ⋆

B ,

−VNTD⋆ VN

=

ρ X i=1

max{0, λ i } vi viT ∈ SN −1

(1262)

§

Once optimal B ⋆ is found, the technique of 5.12 can be used to determine a uniquely corresponding optimal Euclidean distance matrix D⋆ ; a unique correspondence by injectivity arguments in 5.6.2.

§

7.1.2.1

Projection on rank ρ subset of PSD cone

Because Problem 1 is the same as minimize D∈ SN h

k−VNT (D − H)VN k2F

subject to rank VNTDVN ≤ ρ −VNTDVN º 0

(1270)

and because (957) provides invertible mapping to the generic problem, then Problem 1 is truly a Euclidean projection of vectorized −VNTH VN on that generally nonconvex subset of symmetric matrices (belonging to the −1 positive semidefinite cone SN ) having rank no greater than desired affine +

7.1. FIRST PREVALENT PROBLEM:

537

dimension ρ ; 7.9 called rank ρ subset: (201) (234) −1 −1 N −1 SN \ SN | rank X ≤ ρ} + + (ρ + 1) = {X ∈ S+

7.1.3

(239)

Choice of spectral cone

Spectral projection substitutes projection on a polyhedral cone, containing a complete set of eigenspectra, in place of projection on a convex set of diagonalizable matrices; e.g., (1282). In this section we develop a method of spectral projection for constraining rank of positive semidefinite matrices in a proximity problem like (1269). We will see why an orthant turns out to be the best choice of spectral cone, and why presorting is critical. Define a nonlinear permutation-operator π(x) : Rn → Rn that sorts its vector argument x into nonincreasing order. 7.1.3.0.1 Definition. Spectral projection. Let R be an orthogonal matrix and Λ a nonincreasingly ordered diagonal matrix of eigenvalues. Spectral projection means unique minimum-distance projection of a rotated (R , B.5.4) nonincreasingly ordered (π) vector (δ) of eigenvalues ¡ ¢ π δ(RTΛR) (1271)

§

§

on a polyhedral cone containing all eigenspectra corresponding to a rank ρ subset of a positive semidefinite cone ( 2.9.2.1) or the EDM cone (in Cayley-Menger form, 5.11.2.3). △

§

§

In the simplest and most common case, projection on a positive semidefinite cone, orthogonal matrix R equals I ( 7.1.4.0.1) and diagonal matrix Λ is ordered during diagonalization ( A.5.1). Then spectral projection simply means projection of δ(Λ) on a subset of the nonnegative orthant, as we shall now ascertain: It is curious how nonconvex Problem 1 has such a simple analytical solution (1262). Although solution to generic problem (1269) is well known since 1936 [119], its equivalence was observed in 1997 [328, 2] to projection

§

§

7.9

Recall: affine dimension is a lower bound on embedding (§2.3.1), equal to dimension of the smallest affine set in which points from a list X corresponding to an EDM D can be embedded.

538

CHAPTER 7. PROXIMITY PROBLEMS

§

of an ordered vector of eigenvalues (in diagonal matrix Λ) on a subset of the monotone nonnegative cone ( 2.13.9.4.2) −1 KM+ = {υ | υ1 ≥ υ2 ≥ · · · ≥ υN −1 ≥ 0} ⊆ RN +

(401)

Of interest, momentarily, is only the smallest convex subset of the monotone nonnegative cone KM+ containing every nonincreasingly ordered eigenspectrum corresponding to a rank ρ subset of the positive semidefinite −1 cone SN ; id est, + ρ

KM+ , {υ ∈ Rρ | υ1 ≥ υ2 ≥ · · · ≥ υρ ≥ 0} ⊆ Rρ+

(1272)

a pointed polyhedral cone, a ρ-dimensional convex subset of the −1 monotone nonnegative cone KM+ ⊆ RN having property, for λ denoting + eigenspectra, · ρ ¸ KM+ N −1 = π(λ(rank ρ subset)) ⊆ KM+ , KM+ (1273) 0 For each and every elemental eigenspectrum −1 γ ∈ λ(rank ρ subset) ⊆ RN +

(1274)

of the rank ρ subset (ordered or unordered in λ), there is a nonlinear ρ surjection π(γ) onto KM+ . 7.1.3.0.2 Exercise. Smallest spectral cone. ρ Prove that there is no convex subset of KM+ smaller than KM+ containing every ordered eigenspectrum corresponding to the rank ρ subset of a positive semidefinite cone ( 2.9.2.1). H

§

§

§

7.1.3.0.3 Proposition. (Hardy-Littlewood-P´olya) Inequalities. [165, X] [51, 1.2] Any vectors σ and γ in RN −1 satisfy a tight inequality π(σ)T π(γ) ≥ σ T γ ≥ π(σ)T Ξ π(γ)

(1275)

where Ξ is the order-reversing permutation matrix defined in (1654), and permutator π(γ) is a nonlinear function that sorts vector γ into nonincreasing order thereby providing the greatest upper bound and least lower bound with respect to every possible sorting. ⋄

7.1. FIRST PREVALENT PROBLEM:

539

7.1.3.0.4 Corollary. Monotone nonnegative sort. Any given vectors σ , γ ∈ RN −1 satisfy a tight Euclidean distance inequality kπ(σ) − π(γ)k ≤ kσ − γk

(1276)

where nonlinear function π(γ) sorts vector γ into nonincreasing order thereby providing the least lower bound with respect to every possible sorting. ⋄ Given γ ∈ RN −1 inf kσ−γk = inf kπ(σ)−π(γ)k = inf kσ−π(γ)k = inf kσ−π(γ)k

−1 σ∈RN +

−1 σ∈RN +

−1 σ∈RN +

σ∈KM+

(1277) Yet for γ representing an arbitrary vector of eigenvalues, because 2 2 inf · ρ ¸ kσ − γk ≥ inf · ρ ¸ kσ − π(γ)k =

R σ∈ + 0

·inf ¸ kσ ρ KM+ σ∈ 0

R σ∈ + 0

− π(γ)k2

(1278)

then projection of γ on the eigenspectra of a rank ρ subset can be tightened simply by presorting γ into nonincreasing order. Proof. Simply because π(γ)1:ρ º π(γ1:ρ ) 2 T 2 inf · ρ ¸ kσ − γk = γρ+1:N −1 γρ+1:N −1 + inf kσ1:ρ − γ1:ρ k

σ∈

−1 σ∈RN +

R+ 0

T T = γ T γ + inf σ1:ρ σ1:ρ − 2σ1:ρ γ1:ρ −1 σ∈RN +

T T ≥ γ T γ + inf σ1:ρ σ1:ρ − 2σ1:ρ π(γ)1:ρ

(1279)

−1 σ∈RN +

2 2 inf · ρ ¸ kσ − π(γ)k · ρ ¸ kσ − γk ≥ inf

σ∈

R+ 0

σ∈

R+ 0

¨

540

CHAPTER 7. PROXIMITY PROBLEMS

7.1.3.1

Orthant is best spectral cone for Problem 1

This means unique minimum-distance projection of γ on the nearest spectral member of the rank ρ subset is tantamount to presorting γ into nonincreasing order. Only then does unique spectral projection on a subset ρ KM+ of the monotone nonnegative cone become equivalent to unique spectral projection on a subset Rρ+ of the nonnegative orthant (which is simpler); in other words, unique minimum-distance projection of sorted γ on the nonnegative orthant in a ρ-dimensional subspace of RN is indistinguishable ρ from its projection on the subset KM+ of the monotone nonnegative cone in that same subspace.

7.1.4

Closest-EDM Problem 1, “nonconvex” case

§

Proof of solution (1262), for projection on a rank ρ subset of the positive −1 semidefinite cone SN , can be algebraic in nature. [328, 2] Here we derive + that known result but instead using a more geometric argument via spectral projection on a polyhedral cone (subsuming the proof in 7.1.1). In so doing, we demonstrate how nonconvex Problem 1 is transformed to a convex optimization:

§

§

7.1.4.0.1 Proof. Solution (1262), nonconvex case. As explained in 7.1.2, we may instead work with the more facile generic problem (1269). With diagonalization of unknown B , U ΥU T ∈ SN −1

(1280)

given desired affine dimension 0 ≤ ρ ≤ N − 1 and diagonalizable A , QΛQT ∈ SN −1

(1281)

having eigenvalues in Λ arranged in nonincreasing order, by (43) the generic problem is equivalent to minimize B∈ SN −1

kB − Ak2F

subject to rank B ≤ ρ Bº0

minimize R ,Υ

≡

kΥ − RTΛRk2F

subject to rank Υ ≤ ρ Υº0 R −1 = RT

(1282)

7.1. FIRST PREVALENT PROBLEM:

541

where R , QT U ∈ RN −1×N −1

(1283)

in U on the set of orthogonal matrices is a linear bijection. We propose solving (1282) by instead solving the problem sequence: minimize Υ

kΥ − RTΛRk2F

subject to rank Υ ≤ ρ Υº0 minimize R

kΥ⋆ − RTΛRk2F

subject to R −1 = RT

(a) (1284) (b)

Problem (1284a) is equivalent to: (1) orthogonal projection of RTΛR on an N − 1-dimensional subspace of isometrically isomorphic RN (N −1)/2 −1 containing δ(Υ) ∈ RN , (2) nonincreasingly ordering the ·result,¸ (3) unique + Rρ+ minimum-distance projection of the ordered result on . ( E.9.5) 0 Projection on that N− 1-dimensional subspace amounts to zeroing RTΛR at all entries off the main diagonal; thus, the equivalent sequence leading with a spectral projection:

§

¡ ¢ k δ(Υ) − π δ(RTΛR) k2 Υ · ρ ¸ R+ subject to δ(Υ) ∈ 0 minimize

minimize R

kΥ⋆ − RTΛRk2F

subject to R −1 = RT

(a) (1285) (b)

Because any permutation matrix is an orthogonal matrix, it is always feasible that δ(RTΛR) ∈ RN −1 be arranged in nonincreasing order; hence, the permutation operator π . Unique minimum-distance projection of vector · ρ ¸ ¡ ¢ R+ π δ(RTΛR) on the ρ-dimensional subset of nonnegative orthant 0

542

CHAPTER 7. PROXIMITY PROBLEMS

§

−1 RN requires: ( E.9.2.0.1) +

δ(Υ⋆ )ρ+1:N −1 = 0 δ(Υ⋆ ) º 0 ¡ ¢ δ(Υ⋆ )T δ(Υ⋆ ) − π(δ(RTΛR)) = 0

(1286)

δ(Υ⋆ ) − π(δ(RTΛR)) º 0

which are necessary and sufficient conditions. Any value Υ⋆ satisfying conditions (1286) is optimal for (1285a). So n ( ¡ ¢o max 0 , π δ(RTΛR) i , i = 1 . . . ρ ⋆ (1287) δ(Υ )i = 0, i = ρ+1 . . . N − 1 specifies an optimal solution. The lower bound on the objective with respect to R in (1285b) is tight; by (1253) k |Υ⋆ | − |Λ| kF ≤ kΥ⋆ − RTΛRkF

(1288)

§

where | | denotes absolute entry-value. For selection of Υ⋆ as in (1287), this lower bound is attained when (confer C.4.2.2) R⋆ = I which is the known solution. 7.1.4.1

(1289) ¨

Significance

Importance of this well-known [119] optimal solution (1262) for projection on a rank ρ subset of a positive semidefinite cone should not be dismissed:

ˆ Problem 1, as stated, is generally nonconvex. This analytical solution

at once encompasses projection on a rank ρ subset (239) of the positive semidefinite cone (generally, a nonconvex subset of its boundary) from either the exterior or interior of that cone.7.10 By problem transformation to the spectral domain, projection on a rank ρ subset becomes a convex optimization problem.

7.10

Projection on the boundary from the interior of a convex Euclidean body is generally a nonconvex problem.

7.1. FIRST PREVALENT PROBLEM:

543

ˆ This solution is closed form. ˆ This solution is equivalent to projection on a polyhedral cone in the spectral domain (spectral projection, projection on a spectral cone §7.1.3.0.1); a necessary and sufficient condition (§A.3.1) for membership of a symmetric matrix to a rank ρ subset of a positive semidefinite cone (§2.9.2.1). ˆ Because U = Q , a minimum-distance projection on a rank ρ subset ⋆

of the positive semidefinite cone is a positive semidefinite matrix orthogonal (in the Euclidean sense) to direction of projection.7.11

ˆ For the convex case problem, this solution is always unique. Otherwise, distinct eigenvalues (multiplicity 1) in Λ guarantees uniqueness of this solution by the reasoning in §A.5.0.1 . 7.12

7.1.5

Problem 1 in spectral norm, convex case

When instead we pose the matrix 2-norm (spectral norm) in Problem 1 (1260) for the convex case ρ = N − 1, then the new problem minimize D

k−VNT (D − H)VN k2

(1290)

subject to D ∈ EDMN

§

is convex although its solution is not necessarily unique;7.13 giving rise to −1 nonorthogonal projection ( E.1) on the positive semidefinite cone SN . + Indeed, its solution set includes the Frobenius solution (1262) for the convex case whenever −VNTH VN is a normal matrix. [170, 1] [163] [57, 8.1.1] Proximity problem (1290) is equivalent to minimize µ,D

§

µ

subject to −µI ¹ −VNT (D − H)VN ¹ µI D ∈ EDMN 7.11

§

(1291)

But Theorem E.9.2.0.1 for unique projection on a closed convex cone does not apply here because the direction of projection is not necessarily a member of the dual PSD cone. This occurs, for example, whenever positive eigenvalues are truncated. 7.12 Uncertainty of uniqueness prevents the erroneous conclusion that a rank ρ subset (201) were a convex body by the Bunt-Motzkin (§E.9.0.0.1). · theorem ¸ 2 0 7.13 For each and every |t| ≤ 2, for example, has the same spectral-norm value. 0 t

544

CHAPTER 7. PROXIMITY PROBLEMS

by (1631) where ª ¢¯ ©¯ ¡ µ⋆ = max ¯λ −VNT (D⋆ − H)VN i ¯ , i = 1 . . . N − 1 ∈ R+ i

(1292)

the minimized largest absolute eigenvalue (due to matrix symmetry). For lack of a unique solution here, we prefer the Frobenius rather than spectral norm.

7.2 Let

Second prevalent problem: p Projection on EDM cone in dij √ ◦

p N ×N D , [ dij ] ∈ K = SN h ∩ R+

(1293)

be an unknown matrix of absolute distance; id est, √ √ ◦ ◦ D = [dij ] , D ◦ D ∈ EDMN

(1294)

where ◦ denotes Hadamard product. The second prevalent proximity p problem is a Euclidean projection (in the natural coordinates dij ) of matrix H on a nonconvex subset of the boundary of the nonconvex cone of Euclidean p N absolute-distance matrices rel ∂ EDM : ( 6.3, confer Figure 130(b))  √ ◦ 2  minimize D − Hk k √ F  ◦  D T Problem 2 (1295) subject to rank VN DVN ≤ ρ  p √   ◦ D ∈ EDMN

§

where

p √ ◦ EDMN = { D | D ∈ EDMN }

(1128)

§

This statement of the second proximity problem is considered difficult to solve because of the constraint on desired affine dimension ρ ( 5.7.2) and because the objective function Xp √ ◦ ( dij − hij )2 (1296) k D − Hk2F = i,j

is expressed in the natural coordinates; projection on a doubly nonconvex set.

7.2. SECOND PREVALENT PROBLEM:

545

Our solution to this second problem prevalent in the literature requires measurement matrix H to be nonnegative; ×N H = [hij ] ∈ RN +

(1297)

§

If the H matrix given has negative entries, then the technique of solution presented here becomes invalid. As explained in 7.0.1, projection of H N ×N on K = SN (1247) prior to application of this proposed solution is h ∩ R+ incorrect.

7.2.1

Convex case

When ρ = N − 1, the rank constraint vanishes and a convex problem that is equivalent to (1254) emerges:7.14 √ k ◦ D − Hk2F D p √ subject to ◦ D ∈ EDMN minimize √ ◦

X

minimize ⇔

D

i,j

dij − 2hij

subject to D ∈ EDMN

p dij + h2ij

(1298)

For any fixed i and j , the argument of summation is a convex function of dij because (for nonnegative constant hij ) the negative square root is convex in nonnegative dij and because dij + h2ij is affine (convex). Because the sum of any number of convex functions in D remains convex [57, 3.2.1] and because the feasible set is convex in D , we have a convex optimization problem: √ minimize 1T(D − 2H ◦ ◦ D )1 + kHk2F D (1299) subject to D ∈ EDMN

§

The objective function being a sum of strictly convex functions is, moreover, strictly convex in D on the nonnegative orthant. Existence of a unique solution D⋆ for this second prevalent problem depends upon nonnegativity of H and a convex feasible set ( 3.1.2).7.15

§

7.14 still thought to be a nonconvex problem as late as 1997 [329] even though discovered convex by de √ Leeuw in 1993. [94] [49, §13.6] Yet using methods from §3, it can be easily ascertained: k ◦ D − HkF is not convex in D . 7.15 The transformed problem in variable D no longer describes Euclidean projection on p an EDM cone. Otherwise we might erroneously conclude EDMN were a convex body by the Bunt-Motzkin theorem (§E.9.0.0.1).

546 7.2.1.1

CHAPTER 7. PROXIMITY PROBLEMS Equivalent semidefinite program, Problem 2, convex case

§

Convex problem (1298) is numerically solvable for its global minimum using an interior-point method [57, 11] [366] [269] [257] [357] [11] [133]. We translate (1298) to an equivalent semidefinite program (SDP) for a pedagogical reason made clear in 7.2.2.2 and because there exist readily available computer programs for numerical solution [153] [359] [360] [334] [32] [358]. ×N Substituting a new matrix variable Y , [yij ] ∈ RN + p hij dij ← yij (1300)

§

Boyd proposes: problem (1298) is equivalent to the semidefinite program X minimize dij − 2yij + h2ij D,Y

subject to

i,j

"

dij yij yij h2ij

#

º0,

i,j = 1 . . . N

(1301)

D ∈ EDMN

§

To see that, recall dij ≥ 0 is implicit to D ∈ EDMN ( 5.8.1, (856)). So ×N when H ∈ RN is nonnegative as assumed, + # " q q dij yij º 0 ⇔ hij dij ≥ yij2 (1302) 2 yij hij Minimization of the objective function implies maximization of yij that is bounded above. p Hence nonnegativity of yij is implicit to (1301) and, as desired, yij → hij dij as optimization proceeds. ¨ If the given matrix H is now assumed symmetric and nonnegative, ×N H = [hij ] ∈ SN ∩ RN +

(1303)

√ N ×N then Y = H ◦ ◦ D must belong to K = SN (1247). Because Y ∈ SN h ∩ R+ h ( B.4.2 no.20), then X √ ◦ dij − 2yij + h2ij = −N tr(V (D − 2 Y )V ) + kHk2F (1304) k D − Hk2F =

§

i,j

7.2. SECOND PREVALENT PROBLEM:

547

So convex problem (1301) is equivalent to the semidefinite program minimize − tr(V (D − 2 Y )V ) D,Y " # dij yij subject to º 0 , N ≥ j > i = 1... N−1 yij h2ij

(1305)

Y ∈ SN h

D ∈ EDMN where the constants h2ij and N have been dropped arbitrarily from the objective. 7.2.1.2

Gram-form semidefinite program, Problem 2, convex case

There is great advantage to expressing problem statement (1305) in Gram-form because Gram matrix G is a bidirectional bridge between point list X and distance matrix D ; e.g., Example 5.4.2.3.5, Example 6.4.0.0.1. This way, problem convexity can be maintained while simultaneously constraining point list X , Gram matrix G , and distance matrix D at our discretion. Convex problem (1305) may be equivalently written via linear bijective ( 5.6.1) EDM operator D(G) (849);

§

minimize − tr(V (D(G) − 2 Y )V ) # " hΦij , Gi yij º0, subject to yij h2ij

N G∈SN c , Y ∈ Sh

N ≥ j > i = 1... N−1

(1306)

Gº0 where distance-square D = [dij ] ∈ SN h (833) is related to Gram matrix entries N N G = [gij ] ∈ Sc ∩ S+ by dij = gii + gjj − 2gij = hΦij , Gi

(848)

where Φij = (ei − ej )(ei − ej )T ∈ SN +

(835)

548

CHAPTER 7. PROXIMITY PROBLEMS

Confinement of G to the geometric center subspace provides numerical stability and no loss of generality (confer (1133)); implicit constraint G1 = 0 is otherwise unnecessary. To include constraints on the list X ∈ Rn×N , we would first rewrite (1306) minimize

N n×N G∈SN c , Y ∈ Sh , X∈ R

subject to

− tr(V (D(G) − 2 Y )V ) # " hΦij , Gi yij º0, yij h2ij · ¸ I X º 0 XT G

N ≥ j > i = 1... N−1

(1307)

X∈ C

and then add the constraints, realized here in abstract membership to some convex set C . This problem realization includes a convex relaxation of the nonconvex constraint G = X TX and, if desired, more constraints on G could be added. This technique is discussed in 5.4.2.3.5.

§

7.2.2

Minimization of affine dimension in Problem 2

When desired affine dimension ρ is diminished, the rank function becomes reinserted into problem (1301) that is then rendered difficult to solve because N ×N the feasible set {D , Y } loses convexity in SN . Indeed, the rank h ×R function is quasiconcave ( 3.9) on the positive semidefinite cone; ( 2.9.2.6.2) id est, its sublevel sets are not convex.

§

7.2.2.1

§

Rank minimization heuristic

A remedy developed in [244] [125] [126] [124] introduces convex envelope of the quasiconcave rank function: (Figure 146) 7.2.2.1.1 Definition. Convex envelope. [184] Convex envelope cenv f of a function f : C → R is defined to be the largest convex function g such that g ≤ f on convex domain C ⊆ Rn .7.16 △ Provided f 6≡ +∞ and there exists an affine function h ≤ f on Rn , then the convex envelope is equal to the convex conjugate (the Legendre-Fenchel transform) of the convex conjugate of f ; id est, the conjugate-conjugate function f ∗∗ . [185, §E.1] 7.16

7.2. SECOND PREVALENT PROBLEM:

rank X

549

g

cenv rank X

Figure 146: Abstraction of convex envelope of rank function. Rank is a quasiconcave function on positive semidefinite cone, but its convex envelope is the largest convex function whose epigraph contains it. Vertical bar labelled g measures a trace/rank gap; id est, rank found always exceeds estimate; large decline in trace required here for only small decrease in rank.

550

CHAPTER 7. PROXIMITY PROBLEMS

ˆ [125] [124] Convex envelope of rank function: for σ

a singular value, 1X σ(A)i (1308) | kAk2 ≤ κ} = κ i i

cenv(rank A) on {A ∈ Rm×n

1 tr(A) (1309) κ A properly scaled trace thus represents the best convex lower bound on rank for positive semidefinite matrices. The idea, then, is to substitute the convex envelope for rank of some variable A ∈ SM + ( A.6.5) P P rank A ← cenv(rank A) ∝ tr A = σ(A)i = λ(A)i = kλ(A)k1 (1310) n cenv(rank A) on {A ∈ S+ | kAk2 ≤ κ} =

§

i

i

which is equivalent to the sum of all eigenvalues or singular values.

ˆ [124] Convex envelope of the cardinality function is proportional to the 1-norm:

cenv(card x) on {x ∈ Rn | kxk∞ ≤ κ} = 7.2.2.2

1 kxk1 κ

(1311)

Applying trace rank-heuristic to Problem 2

Substituting rank envelope for rank function in Problem 2, for D ∈ EDMN (confer (988)) cenv rank(−VNTDVN ) = cenv rank(−V D V ) ∝ − tr(V D V )

(1312)

and for desired affine dimension ρ ≤ N − 1 and nonnegative H [sic] we get a convex optimization problem √ minimize k ◦ D − Hk2F D

subject to − tr(V D V ) ≤ κ ρ

(1313)

D ∈ EDMN

where κ ∈ R+ is a constant determined by cut-and-try. The equivalent semidefinite program makes κ variable: for nonnegative and symmetric H minimize

κ ρ + 2 tr(V Y V ) " # dij yij subject to º0, yij h2ij D,Y , κ

− tr(V D V ) ≤ κ ρ Y ∈ SN h D ∈ EDMN

N ≥ j > i = 1... N−1

(1314)

7.2. SECOND PREVALENT PROBLEM:

551

which is the same as (1305), the problem with no explicit constraint on affine dimension. As the present problem is stated, the desired affine dimension ρ yields to the variable scale factor κ ; ρ is effectively ignored. Yet this result is an illuminant for problem (1305) and it equivalents (all the way back to (1298)): When the given measurement matrix H is nonnegative and symmetric, finding the closest EDM D as in problem (1298), (1301), or (1305) implicitly entails minimization of affine dimension (confer 5.8.4, 5.14.4). Those non−rank-constrained problems are each inherently equivalent to cenv(rank)-minimization problem (1314), in other words, and their optimal solutions are unique because of the strictly convex objective function in (1298).

§

7.2.2.3

§

Rank-heuristic insight

Minimization of affine dimension by use of this trace rank-heuristic (1312) tends to find the list configuration of least energy; rather, it tends to optimize compaction of the reconstruction by minimizing total distance. (861) It is best used where some physical equilibrium implies such an energy minimization; e.g., [327, 5]. For this Problem 2, the trace rank-heuristic arose naturally in the objective in terms of V . We observe: V (in contrast to VNT ) spreads energy over all available distances ( B.4.2 no.20, contrast no.22) although the rank function itself is insensitive to choice of auxiliary matrix.

§

§

7.2.2.4

Rank minimization heuristic beyond convex envelope

Fazel, Hindi, & Boyd [126] [362] [127] propose a rank heuristic more potent than trace (1310) for problems of rank minimization; rank Y ← log det(Y + εI)

§

(1315)

the concave surrogate function log det in place of quasiconcave rank Y n ( 2.9.2.6.2) when Y ∈ S+ is variable and where ε is a small positive constant. They propose minimization of the surrogate by substituting a sequence comprising infima of a linearized surrogate about the current estimate Yi ; id est, from the first-order Taylor series expansion about Yi on some open interval of kY k2 ( D.1.7) ¡ ¢ log det(Y + εI) ≈ log det(Yi + εI) + tr (Yi + εI)−1 (Y − Yi ) (1316)

§

552

CHAPTER 7. PROXIMITY PROBLEMS

we make the surrogate sequence of infima over bounded convex feasible set C arg inf rank Y ← lim Yi+1

(1317)

¡ ¢ Yi+1 = arg inf tr (Yi + εI)−1 Y

(1318)

Y ∈C

i→∞

where, for i = 0 . . . Y ∈C

Choosing Y0 = I , the first step becomes equivalent to finding the infimum of tr Y ; the trace rank-heuristic (1310). The intuition underlying (1318) is the new term in the argument of trace; specifically, (Yi + εI)−1 weights Y so that relatively small eigenvalues of Y found by the infimum are made even smaller. To see that, substitute the nonincreasingly ordered diagonalizations Yi + εI , Q(Λ + εI )QT

(a)

Y , U ΥU T

(b)

(1319)

into (1318). Then from (1628) we have, T

inf δ((Λ + εI )−1 ) δ(Υ) =

Υ∈U ⋆TCU ⋆

≤ T

¡ ¢ inf tr (Λ + εI )−1 RT ΥR

inf

Υ∈U TCU RT=R−1

inf tr((Yi + εI)−1 Y )

(1320)

Y ∈C

where R , Q U in U on the set of orthogonal matrices is a bijection. The role of ε is, therefore, to limit the maximum weight; the smallest entry on the main diagonal of Υ gets the largest weight. ¨ 7.2.2.5

Applying log det rank-heuristic to Problem 2

When the log det rank-heuristic is inserted into Problem 2, problem (1314) becomes the problem sequence in i minimize

κ ρ + 2 tr(V Y V ) # " djl yjl º0, subject to yjl h2jl D,Y , κ

l > j = 1... N−1 −1

− tr((−V Di V + εI ) V D V ) ≤ κ ρ Y ∈ SN h D ∈ EDMN where Di+1 , D⋆ ∈ EDMN , and D0 , 11T − I .

(1321)

7.2. SECOND PREVALENT PROBLEM: 7.2.2.6

553

Tightening this log det rank-heuristic

Like the trace method, this log det technique for constraining rank offers no provision for meeting a predetermined upper bound ρ . Yet since the eigenvalues of the sum are simply determined, λ(Yi + εI) = δ(Λ + εI ) , we may certainly force selected weights to ε−1 by manipulating diagonalization (1319a). Empirically we find this sometimes leads to better results, although affine dimension of a solution cannot be guaranteed. 7.2.2.7

Cumulative summary of rank heuristics

§

§

We have studied the perturbation method of rank reduction in 4.3, as well as the trace heuristic (convex envelope method 7.2.2.1.1) and log det heuristic in 7.2.2.4. There is another good contemporary method called LMIRank [265] based on alternating projection ( E.10) that does not solve the ball packing problem presented in 5.4.2.3.4.

§

§

§

§

7.2.2.7.1 Example. Unidimensional scaling. We apply the convex iteration method from 4.4.1 to numerically solve an instance of Problem 2; a method empirically superior to the foregoing convex envelope and log det heuristics for rank regularization and enforcing affine dimension. Unidimensional scaling, [96] a historically practical application of multidimensional scaling ( 5.12), entails solution of an optimization problem having local minima whose multiplicity varies as the factorial of point-list cardinality; geometrically, it means reconstructing a list constrained to lie in one affine dimension. In terms of point list, the nonconvex problem is: given ×N nonnegative symmetric matrix H = [hij ] ∈ SN ∩ RN (1303) whose entries + hij are all known,

§

minimize {xi ∈ R}

N X

i , j =1

(|xi − xj | − hij )2

(1254)

called a raw stress problem [49, p.34] which has an implicit constraint on dimensional embedding of points {xi ∈ R , i = 1 . . . N }. This problem has proven NP-hard; e.g., [69].

554

CHAPTER 7. PROXIMITY PROBLEMS

As always, we first transform variables to distance-square D ∈ SN h ; so begin with convex problem (1305) on page 547 minimize − tr(V (D − 2 Y )V ) D,Y # " dij yij º 0 , N ≥ j > i = 1... N−1 subject to yij h2ij Y ∈

(1322)

SN h

D ∈ EDMN rank VNTDVN = 1 that becomes equivalent to (1254) by making explicit the constraint on affine dimension via rank. The iteration is formed by moving the dimensional constraint to the objective: minimize −hV (D − 2 Y )V , I i − whVNTDVN , W i D,Y " # dij yij subject to º 0 , N ≥ j > i = 1... N−1 yij h2ij

(1323)

Y ∈ SN h

D ∈ EDMN where w (≈ 10) is a positive scalar just large enough to make hVNTDVN , W i vanish to within some numerical precision, and where direction matrix W is an optimal solution to semidefinite program (1627a) minimize −hVNTD⋆ VN , W i W

subject to 0 ¹ W ¹ I tr W = N − 1

(1324)

one of which is known in closed form. Semidefinite programs (1323) and (1324) are iterated until convergence in the sense defined on page 295. This iteration is not a projection method. Convex problem (1323) is neither a relaxation of unidimensional scaling problem (1322); instead, problem (1323) is a convex equivalent to (1322) at convergence of the iteration.

7.3. THIRD PREVALENT PROBLEM:

555

Jan de Leeuw provided us with some test data 

   H=   

0.000000 5.235301 5.499274 6.404294 6.486829 6.263265

5.235301 0.000000 3.208028 5.840931 3.559010 5.353489

5.499274 3.208028 0.000000 5.679550 4.020339 5.239842

6.404294 5.840931 5.679550 0.000000 4.862884 4.543120

6.486829 3.559010 4.020339 4.862884 0.000000 4.618718

 6.263265 5.353489   5.239842   4.543120   4.618718  0.000000 (1325)

and a globally optimal solution X ⋆ = [ −4.981494 −2.121026 −1.038738 4.555130 0.764096 2.822032 ] =[

x⋆1

x⋆2

x⋆3

x⋆4

x⋆5

x⋆6

]

(1326) found by searching 6! local minima of (1254) [96]. By iterating convex problems (1323) and (1324) about twenty times (initial W = 0) we find the global infimum 98.12812 of stress problem (1254), and by (1078) we find a corresponding one-dimensional point list that is a rigid transformation in R of X ⋆ . Here we found the infimum to accuracy of the given data, but that ceases to hold as problem size increases. Because of machine numerical precision and an interior-point method of solution, we speculate, accuracy degrades quickly as problem size increases beyond this. 2

7.3

Third prevalent problem: Projection on EDM cone in dij In summary, we find that the solution to problem [(1256.3) p.531] is difficult and depends on the dimension of the space as the geometry of the cone of EDMs becomes more complex.

§

−Hayden, Wells, Liu, & Tarazaga (1991) [171, 3]

556

CHAPTER 7. PROXIMITY PROBLEMS

Reformulating Problem 2 (p.544) in terms of EDM D changes the problem considerably:  minimize kD − Hk2F   D T Problem 3 (1327) subject to rank VN DVN ≤ ρ   N D ∈ EDM This third prevalent proximity problem is a Euclidean projection of given matrix H on a generally nonconvex subset (ρ < N − 1) of ∂ EDMN the boundary of the convex cone of Euclidean distance matrices relative to subspace SN Because coordinates of projection are h (Figure 130(d)). distance-square and H presumably now holds distance-square measurements, numerical solution to Problem 3 is generally different than that of Problem 2. For the moment, we need make no assumptions regarding measurement matrix H .

7.3.1

Convex case minimize D

kD − Hk2F

(1328)

subject to D ∈ EDMN

When the rank constraint disappears (for ρ = N − 1), this third problem becomes obviously convex because the feasible set is then the entire EDM cone and because the objective function X kD − Hk2F = (dij − hij )2 (1329) i,j

is a strictly convex quadratic in D ;7.17 X 2 minimize dij − 2hij dij + h2ij D

(1330)

i,j

subject to D ∈ EDM

N

Optimal solution D⋆ is therefore unique, as expected, for this simple projection on the EDM cone equivalent to (1255). 7.17

For nonzero Y ∈ SN h and some open interval of t ∈ R (§3.8.3.0.2, §D.2.3) d2 k(D + t Y ) − Hk2F = 2 tr Y T Y > 0 dt2

¨

7.3. THIRD PREVALENT PROBLEM: 7.3.1.1

557

Equivalent semidefinite program, Problem 3, convex case

§

In the past, this convex problem was solved numerically by means of alternating projection. (Example 7.3.1.1.1) [142] [135] [171, 1] We translate (1330) to an equivalent semidefinite program because we have a good solver: Assume the given measurement matrix H to be nonnegative and symmetric;7.18 ×N H = [hij ] ∈ SN ∩ RN +

(1303)

We then propose: Problem (1330) is equivalent to the semidefinite program, for 2 (1331) ∂ , [dij ] = D ◦D a matrix of distance-square squared, minimize − tr(V (∂ − 2 H ◦ D)V ) ∂,D # " ∂ij dij º 0 , N ≥ j > i = 1... N−1 subject to dij 1

(1332)

D ∈ EDMN ∂ ∈ SN h where "

∂ij dij dij 1

#

2

º 0 ⇔ ∂ij ≥ dij

(1333)

§

Symmetry of input H facilitates trace in the objective ( B.4.2 no.20), while 2 its nonnegativity causes ∂ij → dij as optimization proceeds. 7.3.1.1.1 Example. Alternating projection on nearest EDM. By solving (1332) we confirm the result from an example given by Glunt, Hayden, Hong, & Wells [142, 6] who found analytical solution to convex optimization problem (1328) for particular cardinality N = 3 by using the

§

7.18

If that H given has negative entries, then the technique of solution presented here becomes invalid. Projection of H on K (1247) prior to application of this proposed technique, as explained in §7.0.1, is incorrect.

558

CHAPTER 7. PROXIMITY PROBLEMS

§

alternating projection method of von Neumann ( E.10):     19 0 19 0 1 1 9 9  76  0 D⋆ =  19 H= 1 0 9 , 9 9  19 76 1 9 0 0 9

(1334)

9

The original problem (1328) of projecting H on the EDM cone is transformed to an equivalent iterative sequence of projections on the two convex cones (1196) from 6.8.1.1. Using ordinary alternating projection, input H goes to D⋆ with an accuracy of four decimal places in about 17 iterations. Affine dimension corresponding to this optimal solution is r = 1. Obviation of semidefinite programming’s computational expense is the principal advantage of this alternating projection technique. 2

§

7.3.1.2

Schur-form semidefinite program, Problem 3 convex case

Semidefinite program (1332) can be reformulated by moving the objective function in minimize kD − Hk2F D (1328) subject to D ∈ EDMN to the constraints. This makes an equivalent epigraph form of the problem: for any measurement matrix H minimize t∈ R , D

t (1335)

subject to kD − Hk2F ≤ t D ∈ EDMN

§

We can transform this problem to an equivalent Schur-form semidefinite program; ( 3.6.2) minimize t∈ R , D

subject to

t ·

tI vec(D − H) T vec(D − H) 1

D ∈ EDMN

¸

º0

(1336)

characterized by great sparsity and structure. The advantage of this SDP is lack of conditions on input H ; e.g., negative entries would invalidate any solution provided by (1332). ( 7.0.1.2)

§

7.3. THIRD PREVALENT PROBLEM: 7.3.1.3

559

Gram-form semidefinite program, Problem 3 convex case

§

Further, this problem statement may be equivalently written in terms of a Gram matrix via linear bijective ( 5.6.1) EDM operator D(G) (849); minimize

G∈ SN c , t∈ R

subject to

t ·

tI vec(D(G) − H) T vec(D(G) − H) 1

¸

(1337)

º0

Gº0

To include constraints on the list X ∈ Rn×N , we would rewrite this: minimize

n×N G∈ SN c , t∈ R , X∈ R

subject to

t ·

tI vec(D(G) − H) T vec(D(G) − H) 1 · ¸ I X º 0 XT G

¸

X∈ C

º0

(1338)

§

where C is some abstract convex set. This technique is discussed in 5.4.2.3.5. 7.3.1.4

Dual interpretation, projection on EDM cone

§

From E.9.1.1 we learn that projection on a convex set has a dual form. In the circumstance K is a convex cone and point x exists exterior to the cone or on its boundary, distance to the nearest point P x in K is found as the optimal value of the objective kx − P xk = maximize aTx a

subject to kak ≤ 1 a∈K

◦

(1939)

◦

where K is the polar cone. Applying this result to (1328), we get a convex optimization for any given symmetric matrix H exterior to or on the EDM cone boundary: minimize D

kD − Hk2F

subject to D ∈ EDMN

maximize hA◦ , H i ◦ A

≡ subject to kA◦ kF ≤ 1

A◦ ∈ EDM

(1339) N◦

560

CHAPTER 7. PROXIMITY PROBLEMS

Then from (1941) projection of H on cone EDMN is D⋆ = H − A◦⋆ hA◦⋆ , H i

(1340)

Critchley proposed, instead, projection on the polar EDM cone in his 1980 thesis [84, p.113]: In that circumstance, by projection on the algebraic complement ( E.9.2.2.1),

§

D⋆ = A⋆ hA⋆ , H i

(1341)

which is equal to (1340) when A⋆ solves maximize hA , H i A

(1342)

subject to kAkF = 1

A ∈ EDMN ◦

This projection of symmetric H on polar cone EDMN can be made a convex problem, of course, by relaxing the equality constraint (kAkF ≤ 1).

7.3.2

Minimization of affine dimension in Problem 3

§

When desired affine dimension ρ is diminished, Problem 3 (1327) is difficult to solve [171, 3] because the feasible set in RN (N −1)/2 loses convexity. By substituting rank envelope (1312) into Problem 3, then for any given H we get a convex problem minimize D

kD − Hk2F

subject to − tr(V D V ) ≤ κ ρ D ∈ EDMN

(1343)

where κ ∈ R+ is a constant determined by cut-and-try. Given κ , problem (1343) is a convex optimization having unique solution in any desired affine dimension ρ ; an approximation to Euclidean projection on that nonconvex subset of the EDM cone containing EDMs with corresponding affine dimension no greater than ρ .

7.3. THIRD PREVALENT PROBLEM:

561

The SDP equivalent to (1343) does not move κ into the variables as on page 550: for nonnegative symmetric input H and distance-square squared variable ∂ as in (1331), minimize − tr(V (∂ − 2 H ◦ D)V ) ∂,D " # ∂ij dij subject to º 0 , N ≥ j > i = 1... N−1 dij 1

(1344)

− tr(V D V ) ≤ κ ρ D ∈ EDMN ∂ ∈ SN h That means we will not see equivalence of this cenv(rank)-minimization problem to the non−rank-constrained problems (1330) and (1332) like we saw for its counterpart (1314) in Problem 2.

§

Another approach to affine dimension minimization is to project instead on the polar EDM cone; discussed in 6.8.1.5.

7.3.3

Constrained affine dimension, Problem 3

When one desires affine dimension diminished further below what can be achieved via cenv(rank)-minimization as in (1344), spectral projection can be considered a natural means in light of its successful application to projection on a rank ρ subset of the positive semidefinite cone in 7.1.4.

§

Yet it is wrong here to zero eigenvalues of −V D V or −V G V or a variant to reduce affine dimension, because that particular method comes from projection on a positive semidefinite cone (1282); zeroing those eigenvalues here in Problem 3 would place an elbow in the projection path (Figure 145) thereby producing a result that is necessarily suboptimal. Problem 3 is instead a projection on the EDM cone whose associated spectral cone is considerably different. ( 5.11.2.3) Proper choice of spectral cone is demanded by diagonalization of that variable argument to the objective:

§

562 7.3.3.1

CHAPTER 7. PROXIMITY PROBLEMS Cayley-Menger form

§

We use Cayley-Menger composition of the Euclidean distance matrix to solve a problem that is the same as Problem 3 (1327): ( 5.7.3.0.1) °· ¸ · ¸°2 T ° ° 0 1T 0 1 ° − minimize ° ° 1 −D 1 −H °F D · ¸ (1345) 0 1T subject to rank ≤ ρ+2 1 −D D ∈ EDMN a projection of H on a generally nonconvex subset (when ρ < N − 1) of the Euclidean distance matrix cone boundary rel ∂ EDMN ; id est, projection from the EDM cone interior or exterior on a subset of its relative boundary ( 6.6, (1125)). Rank of an optimal solution is intrinsically bounded above and below; ¸ · 0 1T ≤ ρ+2 ≤ N +1 (1346) 2 ≤ rank 1 −D⋆

§

Our proposed strategy solution is projection on that subset µ· for low-rank ¸¶ 0 1T of a spectral cone λ ( 5.11.2.3) corresponding to affine 1 −EDMN dimension not in excess of that ρ desired; id est, spectral projection on  ρ+1  R+  ∩ ∂H ⊂ RN +1  0 (1347) R− where ∂H = {λ ∈ RN +1 | 1T λ = 0} (1058)

§

is a hyperplane through the origin. This pointed polyhedral cone (1347), to which membership subsumes the rank constraint, has empty interior. Given desired affine dimension 0 ≤ ρ ≤ N − 1 and diagonalization ( A.5) of unknown EDM D · ¸ 0 1T +1 , U ΥU T ∈ SN (1348) h 1 −D

§

and given symmetric H in diagonalization ¸ · 0 1T , QΛQT ∈ SN +1 1 −H

(1349)

7.3. THIRD PREVALENT PROBLEM:

563

having eigenvalues arranged in nonincreasing order, then by (1071) problem (1345) is equivalent to ° ¡ ¢° °δ(Υ) − π δ(RTΛR) °2 Υ, R  ρ+1  R+  ∩ ∂H subject to δ(Υ) ∈  0 R− minimize

(1350)

δ(QR ΥRTQT ) = 0 R −1 = RT

where π is the permutation operator from argument in nonincreasing order,7.19 where

§7.1.3 arranging its vector

R , QT U ∈ RN +1×N +1

(1351)

in U on the set of orthogonal matrices is a bijection, and where ∂H insures one negative eigenvalue. Hollowness constraint δ(QR ΥRTQT ) = 0 makes problem (1350) difficult by making the two variables dependent. Our plan is to instead divide problem (1350) into two and then iterate their solution: ° ¡ ¢° °δ(Υ) − π δ(RTΛR) °2 Υ  ρ+1  R+  ∩ ∂H subject to δ(Υ) ∈  0 R− minimize

minimize R

kR Υ⋆ RT − Λk2F

subject to δ(QR Υ⋆ RTQT ) = 0 R −1 = RT

7.19

Recall, any permutation matrix is an orthogonal matrix.

(a) (1352)

(b)

564

CHAPTER 7. PROXIMITY PROBLEMS

§

Proof. We justify disappearance of the hollowness constraint in convex optimization problem (1352a): From the arguments in 7.1.3 with regard  ρ+1to π the permutation operator, cone membership constraint R+  ∩ ∂H from (1352a) is equivalent to  δ(Υ) ∈ 0 R−  Rρ+1 +  ∩ ∂H ∩ KM δ(Υ) ∈  0 R− 

(1353)

§

where KM is the monotone cone ( 2.13.9.4.3). Membership of δ(Υ) to the polyhedral cone of majorization (Theorem A.1.2.0.1) ∗

∗

Kλδ = ∂H ∩ KM+

(1370)

§

∗

where KM+ is the dual monotone nonnegative cone ( 2.13.9.4.2), is a condition (in absence of a hollowness that would insure existence · constraint) ¸ T 0 1 of a symmetric hollow matrix . Curiously, intersection of 1 −D  ρ+1  R+  ∩ ∂H ∩ KM from (1353) with the cone of  this feasible superset 0 R− ∗ majorization Kλδ is a benign operation; id est, ∗

∂H ∩ KM+ ∩ KM = ∂H ∩ KM

§

(1354)

verifiable by observing conic dependencies ( 2.10.3) among the aggregate of halfspace-description normals. The cone membership constraint in (1352a) therefore inherently insures existence of a symmetric hollow matrix. ¨

7.3. THIRD PREVALENT PROBLEM:

565

§

Optimization (1352b) would be a Procrustes problem ( C.4) were it not for the hollowness constraint; it is, instead, a minimization over the intersection of the nonconvex manifold of orthogonal matrices with another nonconvex set in variable R specified by the hollowness constraint.

§

We solve problem (1352b) by a method introduced in 4.6.0.0.2: Define R = [ r1 · · · rN +1 ] ∈ RN +1×N +1 and make the assignment 

r1 .. .



T [ r1T · · · rN +1 1 ]

  2   (1355) G = ∈ S(N +1) +1  r  N +1  1     T R11 · · · R1,N +1 r1 r1 r1T · · · r1 rN r1 +1 .. ..   ..  .. ...  ... . .    .  . , = T    T T r r r r r R R r  1,N +1 N +1  N +1,N +1 N +1   N +1 1 N +1 N +1 T T r1T ··· rN 1 r1T ··· rN 1 +1 +1 where Rij , ri rjT ∈ RN +1×N +1 and Υ⋆ii ∈ R . Since R Υ⋆ RT = problem (1352b) is equivalently expressed:

minimize

Rii ∈ S , Rij , ri

subject to

°N +1 °2 °P ⋆ ° ° Υii Rii − Λ° ° ° i=1 F tr Rii = 1 , tr Rij= 0 , R11 · · · R1,N +1 r1 .. .. ...  . .  G= T RN +1,N +1 rN +1  R1,N +1 T rT ··· rN 1 +1 µ N +1 1 ¶ P δ Q Υii⋆ Rii QT = 0

NP +1 i=1

Υii⋆ Rii then

i=1 . . . N + 1 i<j = 2 . . . N + 1 

  (º 0) 

(1356)

i=1

rank G = 1

§

The rank constraint is regularized by method of convex iteration developed in 4.4. Problem (1356) is partitioned into two convex problems:

566

CHAPTER 7. PROXIMITY PROBLEMS

°N +1 °2 °P ⋆ ° minimize ° Υii Rii − Λ° ° ° + hG , W i Rij , ri i=1 F subject to tr Rii = 1 , tr Rij= 0 , R11 · · · R1,N +1 r1 .. .. ...  . .  G= T R R r  1,N +1 N +1,N +1 N +1 T T r ··· rN +1 1 µ N +1 1 ¶ P δ Q Υii⋆ Rii QT = 0

i=1 . . . N + 1 i<j = 2 . . . N + 1 

(1357)

  º 0 

i=1

and

minimize 2

hG⋆ , W i

subject to

0¹W¹I tr W = (N + 1)2

W ∈ S(N +1)

+1

(1358)

then iterated with convex problem (1352a) until a rank-1 G matrix is found and the objective of (1352a) is minimized. An optimal solution to (1358) is known in closed form (p.631). The hollowness constraint in (1357) may cause numerical infeasibility; in that case, it can be moved to the objective within an added weighted norm. Conditions for termination of the iteration would then comprise a vanishing norm of hollowness.

7.4

Conclusion

There has been little progress in spectral projection since the discovery by Eckart & Young in 1936 [119] leading to a formula for projection on a rank ρ subset of a positive semidefinite cone ( 2.9.2.1). [134] The only closed-form spectral method presently available for solving proximity problems, having a constraint on rank, is based on their discovery (Problem 1, 7.1, 5.13). One recourse is intentional misapplication of Eckart & Young’s result by introducing spectral projection on a positive semidefinite cone into Problem 3 via D(G) (849), for example. [69] Since Problem 3 instead demands spectral projection on the EDM cone, any solution acquired that way is necessarily suboptimal.

§

§ §

7.4. CONCLUSION

567

A second recourse is problem redesign: A presupposition to all proximity problems in this chapter is that matrix H is given. We considered H having various properties such as nonnegativity, symmetry, hollowness, or lack thereof. It was assumed that if H did not already belong to the EDM cone, then we wanted an EDM closest to H in some sense; id est, input-matrix H was assumed corrupted somehow. For practical problems, it withstands reason that such a proximity problem could instead be reformulated so that some or all entries of H were unknown but bounded above and below by known limits; the norm objective is thereby eliminated as in the development beginning on page 308. That redesign (the art, p.8), in terms of the Gram-matrix bridge between point-list X and EDM D , at once encompasses proximity and completion problems. A third recourse is to apply the method of convex iteration just like we did in 7.2.2.7.1. This technique is applicable to any semidefinite problem requiring a rank constraint; it places a regularization term in the objective that enforces the rank constraint. The importance and application of solving rank-constrained problems are enormous, a conclusion generally accepted gratis by the mathematics and engineering communities. For example, one might be interested in the minimal order dynamic output feedback which stabilizes a given linear time invariant plant (this problem is considered to be among the most important open problems in control). [245] Rank-constrained semidefinite programs arise in many vital feedback and control problems [158], optics [71], and communications [263] [236]. Rank-constrained problems also appear naturally in combinatorial optimization. ( 4.6.0.0.10)

§

§

568

CHAPTER 7. PROXIMITY PROBLEMS

Appendix A Linear algebra A.1

Main-diagonal δ operator, λ , trace, vec

We introduce notation δ denoting the main-diagonal linear self-adjoint operator. When linear function δ operates on a square matrix A ∈ RN ×N , δ(A) returns a vector composed of all the entries from the main diagonal in the natural order; δ(A) ∈ RN (1359) Operating on a vector y ∈ RN , δ naturally returns a diagonal matrix; δ(y) ∈ SN

(1360)

Operating recursively on a vector Λ ∈ RN or diagonal matrix Λ ∈ SN , δ(δ(Λ)) returns Λ itself; δ 2(Λ) ≡ δ(δ(Λ)) , Λ

§

§

(1361)

§

Defined in this manner, main-diagonal linear operator δ is self-adjoint [211, 3.10, 9.5-1];A.1 videlicet, ( 2.2) ¡ ¢ δ(A)T y = hδ(A) , yi = hA , δ(y)i = tr AT δ(y) (1362)

Linear operator T : Rm×n → RM ×N is self-adjoint when, for each and every X1 , X2 ∈ Rm×n A.1

hT (X1 ) , X2 i = hX1 , T (X2 )i

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

569

570

A.1.1

APPENDIX A. LINEAR ALGEBRA

Identities

§

This δ notation is efficient and unambiguous as illustrated in the following examples where A ◦ B denotes Hadamard product [188] [146, 1.1.4] of matrices of like size, ⊗ Kronecker product [152] ( D.1.2.1), y a vector, X a matrix, ei the i th member of the standard basis for Rn , SN h the symmetric hollow subspace, σ a vector of (nonincreasingly) ordered singular values, and λ(A) denotes a vector of nonincreasingly ordered eigenvalues of matrix A :

§

1. δ(A) = δ(AT ) 2. tr(A) = tr(AT ) = δ(A)T 1 = hI , Ai 3. δ(cA) = c δ(A)

c∈R

4. tr(cA) = c tr(A) = c 1T λ(A)

c∈R

5. vec(cA) = c vec(A)

c∈R

6. A ◦ cB = cA ◦ B

c∈R

7. A ⊗ cB = cA ⊗ B

c∈R

8. δ(A + B) = δ(A) + δ(B) 9. tr(A + B) = tr(A) + tr(B) 10. vec(A + B) = vec(A) + vec(B) 11. (A + B) ◦ C = A ◦ C + B ◦ C A ◦ (B + C ) = A ◦ B + A ◦ C 12. (A + B) ⊗ C = A ⊗ C + B ⊗ C A ⊗ (B + C ) = A ⊗ B + A ⊗ C 13. sgn(c) λ(|c|A) = c λ(A)

c∈R

14. sgn(c) σ(|c|A) = c σ(A) √ √ 15. tr(c ATA ) = c tr ATA = c 1T σ(A)

c∈R

16. π(δ(A)) = λ(I ◦ A) where π is the presorting function. 17. δ(AB) = (A ◦ B T )1 = (B T ◦ A)1 18. δ(AB)T = 1T(AT ◦ B) = 1T(B ◦ AT )

c∈R

A.1. MAIN-DIAGONAL δ OPERATOR, λ , TRACE, VEC  u1 v 1   19. δ(uv T ) =  ...  = u ◦ v , uN v N 

571

u,v ∈ RN

20. tr(ATB) = tr(AB T ) = tr(BAT ) = tr(B TA) = 1T(A ◦ B)1 = 1T δ(AB T ) = δ(ATB)T 1 = δ(BAT )T 1 = δ(B TA)T 1

§

SN (confer B.4.2 no.20) P N tr(−V (D ◦ H)V ) = tr(DTH) = 1T(D ◦ H)1 = tr(11T(D ◦ H)) = dij hij

N 21. D = [dij ] ∈ SN h , H = [hij ] ∈ Sh , V = I −

1 11T ∈ N

i,j

22. tr(ΛA) = δ(Λ)T δ(A) , δ 2(Λ) , Λ ∈ SN ¡ ¢ ¡ ¢ ¡ ¢ 23. y TB δ(A) = tr B δ(A)y T = tr δ(B Ty)A = tr A δ(B Ty) ¡ ¢ ¡ ¢ ¡ ¢ = δ(A)TB Ty = tr y δ(A)TB T = tr AT δ(B Ty) = tr δ(B Ty)AT P T T 24. δ 2(ATA) = ei eT i A Aei ei i

¡

T

25. δ δ(A)1

¢

¢ ¡ = δ 1 δ(A)T = δ(A)

26. δ(A1)1 = δ(A11T ) = A1 ,

δ(y)1 = δ(y1T ) = y

27. δ(I 1) = δ(1) = I T 28. δ(ei eT j 1) = δ(ei ) = ei ei

29. For ζ = [ζ i ] ∈ Rk and x = [xi ] ∈ Rk , 30.

P

ζ i /xi = ζ T δ(x)−1 1

i

vec(A ◦ B) = vec(A) ◦ vec(B) = δ(vec A) vec(B) = vec(B) ◦ vec(A) = δ(vec B) vec(A)

(37)(1711)

31. vec(A XB) = (B T ⊗ A) vec X 32. vec(B XA) = (AT ⊗ B) vec X

¡ ¢ not H

tr(A XBX T ) = vec(X)T vec(A XB) = vec(X)T (B T ⊗ A) vec X ¡ ¢T 33. = δ vec(X) vec(X)T (B T ⊗ A) 1 34.

tr(A X TBX) = vec(X)T vec(B XA) = vec(X)T (AT ⊗ B) vec X ¡ ¢T = δ vec(X) vec(X)T (AT ⊗ B) 1

[152]

572

APPENDIX A. LINEAR ALGEBRA

35. For any permutation matrix Ξ and dimensionally compatible vector y or matrix A δ(Ξ y) = Ξ δ(y) ΞT δ(Ξ A ΞT ) = Ξ δ(A)

(1363) (1364)

So given any permutation matrix Ξ and any dimensionally compatible matrix B , for example, δ 2(B) = Ξ δ 2(ΞTB Ξ)ΞT

(1365)

36. A ⊗ (B ⊗ C ) = (A ⊗ B) ⊗ C 37. (A ⊗ B)(C ⊗ D) = AC ⊗ BD 38. For A a vector, (A ⊗ B) = (A ⊗ I )B 39. For B a row vector, (A ⊗ B) = A(I ⊗ B) 40. (A ⊗ B)T = AT ⊗ B T 41. (A ⊗ B)−1 = A−1 ⊗ B −1 42. tr(A ⊗ B) = tr A tr B 43. For A ∈ Rm×m , B ∈ Rn×n , det(A ⊗ B) = detn (A) detm (B) 44. There exist permutation matrices Ξ1 and Ξ2 such that [152, p.28] A ⊗ B = Ξ1 (B ⊗ A)Ξ2

(1366)

45. For eigenvalues λ(A) ∈ C n and eigenvectors v(A) ∈ C n×n such that A = vδ(λ)v −1 ∈ Rn×n λ(A ⊗ B) = λ(A) ⊗ λ(B) ,

v(A ⊗ B) = v(A) ⊗ v(B)

(1367)

46. Given analytic function [78] f : C n×n → C n×n , then f (I ⊗A) = I ⊗f (A) and f (A ⊗ I) = f (A) ⊗ I [152, p.28]

A.1. MAIN-DIAGONAL δ OPERATOR, λ , TRACE, VEC

A.1.2

Majorization

573

§

§

§

A.1.2.0.1 Theorem. (Schur) Majorization. [368, 7.4] [188, 4.3] N [189, 5.5] Let λ ∈ R denote a given vector of eigenvalues and let δ ∈ RN denote a given vector of main diagonal entries, both arranged in nonincreasing order. Then ∗

∃ A ∈ SN Ä λ(A) = λ and δ(A) = δ ⇐ λ − δ ∈ Kλδ and conversely

∗

A ∈ SN ⇒ λ(A) − δ(A) ∈ Kλδ

(1368) (1369)

The difference belongs to the pointed polyhedral cone of majorization (empty interior, confer (286)) ∗

∗

∗

Kλδ , KM+ ∩ {ζ 1 | ζ ∈ R}

(1370)

∗

where KM+ is the dual monotone nonnegative cone (407), and where the ∗ dual of the line is a hyperplane; ∂H = {ζ 1 | ζ ∈ R} = 1⊥ . ⋄ ∗

Majorization cone Kλδ is naturally consequent to the definition of majorization; id est, vector y ∈ RN majorizes vector x if and only if k X i=1

xi ≤

k X i=1

yi

∀1 ≤ k ≤ N

(1371)

and 1T x = 1T y

(1372)

Under these circumstances, rather, vector x is majorized by vector y . In the particular circumstance δ(A) = 0, we get: A.1.2.0.2 Corollary. Symmetric hollow majorization. N Let λ ∈ R denote a given vector of eigenvalues arranged in nonincreasing order. Then ∗ ∃ A ∈ SN (1373) h Ä λ(A) = λ ⇐ λ ∈ Kλδ and conversely where

∗ Kλδ

∗

A ∈ SN h ⇒ λ(A) ∈ Kλδ is defined in (1370).

(1374) ⋄

574

APPENDIX A. LINEAR ALGEBRA

A.2

Semidefiniteness: domain of test

§

The most fundamental necessary, sufficient, and definitive test for positive semidefiniteness of matrix A ∈ Rn×n is: [189, 1] xTA x ≥ 0 for each and every x ∈ Rn such that kxk = 1

(1375)

Traditionally, authors demand evaluation over broader domain; namely, over all x ∈ Rn which is sufficient but unnecessary. Indeed, that standard textbook requirement is far over-reaching because if xTA x is nonnegative for particular x = xp , then it is nonnegative for any αxp where α ∈ R . Thus, only normalized x in Rn need be evaluated. Many authors add the further requirement that the domain be complex; the broadest domain. By so doing, only Hermitian matrices (AH = A where superscript H denotes conjugate transpose)A.2 are admitted to the set of positive semidefinite matrices (1378); an unnecessary prohibitive condition.

A.2.1

Symmetry versus semidefiniteness

We call (1375) the most fundamental test of positive semidefiniteness. Yet some authors instead say, for real A and complex domain {x ∈ C n } , the complex test xHAx ≥ 0 is most fundamental. That complex broadening of the domain of test causes nonsymmetric real matrices to be excluded from the set of positive semidefinite matrices. Yet admitting nonsymmetric real matrices or not is a matter of preferenceA.3 unless that complex test is adopted, as we shall now explain. Any real square matrix A has a representation in terms of its symmetric and antisymmetric parts; id est, A=

(A + AT ) (A − AT ) + 2 2

(48)

Because, for all real A , the antisymmetric part vanishes under real test, xT A.2

(A − AT ) x=0 2

(1376)

Hermitian symmetry is the complex analogue; the real part of a Hermitian matrix is symmetric while its imaginary part is antisymmetric. A Hermitian matrix has real eigenvalues and real main diagonal. A.3 Golub & Van Loan [146, §4.2.2], for example, admit nonsymmetric real matrices.

A.2. SEMIDEFINITENESS: DOMAIN OF TEST

575

only the symmetric part of A , (A + AT )/2, has a role determining positive semidefiniteness. Hence the oft-made presumption that only symmetric matrices may be positive semidefinite is, of course, erroneous under (1375). Because eigenvalue-signs of a symmetric matrix translate unequivocally to its semidefiniteness, the eigenvalues that determine semidefiniteness are always those of the symmetrized matrix. ( A.3) For that reason, and because symmetric (or Hermitian) matrices must have real eigenvalues, the convention adopted in the literature is that semidefinite matrices are synonymous with symmetric semidefinite matrices. Certainly misleading under (1375), that presumption is typically bolstered with compelling examples from the physical sciences where symmetric matrices occur within the mathematical exposition of natural phenomena.A.4 [130, 52] Perhaps a better explanation of this pervasive presumption of symmetry comes from Horn & Johnson [188, 7.1] whose perspectiveA.5 is the complex matrix, thus necessitating the complex domain of test throughout their treatise. They explain, if A ∈ C n×n

§

§

§

. . . and if xHAx is real for all x ∈ C n , then A is Hermitian. Thus, the assumption that A is Hermitian is not necessary in the definition of positive definiteness. It is customary, however. Their comment is best explained by noting, the real part of xHAx comes from the Hermitian part (A + AH )/2 of A ; Re(xHAx) = xH

A + AH x 2

(1377)

rather, xHAx ∈ R ⇔ AH = A

(1378)

because the imaginary part of xHAx comes from the anti-Hermitian part (A − AH )/2 ; A − AH Im(xHAx) = xH x (1379) 2 that vanishes for nonzero x if and only if A = AH . So the Hermitian symmetry assumption is unnecessary, according to Horn & Johnson, not A.4

Symmetric matrices are certainly pervasive in the our chosen subject as well. A totally complex perspective is not necessarily more advantageous. The positive semidefinite cone, for example, is not self-dual (§2.13.5) in the ambient space of Hermitian matrices. [182, §II] A.5

576

APPENDIX A. LINEAR ALGEBRA

because nonHermitian matrices could be regarded positive semidefinite, rather because nonHermitian (includes nonsymmetric real) matrices are not comparable on the real line under xHAx . Yet that complex edifice is dismantled in the test of real matrices (1375) because the domain of test is no longer necessarily complex; meaning, xTAx will certainly always be real, regardless of symmetry, and so real A will always be comparable. In summary, if we limit the domain of test to all x in Rn as in (1375), then nonsymmetric real matrices are admitted to the realm of semidefinite matrices because they become comparable on the real line. One important exception occurs for rank-one matrices Ψ = uv T where u and v are real vectors: Ψ is positive semidefinite if and only if Ψ = uuT . ( A.3.1.0.7) We might choose to expand the domain of test to all x in C n so that only symmetric matrices would be comparable. The alternative to expanding domain of test is to assume all matrices of interest to be symmetric; that is commonly done, hence the synonymous relationship with semidefinite matrices.

§

A.2.1.0.1 Example. Nonsymmetric positive definite product. Horn & Johnson assert and Zhang agrees: If A,B ∈ C n×n are positive definite, then we know that the product AB is positive definite if and only if AB is Hermitian. [188, 7.6, prob.10] [368, 6.2, 3.2]

§

§ §

Implicitly in their statement, A and B are assumed individually Hermitian and the domain of test is assumed complex. We prove that assertion to be false for real matrices under (1375) that adopts a real domain of test.     3 0 −1 0 5.9  0 5  4.5  1 0  ,  AT = A =  λ(A) =  (1380)  −1 1   3.4  4 1 0 0 1 4 2.0 

4  4 BT = B =   −1 −1

 4 −1 −1 5 0 0  , 0 5 1  0 1 4



 8.8  5.5   λ(B) =   3.3  0.24

(1381)

A.3. PROPER STATEMENTS



 13 12 −8 −4  19 25 5 1  , (AB)T 6= AB =   −5 1 22 9  −5 0 9 17  13 15.5 −6.5 −4.5  15.5 25 3 0.5  1 , (AB + (AB)T ) =  2  −6.5 3 22 9  −4.5 0.5 9 17 

577



 36.  29.   λ(AB) =   10.  0.72

(1382)

 36. ¡ ¢  30.   λ 12 (AB + (AB)T ) =   10.  0.014 

(1383) √ √ n n Whenever A ∈ S+ and B ∈ S+ , then λ(AB) = λ( AB A) will always be a nonnegative vector by (1407) and Corollary A.3.1.0.5. Yet positive definiteness of product ¡1 ¢ AB is certified instead by the nonnegative eigenvalues T λ 2 (AB + (AB) ) in (1383) ( A.3.1.0.1) despite the fact that AB is not symmetric.A.6 Horn & Johnson and Zhang resolve the anomaly by choosing to exclude nonsymmetric matrices and products; they do so by expanding the domain of test to C n . 2

§

A.3

Proper statements of positive semidefiniteness

Unlike Horn & Johnson and Zhang, we never adopt a complex domain of test with real matrices. So motivated is our consideration of proper statements of positive semidefiniteness under real domain of test. This restriction, ironically, complicates the facts when compared to the corresponding statements for the complex case (found elsewhere, [188] [368]). We state several fundamental facts regarding positive semidefiniteness of real matrix A and the product AB and sum A + B of real matrices under fundamental real test (1375); a few require proof as they depart from the standard texts, while those remaining are well established or obvious. It is a little more difficult to find a counter-example in R2×2 or R3×3 ; which may have served to advance any confusion. A.6

578

APPENDIX A. LINEAR ALGEBRA

A.3.0.0.1 Theorem. Positive (semi)definite matrix. A ∈ SM is positive semidefinite if and only if for each and every vector x ∈ RM of unit norm, kxk = 1 ,A.7 we have xTA x ≥ 0 (1375); A º 0 ⇔ tr(xxTA) = xTA x ≥ 0 ∀ xxT

(1384)

Matrix A ∈ SM is positive definite if and only if for each and every kxk = 1 we have xTA x > 0 ; A ≻ 0 ⇔ tr(xxTA) = xTA x > 0 ∀ xxT , xxT 6= 0

(1385) ⋄

§

Proof. Statements (1384) and (1385) are each a particular instance of dual generalized inequalities ( 2.13.2) with respect to the positive semidefinite cone; videlicet, [335] A º 0 ⇔ hxxT , Ai ≥ 0 ∀ xxT(º 0) A ≻ 0 ⇔ hxxT , Ai > 0 ∀ xxT(º 0) , xxT 6= 0

(1386)

This says: positive semidefinite matrix A must belong to the normal side of every hyperplane whose normal is an extreme direction of the positive semidefinite cone. Relations (1384) and (1385) remain true when xxT is replaced with “for each and every” positive semidefinite matrix X ∈ SM + ( 2.13.5) of unit norm, kXk = 1, as in

§

A º 0 ⇔ tr(XA) ≥ 0 ∀ X ∈ SM +

A ≻ 0 ⇔ tr(XA) > 0 ∀ X ∈ SM + , X 6= 0

(1387)

§

But that condition is more than what is necessary. By the discretized membership theorem in 2.13.4.2.1, the extreme directions xxT of the positive semidefinite cone constitute a minimal set of generators necessary and sufficient for discretization of dual generalized inequalities (1387) certifying membership to that cone. ¨ The traditional condition requiring all x ∈ RM for defining positive (semi)definiteness is actually more than what is necessary. The set of norm-1 vectors is necessary and sufficient to establish positive semidefiniteness; actually, any particular norm and any nonzero norm-constant will work. A.7

A.3. PROPER STATEMENTS

A.3.1

579

Semidefiniteness, eigenvalues, nonsymmetric

¢ ¡ When A ∈ Rn×n , let λ 21 (A + AT ) ∈ Rn denote eigenvalues of the symmetrized matrixA.8 arranged in nonincreasing order.

ˆ By positive semidefiniteness of A ∈ R (confer §A.3.1.0.1)

n×n

§

we mean,A.9 [253, 1.3.1]

xTA x ≥ 0 ∀ x ∈ Rn ⇔ A + AT º 0 ⇔ λ(A + AT ) º 0

ˆ (§2.9.0.1)

A º 0 ⇒ AT = A

(1389)

A º B ⇔ A − B º 0 ; A º 0 or B º 0 T

T

T

(1390)

x A x≥ 0 ∀ x ; A = A

(1391)

AT = A , λ(A) º 0 ⇒ xTA x ≥ 0 ∀ x

(1392)

λ(A) º 0 ⇐ AT = A , xTA x ≥ 0 ∀ x

(1393)

ˆ Matrix symmetry is not intrinsic to positive semidefiniteness; ˆ If A

(1388)

= A then λ(A) º 0 ⇔ A º 0

(1394)

meaning, matrix A belongs to the positive semidefinite cone in the subspace of symmetric matrices if and only if its eigenvalues belong to the nonnegative orthant.

ˆ For

hA , Ai = hλ(A) , λ(A)i

(40)

µ ∈ R , A ∈ Rn×n , and vector λ(A) ∈ C n holding the ordered eigenvalues of A λ(µI + A) = µ1 + λ(A) (1395)

Proof: A = M JM −1 and µI + A = M (µI + J )M −1 where J is the Jordan form for A ; [309, 5.6, App.B] id est, δ(J ) = λ(A) , so λ(µI + A) = δ(µI + J ) because µI + J is also a Jordan form. ¨

§

¡ ¢ The symmetrization of A is (A + AT )/2. λ 21 (A + AT ) = λ(A + AT )/2. A.9 Strang agrees [309, p.334] it is not λ(A) that requires observation. Yet he is mistaken by proposing the Hermitian part alone xH(A+AH )x be tested, because the anti-Hermitian part does not vanish under complex test unless A is Hermitian. (1379) A.8

580

APPENDIX A. LINEAR ALGEBRA By similar reasoning, λ(I + µA) = 1 + λ(µA)

(1396)

For vector σ(A) holding the singular values of any matrix A σ(I + µATA) = π(|1 + µσ(ATA)|) σ(µI + ATA) = π(|µ1 + σ(ATA)|)

(1397) (1398)

where π is the nonlinear permutation-operator sorting its vector argument into nonincreasing order.

ˆ For A ∈ S

M

§

and each and every kwk = 1 [188, 7.7, prob.9] wTAw ≤ µ ⇔ A ¹ µI ⇔ λ(A) ¹ µ1

(1399)

A is normal ⇔ kAk2F = λ(A)T λ(A)

(1400)

ˆ [188, §2.5.4] (confer (39)) ˆ For A ∈ R

m×n

ATA º 0 ,

AAT º 0

(1401) xTATAx = kAxk22 ,

because, for dimensionally compatible vector x , xTAATx = kATxk22 .

ˆ For A ∈ R

n×n

and c ∈ R tr(cA) = c tr(A) = c 1T λ(A)

§

( A.1.1 no.4)

For m a nonnegative integer, (1792) m

det(A ) = tr(Am ) =

n Y

i=1 n X i=1

λ(A)m i

(1402)

λ(A)m i

(1403)

A.3. PROPER STATEMENTS

581

ˆ For A diagonalizable (§A.5), A = SΛS

−1

, (confer [309, p.255])

rank A = rank δ(λ(A)) = rank Λ

(1404)

meaning, rank is equal to the number of nonzero eigenvalues in vector λ(A) , δ(Λ)

(1405)

§

by the 0 eigenvalues theorem ( A.7.3.0.1).

ˆ (Ky Fan) For A , B ∈ S

n

§

[51, 1.2] (confer (1667))

tr(AB) ≤ λ(A)T λ(B)

(1653)

with equality (Theobald) when A and B are simultaneously diagonalizable [188] with the same ordering of eigenvalues.

ˆ For A ∈ R

m×n

and B ∈ Rn×m tr(AB) = tr(BA)

(1406)

§

and η eigenvalues of the product and commuted product are identical, including their multiplicity; [188, 1.3.20] id est, η , min{m , n}

λ(AB)1:η = λ(BA)1:η ,

(1407)

§

Any eigenvalues remaining are zero. By the 0 eigenvalues theorem ( A.7.3.0.1), rank(AB) = rank(BA) ,

AB and BA diagonalizable

(1408)

ˆ For any compatible matrices A , B [188, §0.4] ˆ For A,B ∈ S

min{rank A , rank B} ≥ rank(AB)

(1409)

n +

rank A + rank B ≥ rank(A + B) ≥ min{rank A , rank B} ≥ rank(AB) (1410)

ˆ For A,B ∈ S

n +

§

linearly independent ( B.1.1),

rank A + rank B = rank(A + B) > min{rank A , rank B} ≥ rank(AB) (1411)

582

APPENDIX A. LINEAR ALGEBRA

ˆ Because

R(ATA) = R(AT ) and R(AAT ) = R(A) (p.610), for any

A ∈ Rm×n

rank(AAT ) = rank(ATA) = rank A = rank AT

ˆ For A ∈ R

m×n

(1412)

having no nullspace, and for any B ∈ Rn×k rank(AB) = rank(B)

(1413)

Proof. For any compatible matrix C , N (CAB) ⊇ N (AB) ⊇ N (B) is obvious. By assumption ∃ A† Ä A† A = I . Let C = A† , then N (AB) = N (B) and the stated result follows by conservation of ¨ dimension (1515).

ˆ For A ∈ S

n

and any nonsingular matrix Y inertia(A) = inertia(YAY T )

(1414)

§

a.k.a, Sylvester’s law of inertia. (1456) [103, 2.4.3]

ˆ For A , B ∈ R

n×n

§

square, [188, 0.3.5] det(AB) = det(BA) det(AB) = det A det B

(1415) (1416)

Yet for A ∈ Rm×n and B ∈ Rn×m [73, p.72] det(I + AB) = det(I + BA)

ˆ For A,B ∈ S , n

(1417)

product AB is symmetric if and only if AB is

commutative;

(AB)T = AB ⇔ AB = BA Proof. (⇒) Suppose T AB = (AB) ⇒ AB = BA . (⇐) Suppose AB = BA . AB = (AB)T .

AB = (AB)T .

(1418)

(AB)T = B TAT = BA .

BA = B TAT = (AB)T .

AB = BA ⇒ ¨

Commutativity alone is insufficient for symmetry of the product. [309, p.26]

A.3. PROPER STATEMENTS

583

ˆ Diagonalizable matrices A , B ∈ R

n×n

§

commute if and only if they are simultaneously diagonalizable. [188, 1.3.12] A product of diagonal matrices is always commutative.

ˆ For A,B ∈ R

n×n

and AB = BA

xTA x ≥ 0 , xTB x ≥ 0 ∀ x ⇒ λ(A+AT )i λ(B+B T )i ≥ 0 ∀ i < xTAB x ≥ 0 ∀ x (1419) the negative result arising because of the schism between the product of eigenvalues λ(A + AT )i λ(B + B T )i and the eigenvalues of the symmetrized matrix product λ(AB + (AB)T )i . For example, X 2 is generally not positive semidefinite unless matrix X is symmetric; then (1401) applies. Simply substituting symmetric matrices changes the outcome:

ˆ For A,B ∈ S

n

and AB = BA

A º 0 , B º 0 ⇒ λ(AB)i = λ(A)i λ(B)i ≥ 0 ∀ i ⇔ AB º 0 (1420) Positive semidefiniteness of A and B is sufficient but not a necessary condition for positive semidefiniteness of the product AB .

§

§

Proof. Because all symmetric matrices are diagonalizable, ( A.5.1) [309, 5.6] we have A = SΛS T and B = T ∆T T , where Λ and ∆ are real diagonal matrices while S and T are orthogonal matrices. Because (AB)T = AB , then T must equal S , [188, 1.3] and the eigenvalues of A are ordered in the same way as those of B ; id est, λ(A)i = δ(Λ)i and λ(B)i = δ(∆)i correspond to the same eigenvector. (⇒) Assume λ(A)i λ(B)i ≥ 0 for i = 1 . . . n . AB = SΛ∆S T is symmetric and has nonnegative eigenvalues contained in diagonal matrix Λ∆ by assumption; hence positive semidefinite by (1388). Now assume A,B º 0. That, of course, implies λ(A)i λ(B)i ≥ 0 for all i because all the individual eigenvalues are nonnegative. (⇐) Suppose AB = SΛ∆S T º 0. Then Λ∆ º 0 by (1388), and so all products λ(A)i λ(B)i must be nonnegative; meaning, sgn(λ(A)) = sgn(λ(B)). We may, therefore, conclude nothing about the semidefiniteness of A and B . ¨

§

584

APPENDIX A. LINEAR ALGEBRA

ˆ For A,B ∈ S

n

and A º 0 , B º 0 (Example A.2.1.0.1)

AB = BA ⇒ λ(AB)i = λ(A)i λ(B)i ≥ 0 ∀ i ⇒ AB º 0

(1421)

AB = BA ⇒ λ(AB)i ≥ 0 , λ(A)i λ(B)i ≥ 0 ∀ i ⇔ AB º 0 (1422)

ˆ For A,B ∈ S

n

§

[189, 4.2.13] A º 0, B º 0 ⇒ A ⊗ B º 0 A ≻ 0, B ≻ 0 ⇒ A ⊗ B ≻ 0

(1423) (1424)

and the Kronecker product is symmetric.

ˆ For A,B ∈ S

n

§

[368, 6.2] A º 0 ⇒ tr A ≥ 0

(1425)

A º 0 , B º 0 ⇒ tr A tr B ≥ tr(AB) ≥ 0 (1426) √ √ Because A º 0 , B º 0 ⇒ λ(AB) = λ( AB A) º 0 by (1407) and Corollary A.3.1.0.5, then we have tr(AB) ≥ 0. A º 0 ⇔ tr(AB) ≥ 0 ∀ B º 0

ˆ For A,B,C ∈ S

n

(350)

(L¨owner)

A¹B, B ¹C ⇒ A¹ C A¹B ⇔ A+C ¹ B+C A¹B, A ºB ⇒ A=B A¹A

(transitivity) (additivity) (antisymmetry) (reflexivity)

(1427)

ˆ For A,B ∈ R

n×n

xTA x ≥ xTB x ∀ x ⇒ tr A ≥ tr B

(1428)

Proof. xTA x ≥ xTB x ∀ x ⇔ λ((A −B) + (A −B)T )/2 º 0 ⇒ tr(A +AT − (B +B T ))/2 = tr(A −B) ≥ 0. There is no converse. ¨

A.3. PROPER STATEMENTS

ˆ For A,B ∈ S

n

585

§

[368, 6.2] (Theorem A.3.1.0.4) A º B ⇒ tr A ≥ tr B

(1429)

A º B ⇒ δ(A) º δ(B)

(1430)

There is no converse, and restriction to the positive semidefinite cone does not improve the situation. All-strict versions hold.

ˆ For A,B ∈ int S

A º B º 0 ⇒ rank A ≥ rank B

(1431)

A º B º 0 ⇒ det A ≥ det B ≥ 0 A ≻ B º 0 ⇒ det A > det B ≥ 0

(1432) (1433)

n +

§

§

[32, 4.2] [188, 7.7.4]

A º B ⇔ A−1 ¹ B −1 ,

ˆ For A,B ∈ S

n

ˆ For A,B ∈ S

§

(1434)

[368, 6.2] AºBº0 ⇒

n

A ≻ 0 ⇔ A−1 ≻ 0 √ √ A º B

(1435)

§

and AB = BA [368, 6.2, prob.3] A º B º 0 ⇒ Ak º B k ,

k = 1, 2, . . .

(1436)

A.3.1.0.1 Theorem. Positive semidefinite ordering of eigenvalues. For A , B ∈ RM ×M , place matrix into ¢ ¡ of each symmetrized ¢ ¡ the eigenvalues the respective vectors λ 21 (A + AT ) , λ 21 (B + B T ) ∈ RM . Then, [309, 6] ¡ ¢ xTA x ≥ 0 ∀ x ⇔ λ A + AT º 0 (1437) ¡ ¢ T T x A x > 0 ∀ x 6= 0 ⇔ λ A + A ≻ 0 (1438) ¢ ¡ because¡ xT(A − AT¢)x = 0. (1376) Now arrange the ¡entries of λ¢ 21 (A + AT ) and λ 21 (B + B T ) in nonincreasing order so λ 12 (A + AT ) 1 holds the ¡ ¢ largest eigenvalue of symmetrized A while λ 21 (B + B T ) 1 holds the largest eigenvalue of symmetrized B , and so on. Then [188, 7.7, prob.1, prob.9] for κ ∈ R ¡ ¢ ¡ ¢ xTA x ≥ xTB x ∀ x ⇒ λ A + AT º λ B + B T (1439) ¢ ¡ xTA x ≥ xTI x κ ∀ x ⇔ λ 21 (A + AT ) º κ1

§

§

586

APPENDIX A. LINEAR ALGEBRA

Now let A , B ∈ SM have diagonalizations A = QΛQT and B = U ΥU T with λ(A) = δ(Λ) and λ(B) = δ(Υ) arranged in nonincreasing order. Then AºB AºB AºB T S AS º B

§

⇔ ⇒ : ⇐

λ(A−B) º 0 λ(A) º λ(B) λ(A) º λ(B) λ(A) º λ(B)

where S = QU T . [368, 7.5]

(1440) (1441) (1442) (1443) ⋄

§

A.3.1.0.2 Theorem. (Weyl) Eigenvalues of sum. [188, 4.3.1] M ×M For A , B ∈ R , place¡ the eigenvalues matrix into ¢ ¡ 1 of eachTsymmetrized ¢ M 1 T the respective vectors λ (A + A ) , λ (B + B ) ∈ R in nonincreasing 2 ¡ ¢ 2 order so λ 12 (A + AT ) 1 holds the largest eigenvalue of symmetrized A while ¡ ¢ λ 21 (B + B T ) 1 holds the largest eigenvalue of symmetrized B , and so on. Then, for any k ∈ {1 . . . M } ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ λ A + AT k + λ B + B T M ≤ λ (A + AT ) + (B + B T ) k ≤ λ A + AT k + λ B + B T 1 (1444) ⋄

Weyl’s theorem establishes positive semidefiniteness of a sum of positive semidefinite matrices; for A , B ∈ SM + λ(A)k + λ(B)M ≤ λ(A + B)k ≤ λ(A)k + λ(B)1

§

(1445)

Because SM + is a convex cone ( 2.9.0.0.1), then by (160) A , B º 0 ⇒ ζ A + ξ B º 0 for all ζ , ξ ≥ 0

(1446)

§

A.3.1.0.3 Corollary. Eigenvalues of sum and difference. [188, 4.3] For A ∈ SM and B ∈ SM , place the eigenvalues of each matrix into the + M respective vectors λ(A), λ(B) ∈ R in nonincreasing order so λ(A)1 holds the largest eigenvalue of A while λ(B)1 holds the largest eigenvalue of B , and so on. Then, for any k ∈ {1 . . . M } λ(A − B)k ≤ λ(A)k ≤ λ(A + B)k

(1447) ⋄

A.3. PROPER STATEMENTS

587

When B is rank-one positive semidefinite, the eigenvalues interlace; id est, for B = qq T λ(A)k−1 ≤ λ(A − qq T )k ≤ λ(A)k ≤ λ(A + qq T )k ≤ λ(A)k+1

(1448)

Theorem. Positive (semi)definite principal submatrices.A.10

A.3.1.0.4

ˆ A ∈ S is positive semidefinite if and only if all M principal submatrices of dimension M −1 are positive semidefinite and det A is nonnegative. ˆ A ∈ S is positive definite if and only if any one principal submatrix M

M

of dimension M − 1 is positive definite and det A is positive.

⋄

If any one principal submatrix of dimension M−1 is not positive definite, conversely, then A can neither be. Regardless of symmetry, if A ∈ RM ×M is positive (semi)definite, then the determinant of each and every principal submatrix is (nonnegative) positive. [253, 1.3.1]

§

A.3.1.0.5 Corollary. Positive (semi)definite symmetric products. [188, p.399]

ˆ If A ∈ S is positive definite and any particular dimensionally compatible matrix Z has no nullspace, then Z AZ is positive definite. ˆ If matrix A ∈ S is positive (semi)definite then, for any matrix Z of compatible dimension, Z AZ is positive semidefinite. ˆ A ∈ S is positive (semi)definite if and only if there exists a nonsingular Z such that Z AZ is positive (semi)definite. ˆ If A ∈ S is positive semidefinite and singular it remains possible, for M

T

M

T

M

T

M

some skinny Z ∈ RM ×N with N < M , that Z TAZ becomes positive definite.A.11 ⋄

A.10

A recursive condition for positive (semi)definiteness, this theorem is a synthesis of facts from [188, §7.2] [309, §6.3] (confer [253, §1.3.1]). A principal submatrix is formed by discarding any subset of rows and columns having the same indices. There are M !/(1!(M − 1)!) principal 1×1 submatrices, M !/(2!(M − 2)!) principal 2×2 submatrices, and so on, totaling 2M − 1 principal submatrices including A itself. By loading y in y TA y with various patterns of ones and zeros, it follows that any principal submatrix must be positive (semi)definite whenever A is. A.11 Using the interpretation in §E.6.4.3, this means coefficients of orthogonal projection of vectorized A on a subset of extreme directions from SM + determined by Z can be positive.

588

APPENDIX A. LINEAR ALGEBRA

We can deduce from these, given nonsingular matrix Z and any particular M dimensionally · ¸ compatible Y : matrix A ∈ S is positive semidefinite if and ZT only if A [ Z Y ] is positive semidefinite. In other words, from the YT Corollary it follows: for dimensionally compatible Z

ˆ A º 0 ⇔ Z AZ º 0 T

and Z T has a left inverse

Products such as Z †Z and ZZ † are symmetric and positive semidefinite although, given A º 0, Z †AZ and ZAZ † are neither necessarily symmetric or positive semidefinite.

§

§

A.3.1.0.6 Theorem. Symmetric projector semidefinite. [20, III] [21, 6] [207, p.55] For symmetric idempotent matrices P and R P,R º 0

(1449)

P º R ⇔ R(P ) ⊇ R(R) ⇔ N (P ) ⊆ N (R)

§

Projector P is never positive definite [311, 6.5, prob.20] unless it is the identity matrix. ⋄ A.3.1.0.7 Theorem. Symmetric positive semidefinite. Given real matrix Ψ with rank Ψ = 1 Ψ º 0 ⇔ Ψ = uuT

(1450)

where u is some real vector; id est, symmetry is necessary and sufficient for positive semidefiniteness of a rank-1 matrix. ⋄

§

Proof. Any rank-one matrix must have the form Ψ = uv T . ( B.1) Suppose Ψ is symmetric; id est, v = u . For all y ∈ RM , y Tu uTy ≥ 0. Conversely, suppose uv T is positive semidefinite. We know that can hold if and only if uv T + vuT º 0 ⇔ for all normalized y ∈ RM , 2 y Tu v Ty ≥ 0 ; but that is possible only if v = u . ¨ The same does not hold true for matrices of higher rank, as Example A.2.1.0.1 shows.

A.4. SCHUR COMPLEMENT

A.4

589

Schur complement

Consider Schur-form partitioned matrix G : Given AT = A and C T = C , then [55] · ¸ A B G= º0 BT C (1451) ⇔ A º 0 , B T(I −AA† ) = 0 , C −B TA† B º 0 ⇔ C º 0 , B(I − CC † ) = 0 , A−B C † B T º 0

§

where A† denotes the Moore-Penrose (pseudo)inverse ( E). In the first instance, I − AA† is a symmetric projection matrix orthogonally projecting on N (AT ). (1834) It is apparently required R(B) ⊥ N (AT )

(1452)

which precludes A = 0 when B is any nonzero matrix. Note that A ≻ 0 ⇒ A† = A−1 ; thereby, the projection matrix vanishes. Likewise, in the second instance, I − CC † projects orthogonally on N (C T ). It is required R(B T ) ⊥ N (C T )

(1453)

†

which precludes C = 0 for B nonzero. Again, C ≻ 0 ⇒ C = C get, for A or C nonsingular, · ¸ A B G= º0 BT C ⇔

−1

. So we

(1454)

A ≻ 0 , C −B TA−1 B º 0 or C ≻ 0 , A−B C −1 B T º 0

When A is full-rank then, for all B of compatible dimension, R(B) is in R(A). Likewise, when C is full-rank, R(B T ) is in R(C ). Thus the flavor, for A and C nonsingular, ¸ · A B ≻0 G= BT C (1455) ⇔ A ≻ 0 , C −B TA−1B ≻ 0 ⇔ C ≻ 0 , A−BC −1B T ≻ 0

where C − B TA−1B is called the Schur complement of A in G , while the Schur complement of C in G is A − BC −1B T . [139, 4.8]

§

590

APPENDIX A. LINEAR ALGEBRA

§

Origin of the term Schur complement is from complementary inertia: [103, 2.4.4] Define ¡ ¢ inertia G ∈ SM , {p , z , n}

(1456)

where p , z , n respectively represent number of positive, zero, and negative eigenvalues of G ; id est, M = p+z+n (1457) Then, when A is invertible, inertia(G) = inertia(A) + inertia(C − B TA−1B)

(1458)

and when C is invertible, inertia(G) = inertia(C ) + inertia(A − BC −1B T )

(1459)

When A = C = 0, denoting by σ(B) ∈ Rm + the nonincreasingly ordered m×m singular values of matrix B ∈ R , then we have the eigenvalues [51, 1.2, prob.17]

§

λ(G) = λ

µ·

0 B BT 0

¸¶

=

·

σ(B) −Ξ σ(B)

¸

(1460)

and inertia(G) = inertia(B TB) + inertia(−B TB)

(1461)

where Ξ is the order-reversing permutation matrix defined in (1654). A.4.0.0.1 Example. Nonnegative polynomial. [32, p.163] Schur-form positive semidefiniteness is sufficient for quadratic polynomial convexity in x , but it is necessary and sufficient for nonnegativity; videlicet, for all compatible x · ¸· ¸ [ xT 1 ] A b x ≥ 0 ⇔ xTA x + 2bTx + c ≥ 0 (1462) bT c 1 Sublevel set {x | xTA x + 2bTx + c ≤ 0} is convex if A º 0, but the quadratic polynomial is a convex function if and only if A º 0. 2

A.4. SCHUR COMPLEMENT

591

A.4.0.0.2 Example. Schur conditions. From (1425), ·

A B BT C

¸

º0

⇒ tr(A + C ) ≥ 0

m ·

A 0 T 0 C −B TA−1B

¸

º 0 ⇒ tr(C −B TA−1B) ≥ 0

¸

º 0 ⇒ tr(A−BC −1B T ) ≥ 0

(1463)

m ·

A−BC −1B T 0 0T C

Since tr(C −B TA−1B) ≥ 0 ⇔ tr C ≥ tr(B TA−1B) ≥ 0 for example, then minimization of tr C is necessary and sufficient for minimization of · ¸ A B tr(C −B TA−1B) when both are under constraint º 0. 2 BT C A.4.0.0.3 Example. Sparse Schur conditions. Setting matrix A to the identity simplifies the Schur conditions. consequence relates the definiteness of three quantities: ·

I B BT C

¸

T

º0 ⇔ C−B Bº0 ⇔

·

I 0 T 0 C −B TB

¸

One

º 0 (1464) 2

A.4.0.0.4 Exercise. Eigenvalues λ of sparse Schur-form. Prove: given C −B TB = 0, for B ∈ Rm×n and C ∈ S n λ

µ·

I B BT C

¸¶

i

   1 + λ(C )i , 1 ≤ i ≤ n 1, n
(1465) H

592

APPENDIX A. LINEAR ALGEBRA

A.4.0.0.5 Theorem. When symmetric matrix · A rank BT

Rank of partitioned matrices. A is invertible and C is symmetric, ¸ ¸ · A 0 B = rank 0T C −B TA−1B C

(1466)

= rank A + rank(C −B TA−1B)

§

equals rank of a block on the main diagonal plus rank of its Schur complement [368, 2.2, prob.7]. Similarly, when symmetric matrix C is invertible and A is symmetric, ¸ ¸ · · A − BC −1B T 0 A B = rank rank 0T C BT C (1467) = rank(A − BC −1B T ) + rank C

§

⋄

Proof. The first assertion (1466) holds if and only if [188, 0.4.6(c)] ¸ ¸ · · A 0 A B (1468) Y = ∃ nonsingular X, Y Ä X 0T C −B TA−1B BT C

§

Let [188, 7.7.6] T

Y = X =

·

I −A−1B 0T I

¸

(1469) ¨

A.4.0.0.6 Lemma. Rank of Schur-form block. [126] [124] Matrix B ∈ Rm×n has rank B ≤ ρ if and only if there exist matrices A ∈ Sm and C ∈ S n such that · ¸ · ¸ A 0 A B rank ≤ 2ρ and G= º0 (1470) 0T C BT C ⋄ Schur-form positive semidefiniteness alone implies rank A ≥ rank B and rank C ≥ rank B . But, even in absence of semidefiniteness, we must always have rank G ≥ rank A , rank B , rank C by fundamental linear algebra.

A.4. SCHUR COMPLEMENT

A.4.1

593

Determinant G=

·

A B BT C

¸

(1471)

We consider again a matrix G partitioned like (1451), but not necessarily positive (semi)definite, where A and C are symmetric.

ˆ When A is invertible,

det G = det A det(C − B TA−1B)

(1472)

When C is invertible, det G = det C det(A − B C −1B T )

(1473)

det G 6= 0 ⇔ A + BB T ≻ 0

(1474)

ˆ When B is full-rank and skinny, C = 0, and A º 0, then [57, §10.1.1] When B is a (column) vector, then for all C ∈ R and all A of dimension compatible with G det G = det(A)C − B TAT cof B while for C 6= 0 where Acof

(1475)

1 BB T ) (1476) C is the matrix of cofactors [309, 4] corresponding to A . det G = C det(A −

§

ˆ When B is full-rank and fat, A = 0, and C º 0, then det G 6= 0 ⇔ C + B TB ≻ 0

(1477)

When B is a row-vector, then for A 6= 0 and all C of dimension compatible with G det G = A det(C −

1 T B B) A

(1478)

while for all A ∈ R T det G = det(C )A − B Ccof BT

where Ccof is the matrix of cofactors corresponding to C .

(1479)

594

APPENDIX A. LINEAR ALGEBRA

A.5

Eigen decomposition

§

When a square matrix X ∈ Rm×m is diagonalizable, [309, 5.6] then X = SΛS −1

 w1T m X . λ i si wiT = [ s1 · · · sm ] Λ  ..  = T i=1 wm 

(1480)

where {si ∈ N (X − λ i I ) ⊆ Cm } are l.i. (right-)eigenvectorsA.12 constituting the columns of S ∈ Cm×m defined by XS = SΛ

(1481)

{wi ∈ N (X T − λ i I ) ⊆ Cm } are linearly independent left-eigenvectors of X (eigenvectors of X T ) constituting the rows of S −1 defined by [188] S −1X = ΛS −1

(1482)

and where {λ i ∈ C} are eigenvalues (1405) δ(λ(X)) = Λ ∈ Cm×m

(1483)

corresponding to both left and right eigenvectors; id est, λ(X) = λ(X T ). There is no connection between diagonalizability and invertibility of X . [309, 5.2] Diagonalizability is guaranteed by a full set of linearly independent eigenvectors, whereas invertibility is guaranteed by all nonzero eigenvalues.

§

distinct eigenvalues ⇒ l.i. eigenvectors ⇔ diagonalizable not diagonalizable ⇒ repeated eigenvalue

(1484)

A.5.0.0.1 Theorem. Real eigenvector. Eigenvectors of a real matrix corresponding to real eigenvalues must be real. ⋄ Proof. Ax = λx . Given λ = λ∗ , xHAx = λxHx = λkxk2 = xTA x∗ ⇒ x = x∗ , where xH = x∗T . The converse is equally simple. ¨ A.12

Eigenvectors must, of course, be nonzero. The prefix eigen is from the German; in this context meaning, something akin to “characteristic”. [306, p.14]

A.5. EIGEN DECOMPOSITION A.5.0.1

595

Uniqueness

From the fundamental theorem of algebra, [204] which guarantees existence of zeros for a given polynomial, it follows: eigenvalues, including their multiplicity, for a given square matrix are unique; meaning, there is no other set of eigenvalues for that matrix. (Conversely, many different matrices may share the same unique set of eigenvalues; e.g., for any X , λ(X) = λ(X T ).) Uniqueness of eigenvectors, in contrast, disallows multiplicity of the same direction: A.5.0.1.1 Definition. Unique eigenvectors. When eigenvectors are unique, we mean: unique to within a real nonzero scaling, and their directions are distinct. △ If S is a matrix of eigenvectors of X as in (1480), for example, then −S is certainly another matrix of eigenvectors decomposing X with the same eigenvalues. For any square matrix, the eigenvector corresponding to a distinct eigenvalue is unique; [306, p.220] distinct eigenvalues ⇒ eigenvectors unique

(1485)

Eigenvectors corresponding to a repeated eigenvalue are not unique for a diagonalizable matrix; repeated eigenvalue ⇒ eigenvectors not unique

(1486)

Proof follows from the observation: any linear combination of distinct eigenvectors of diagonalizable X , corresponding to a particular eigenvalue, produces another eigenvector. For eigenvalue λ whose multiplicityA.13 dim N (X − λI ) exceeds 1, in other words, any choice of independent vectors from N (X − λI ) (of the same multiplicity) constitutes eigenvectors corresponding to λ . ¨ A.13

A matrix is diagonalizable iff algebraic multiplicity (number of occurrences of same eigenvalue) equals geometric multiplicity dim N (X − λI ) = m − rank(X − λI ) [306, p.15] (number of Jordan blocks w.r.t λ or number of corresponding l.i. eigenvectors).

596

APPENDIX A. LINEAR ALGEBRA

Caveat diagonalizability insures linear independence which implies existence of distinct eigenvectors. We may conclude, for diagonalizable matrices, distinct eigenvalues ⇔ eigenvectors unique (1487) A.5.0.2

Invertible matrix

When diagonalizable matrix X ∈ Rm×m is nonsingular (no zero eigenvalues), then it has an inverse obtained simply by inverting eigenvalues in (1480): X −1 = S Λ−1 S −1 A.5.0.3

(1488)

eigenmatrix

The (right-)eigenvectors {si } (1480) are naturally orthogonal wiTsj = 0 to left-eigenvectors {wi } except, for i = 1 . . . m , wiTsi = 1 ; called a biorthogonality condition [340, 2.2.4] [188] because neither set of left or right eigenvectors is necessarily an orthogonal set. Consequently, each dyad from a diagonalization is an independent ( B.1.1) nonorthogonal projector because

§

§

si wiT si wiT = si wiT

(1489)

(whereas the dyads of singular value decomposition are not inherently projectors (confer (1495))). The dyads of eigen decomposition can be termed eigenmatrices because X si wiT = λ i si wiT

(1490)

Sum of the eigenmatrices is the identity; m X

si wiT = I

(1491)

i=1

A.5.1

Symmetric matrix diagonalization

§

The set of normal matrices is, precisely, that set of all real matrices having a complete orthonormal set of eigenvectors; [368, 8.1] [311, prob.10.2.31] id est, any matrix X for which XX T = X TX ; [146, 7.1.3] [306, p.3] e.g., orthogonal and circulant matrices [154]. All normal matrices are diagonalizable. A symmetric matrix is a special normal matrix whose

§

A.6. SINGULAR VALUE DECOMPOSITION, SVD

597

§

eigenvalues must be realA.14 and whose eigenvectors can be chosen to make a real orthonormal set; [311, 6.4] [309, p.315] id est, for X ∈ Sm  T  s1 m X . T . λ i si sT (1492) X = SΛS = [ s1 · · · sm ] Λ  .  = i T i=1 sm

§

where δ 2(Λ) = Λ ∈ Sm ( A.1) and S −1 = S T ∈ Rm×m (orthogonal matrix, B.5) because of symmetry: SΛS −1 = S −TΛS T . Because arrangement of eigenvectors and their corresponding eigenvalues is arbitrary, we almost always arrange eigenvalues in nonincreasing order as is the convention for singular value decomposition. Then to diagonalize a symmetric matrix that is already a diagonal matrix, orthogonal matrix S becomes a permutation matrix.

§

A.5.1.1

Invertible symmetric matrix

When symmetric matrix X ∈ Sm is nonsingular (invertible), then its inverse (obtained by inverting eigenvalues in (1492)) is also symmetric: X −1 = S Λ−1 S T ∈ Sm A.5.1.2

(1493)

Positive semidefinite matrix square root

When X ∈ Sm + , its unique positive semidefinite matrix square root is defined √ √ X , S Λ S T ∈ Sm (1494) + √ where the square root of nonnegative diagonal matrix Λ is taken entrywise √ √ and positive. Then X = X X .

A.6 A.6.1

§

Singular value decomposition, SVD Compact SVD

[146, 2.5.4] For any A ∈ Rm×n Proof. Suppose λi is an eigenvalue corresponding to eigenvector si of real A = AT . T ∗ ∗T T ∗ ∗ ∗ ∗ Then sH i Asi = si Asi (by transposition) ⇒ si λi si = si λi si because (Asi ) = (λi si ) by 2 ∗ 2 assumption. So we have λi ksi k = λi ksi k . There is no converse. ¨

A.14

598

APPENDIX A. LINEAR ALGEBRA  q1T η P . A = U ΣQT = [ u1 · · · uη ] Σ  ..  = σi ui qiT i=1 qηT 

U ∈ Rm×η ,

Σ ∈ Rη×η ,

(1495)

Q ∈ Rn×η

where U and Q are always skinny-or-square each having orthonormal columns, and where η , min{m , n} (1496)

§

Square matrix Σ is diagonal ( A.1.1) δ 2(Σ) = Σ ∈ Rη×η

(1497)

ρ , rank A = rank Σ

(1499)

holding the singular values {σi ∈ R} of A which are always arranged in nonincreasing order by convention and are related to eigenvalues λ byA.15 p ³√ ´ ´ ³√ p  λ(ATA)i = λ(AAT )i = λ ATA = λ AAT > 0 , 1 ≤ i ≤ ρ i i σ(A)i = σ(AT )i =  0, ρ
§

A.6.2

Subcompact SVD

Some authors allow only nonzero singular values. In that case the compact decomposition can be made smaller; it can be redimensioned in terms of rank ρ because, for any A ∈ Rm×n ρ = rank A = rank Σ = max {i ∈ {1 . . . η} | σi 6= 0} ≤ η

A.15

When matrix A is normal, σ(A) =³|λ(A)|.´[368, §8.1] q p T For η = n , σ(A) = λ(A A) = λ ATA . q ´ ³p For η = m , σ(A) = λ(AAT ) = λ AAT .

A.16

(1501)

A.6. SINGULAR VALUE DECOMPOSITION, SVD

599

ˆ There are η singular values. For any flavor SVD, rank is equivalent to the number of nonzero singular values on the main diagonal of Σ .

Now  q1T ρ P . A = U ΣQT = [ u1 · · · uρ ] Σ  ..  = σi ui qiT i=1 T qρ 

U ∈ Rm×ρ ,

Σ ∈ Rρ×ρ ,

(1502)

Q ∈ Rn×ρ

where the main diagonal of diagonal matrix Σ has no 0 entries, and R{ui } = R(A) R{qi } = R(AT )

A.6.3

(1503)

Full SVD

Another common and useful expression of the SVD makes U and Q square; making the decomposition larger than compact SVD. Completing the nullspace bases in U and Q from (1500) provides what is called the full singular value decomposition of A ∈ Rm×n [309, App.A]. Orthonormal matrices U and Q become orthogonal matrices ( B.5):

§

R{ui | σi 6= 0} = R{ui | σi = 0} = R{qi | σi 6= 0} = R{qi | σi = 0} =

R(A) N (AT ) R(AT ) N (A)

For any matrix A having rank ρ (= rank Σ)  T  q1 η P ..  T  A = U ΣQ = [ u1 · · · um ] Σ σi ui qiT = . i=1 qnT  £ = m×ρ basis R(A)

¤  m×m−ρ basis N (AT )   U ∈ Rm×m ,

Σ ∈ Rm×n ,

(1504)

σ1 σ2

...

Q ∈ Rn×n



 ¡ ¢  T T  n×ρ basis R(A )     (n×n−ρ basis N (A))T (1505)

600

APPENDIX A. LINEAR ALGEBRA

where upper limit of summation η is defined in (1496). Matrix Σ is no longer necessarily square, now padded with respect to (1497) by m−η zero rows or n−η zero columns; the nonincreasingly ordered (possibly 0) singular values appear along its main diagonal as for compact SVD (1498). An important geometrical interpretation of SVD is given in Figure 147 for m = n = 2 : The image of the unit sphere under any m × n matrix multiplication is an ellipse. Considering the three factors of the SVD separately, note that QT is a pure rotation of the circle. Figure 147 shows how the axes q1 and q2 are first rotated by QT to coincide with the coordinate axes. Second, the circle is stretched by Σ in the directions of the coordinate axes to form an ellipse. The third step rotates the ellipse by U into its final position. Note how q1 and q2 are rotated to end up as u1 and u2 , the principal axes of the final ellipse. A direct calculation shows that Aqj = σj uj . Thus qj is first rotated to coincide with the j th coordinate axis, stretched by a factor σj , and then rotated to point in the direction of uj . All of this is beautifully illustrated for 2 × 2 matrices by the Matlab code eigshow.m (see [312]). A direct consequence of the geometric interpretation is that the largest singular value σ1 measures the “magnitude” of A (its 2-norm): kAk2 = sup kAxk2 = σ1

(1506)

kxk2 =1

This means that kAk2 is the length of the longest principal semiaxis of the ellipse. Expressions for U , Q , and Σ follow readily from (1505), AAT U = U ΣΣT and ATAQ = QΣT Σ

(1507)

demonstrating that the columns of U are the eigenvectors of AAT and the columns of Q are the eigenvectors of ATA . −Muller, Magaia, & Herbst [252]

A.6.4

Pseudoinverse by SVD

§

Matrix pseudoinverse ( E) is nearly synonymous with singular value decomposition because of the elegant expression, given A = U ΣQT ∈ Rm×n A† = QΣ†T U T ∈ Rn×m

(1508)

A.6. SINGULAR VALUE DECOMPOSITION, SVD

601

q2

q1 QT

q2

q1

q2

u2 U

Σ

u1

q1

Figure 147: Geometrical interpretation of full SVD [252]: Image of circle {x ∈ R2 | kxk2 = 1} under matrix multiplication Ax is, in general, an ellipse. For the example illustrated, U , [ u1 u2 ] ∈ R2×2 , Q , [ q1 q2 ] ∈ R2×2 .

602

APPENDIX A. LINEAR ALGEBRA

that applies to all three flavors of SVD, where Σ† simply inverts nonzero entries of matrix Σ . Given symmetric matrix A ∈ S n and its diagonalization A = SΛS T ( A.5.1), its pseudoinverse simply inverts all nonzero eigenvalues:

§

A† = SΛ† S T ∈ S n

A.6.5

(1509)

SVD of symmetric matrices

From (1498) and (1494) for A = AT p ³√ ´  λ(A2 )i = λ A2 = |λ(A)i | > 0 , i σ(A)i =  0,

1≤i≤ρ ρ
(1510)

§

A.6.5.0.1 Definition. Step function. (confer 4.3.2.0.1) n n Define the signum-like quasilinear function ψ : R → R that takes value 1 corresponding to a 0-valued entry in its argument: ¸ ½ · xi 1 , ai ≥ 0 , i = 1 . . . n ∈ Rn (1511) = ψ(a) , lim −1 , ai < 0 xi →ai |xi | △ Eigenvalue signs of a symmetric matrix having diagonalization A = SΛS T (1492) can be absorbed either into real U or real Q from the full SVD; [325, p.34] (confer C.4.2.1)

§

or

A = SΛS T = Sδ(ψ(δ(Λ))) |Λ|S T , U ΣQT ∈ S n

(1512)

A = SΛS T = S|Λ| δ(ψ(δ(Λ)))S T , U Σ QT ∈ S n

(1513)

where matrix of singular values Σ = |Λ| denotes entrywise absolute value of diagonal eigenvalue matrix Λ .

A.7 A.7.1

Zeros norm zero

For any given norm, by definition, kxkℓ = 0

⇔

x=0

(1514)

A.7. ZEROS

603

Consequently, a generally nonconvex constraint in x like kAx − bk = κ becomes convex when κ = 0.

A.7.2

0 entry

If a positive semidefinite matrix A = [Aij ] ∈ Rn×n has a 0 entry Aii on its main diagonal, then Aij + Aji = 0 ∀ j . [253, 1.3.1] Any symmetric positive semidefinite matrix having a 0 entry on its main diagonal must be 0 along the entire row and column to which that 0 entry belongs. [146, 4.2.8] [188, 7.1, prob.2]

§

§

A.7.3

§

0 eigenvalues theorem

This theorem is simple, powerful, and widely applicable: A.7.3.0.1 Theorem. Number of 0 eigenvalues. For any matrix A ∈ Rm×n rank(A) + dim N (A) = n

§

(1515)

by conservation of dimension. [188, 0.4.4] For any square matrix A ∈ Rm×m , the number of 0 eigenvalues is at least equal to dim N (A) dim N (A) ≤ number of 0 eigenvalues ≤ m

(1516)

§

while the eigenvectors corresponding to those 0 eigenvalues belong to N (A). [309, 5.1]A.17 A.17

We take as given the well-known fact that the number of 0 eigenvalues cannot be less than dimension of the nullspace. We offer an example of the converse:   1 0 1 0  0 0 1 0   A=  0 0 0 0  1 0 0 0

dim N (A) = 2, λ(A) = [ 0 0 0 1 ]T ; three eigenvectors in the nullspace but only two are independent. The right-hand side of (1516) is tight for nonzero matrices; e.g., (§B.1) dyad uv T ∈ Rm×m has m 0-eigenvalues when u ∈ v ⊥ .

604

APPENDIX A. LINEAR ALGEBRA

§

For diagonalizable matrix A ( A.5), the number of 0 eigenvalues is precisely dim N (A) while the corresponding eigenvectors span N (A). The real and imaginary parts of the eigenvectors remaining span R(A). (TRANSPOSE.) Likewise, for any matrix A ∈ Rm×n rank(AT ) + dim N (AT ) = m

(1517)

For any square A ∈ Rm×m , the number of 0 eigenvalues is at least equal to dim N (AT ) = dim N (A) while the left-eigenvectors (eigenvectors of AT ) corresponding to those 0 eigenvalues belong to N (AT ). For diagonalizable A , the number of 0 eigenvalues is precisely dim N (AT ) while the corresponding left-eigenvectors span N (AT ). The real and imaginary parts of the left-eigenvectors remaining span R(AT ). ⋄ Proof. First we show, for a diagonalizable matrix, the number of 0 eigenvalues is precisely the dimension of its nullspace while the eigenvectors corresponding to those 0 eigenvalues span the nullspace: Any diagonalizable matrix A ∈ Rm×m must possess a complete set of linearly independent eigenvectors. If A is full-rank (invertible), then all m = rank(A) eigenvalues are nonzero. [309, 5.1] Suppose rank(A) < m . Then dim N (A) = m−rank(A). Thus there is a set of m−rank(A) linearly independent vectors spanning N (A). Each of those can be an eigenvector associated with a 0 eigenvalue because A is diagonalizable ⇔ ∃ m linearly independent eigenvectors. [309, 5.2] Eigenvectors of a real matrix corresponding to 0 eigenvalues must be real.A.18 Thus A has at least m−rank(A) eigenvalues equal to 0. Now suppose A has more than m−rank(A) eigenvalues equal to 0. Then there are more than m−rank(A) linearly independent eigenvectors associated with 0 eigenvalues, and each of those eigenvectors must be in N (A). Thus there are more than m−rank(A) linearly independent vectors in N (A) ; a contradiction. Diagonalizable A therefore has rank(A) nonzero eigenvalues and exactly m−rank(A) eigenvalues equal to 0 whose corresponding eigenvectors span N (A).

§

§

Proof. Let ∗ denote complex conjugation. Suppose A = A∗ and Asi = 0. Then si = s∗i ⇒ Asi = As∗i ⇒ As∗i = 0. Conversely, As∗i = 0 ⇒ Asi = As∗i ⇒ si = s∗i . ¨

A.18

A.7. ZEROS

605

By similar argument, the left-eigenvectors corresponding to 0 eigenvalues span N (AT ). Next we show when A is diagonalizable, the real and imaginary parts of its eigenvectors (corresponding to nonzero eigenvalues) span R(A) : The right-eigenvectors of a diagonalizable matrix A ∈ Rm×m are linearly independent if and only if the left-eigenvectors are. So, matrix A has a representation in terms of its right- and left-eigenvectors; from the diagonalization (1480), assuming 0 eigenvalues are ordered last, A =

m X i=1

λ i si wiT

=

kX ≤m

λ i si wiT

(1518)

i=1 λ i 6= 0

§

From the linearly independent dyads theorem ( B.1.1.0.2), the dyads {si wiT } must be independent because each set of eigenvectors are; hence rank A = k , the number of nonzero eigenvalues. Complex eigenvectors and eigenvalues are common for real matrices, and must come in complex conjugate pairs for the summation to remain real. Assume that conjugate pairs of eigenvalues appear in sequence. Given any particular conjugate pair from (1518), we get the partial summation λ i si wiT + λ∗i s∗i wi∗T = 2¡Re(λ i si wiT ) ¢ = 2 Re si Re(λ i wiT ) − Im si Im(λ i wiT )

whereA.19 λ∗i , λ i+1 , equivalently written A = 2

X

i λ∈C λ i 6= 0

s∗i , si+1 , and wi∗ , wi+1 .

T T Re s2i Re(λ2i w2i ) − Im s2i Im(λ2i w2i ) +

X

(1519)

Then (1518) is

λ j sj wjT (1520)

j λ∈R λ j 6= 0

The summation (1520) shows: A is a linear combination of real and imaginary parts of its right-eigenvectors corresponding to nonzero eigenvalues. The k vectors {Re si ∈ Rm , Im si ∈ Rm | λ i 6= 0, i ∈ {1 . . . m}} must therefore span the range of diagonalizable matrix A . The argument is similar regarding the span of the left-eigenvectors. ¨ A.19

Complex conjugate of w is denoted w∗ . Conjugate transpose is denoted wH = w∗T .

606

APPENDIX A. LINEAR ALGEBRA

A.7.4

0 trace and matrix product

×N For X,A ∈ RM (34) +

tr(X TA) = 0 ⇔ X ◦ A = A ◦ X = 0

§

(1521)

§

For X,A ∈ SM + [32, 2.6.1, exer.2.8] [334, 3.1] tr(XA) = 0 ⇔ XA = AX = 0

(1522)

Proof. (⇐) Suppose XA = AX = 0. Then tr(XA) = 0 is obvious. √ √ (⇒) Suppose tr(XA) = 0. tr(XA) = tr( A X A) whose argument is positive semidefinite by Corollary A.3.1.0.5. Trace of any square matrix is equivalent to the sum of its eigenvalues. Eigenvalues of a positive semidefinite matrix can total 0 if and only if each and every nonnegative eigenvalue is 0. The only feasible positive semidefinite matrix, having all 0 eigenvalues, resides at the origin; (confer (1546)) id est, ³√ √ ´T √ √ √ √ AX A = X A X A = 0 (1523) √ √ √ √ √ √ implying X A = 0 which in turn implies X( X A) A = XA = 0. Arguing similarly yields AX = 0. ¨

§

Diagonalizable matrices A and X are simultaneously diagonalizable if and only if they are commutative under multiplication; [188, 1.3.12] id est, iff they share a complete set of eigenvectors. A.7.4.0.1 Example. An equivalence in nonisomorphic spaces. Identity (1522) leads to an unusual equivalence relating convex geometry to traditional linear algebra: The convex sets, given A º 0 {X | hX , Ai = 0} ∩ {X º 0} ≡ {X | N (X) ⊇ R(A)} ∩ {X º 0} (1524) (one expressed in terms of a hyperplane, the other in terms of nullspace and range) are equivalent only when symmetric matrix A is positive semidefinite. We might apply this equivalence to the geometric center subspace, for example, M SM | Y 1 = 0} c = {Y ∈ S

= {Y ∈ SM | N (Y ) ⊇ 1} = {Y ∈ SM | R(Y ) ⊆ N (1T )}

(1921)

A.7. ZEROS

607

from which we derive (confer (940)) M T SM c ∩ S+ ≡ {X º 0 | hX , 11 i = 0}

(1525) 2

A.7.5

Zero definite

The domain over which an arbitrary real matrix A is zero definite can exceed its left and right nullspaces. For any positive semidefinite matrix A ∈ RM ×M (for A + AT º 0) {x | xTAx = 0} = N (A + AT ) (1526) because ∃ R Ä A+AT = RTR , kRxk = 0 ⇔ Rx = 0, and N (A+AT ) = N (R). T Then given any particular vector xp , xT p Axp = 0 ⇔ xp ∈ N (A + A ). For any positive definite matrix A (for A + AT ≻ 0)

§

{x | xTAx = 0} = 0

(1527)

{x | xTAx = 0} = RM ⇔ AT = −A

(1528)

{x | xHAx = 0} = CM ⇔ A = 0

(1529)

Further, [368, 3.2, prob.5]

while The positive semidefinite matrix ¸ · 1 2 A= 0 1

(1530)

for example, has no nullspace. Yet {x | xTAx = 0} = {x | 1T x = 0} ⊂ R2

(1531)

which is the nullspace of the symmetrized matrix. Symmetric matrices are not spared from the excess; videlicet, · ¸ 1 2 B= (1532) 2 1

608

APPENDIX A. LINEAR ALGEBRA

has eigenvalues {−1, 3} , no nullspace, but is zero definite onA.20 √ X , {x ∈ R2 | x2 = (−2 ± 3 )x1 }

(1533)

§

A.7.5.0.1 Proposition. (Sturm/Zhang) Dyad-decompositions. [314, 5.2] Let positive semidefinite matrix X ∈ SM + have rank ρ . Then given symmetric M matrix A ∈ S , hA , X i = 0 if and only if there exists a dyad-decomposition X=

ρ X

xj xjT

(1534)

j=1

satisfying

hA , xj xjT i = 0 for each and every j ∈ {1 . . . ρ}

(1535) ⋄

The dyad-decomposition of X proposed is generally not that obtained from a standard diagonalization by eigen decomposition, unless ρ = 1 or the given matrix A is diagonalizable simultaneously ( A.7.4) with X . That means, elemental dyads in decomposition (1534) constitute a generally nonorthogonal set. Sturm & Zhang give a simple procedure for constructing the dyad-decomposition [Wıκımization]; matrix A may be regarded as a parameter.

§

§

A.7.5.0.2 Example. Dyad. The dyad uv T ∈ RM ×M ( B.1) is zero definite on all x for which either xTu = 0 or xTv = 0 ; {x | xT uv T x = 0} = {x | xTu = 0} ∪ {x | v Tx = 0}

(1536)

id est, on u⊥ ∪ v ⊥ . Symmetrizing the dyad does not change the outcome: {x | xT(uv T + vuT )x/2 = 0} = {x | xTu = 0} ∪ {x | v Tx = 0}

(1537) 2

A.20

These two lines represent the limit in the union of two generally distinct hyperbolae; id est, for matrix B and set X as defined lim {x ∈ R2 | xTB x = ε} = X

ε→0+

Appendix B Simple matrices Mathematicians also attempted to develop algebra of vectors but there was no natural definition of the product of two vectors that held in arbitrary dimensions. The first vector algebra that involved a noncommutative vector product (that is, v×w need not equal w×v) was proposed by Hermann Grassmann in his book Ausdehnungslehre (1844). Grassmann’s text also introduced the product of a column matrix and a row matrix, which resulted in what is now called a simple or a rank-one matrix. In the late 19th century the American mathematical physicist Willard Gibbs published his famous treatise on vector analysis. In that treatise Gibbs represented general matrices, which he called dyadics, as sums of simple matrices, which Gibbs called dyads. Later the physicist P. A. M. Dirac introduced the term “bra-ket” for what we now call the scalar product of a “bra” (row) vector times a “ket” (column) vector and the term “ket-bra” for the product of a ket times a bra, resulting in what we now call a simple matrix, as above. Our convention of identifying column matrices and vectors was introduced by physicists in the 20th century. −Marie A. Vitulli [343]

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

609

610

APPENDIX B. SIMPLE MATRICES

B.1

Rank-one matrix (dyad)

Any matrix formed from the unsigned outer product of two vectors, Ψ = uv T ∈ RM ×N

(1538)

where u ∈ RM and v ∈ RN , is rank-one and called a dyad. Conversely, any rank-one matrix must have the form Ψ . [188, prob.1.4.1] Product −uv T is a negative dyad. For matrix products AB T , in general, we have R(AB T ) ⊆ R(A) ,

§ §

N (AB T ) ⊇ N (B T )

(1539)

with equality when B = A [309, 3.3, 3.6]B.1 or respectively when B is invertible and N (A) = 0. Yet for all nonzero dyads we have R(uv T ) = R(u) ,

N (uv T ) = N (v T ) ≡ v ⊥

(1540)

where dim v ⊥ = N − 1. It is obvious a dyad can be 0 only when u or v is 0; Ψ = uv T = 0 ⇔ u = 0 or v = 0

(1541)

The matrix 2-norm for Ψ is equivalent to the Frobenius norm; kΨk2 = σ1 = kuv T kF = kuv T k2 = kuk kvk

(1542)

When u and v are normalized, the pseudoinverse is the transposed dyad. Otherwise, vuT † T † Ψ = (uv ) = (1543) kuk2 kvk2 B.1

Proof. R(AAT ) ⊆ R(A) is obvious. R(AAT ) = {AAT y | y ∈ Rm } ⊇ {AAT y | ATy ∈ R(AT )} = R(A) by (135)

¨

B.1. RANK-ONE MATRIX (DYAD) R(v)

611

r0

0r

R(Ψ) = R(u)

N (uT )

N (Ψ) = N (v T ) RN = R(v) ⊕ N (uv T )

N (uT ) ⊕ R(uv T ) = RM

§

Figure 148: The four fundamental subspaces [311, 3.6] of any dyad Ψ = uv T ∈ RM ×N . Ψ(x) , uv T x is a linear mapping from RN to RM . The map from R(v) to R(u) is bijective. [309, 3.1]

§

When dyad uv T ∈ RN ×N is square, uv T has at least N − 1 0-eigenvalues and corresponding eigenvectors spanning v ⊥ . The remaining eigenvector u spans the range of uv T with corresponding eigenvalue λ = v Tu = tr(uv T ) ∈ R

(1544)

Determinant is a product of the eigenvalues; so, it is always true that det Ψ = det(uv T ) = 0

(1545)

§

When λ = 1, the square dyad is a nonorthogonal projector projecting on its range (Ψ2 = Ψ , E.6); a projector dyad. It is quite possible that u ∈ v ⊥ making the remaining eigenvalue instead 0 ;B.2 λ = 0 together with the first N − 1 0-eigenvalues; id est, it is possible uv T were nonzero while all its eigenvalues are 0. The matrix · ¸ · ¸ 1 1 1 [1 1] = (1546) −1 −1 −1 for example, has two 0-eigenvalues. In other words, eigenvector u may simultaneously be a member of the nullspace and range of the dyad. The explanation is, simply, because u and v share the same dimension, dim u = M = dim v = N : B.2

A dyad is not always diagonalizable (§A.5) because its eigenvectors are not necessarily independent.

612

APPENDIX B. SIMPLE MATRICES

Proof. Figure 148 shows the four fundamental subspaces for the dyad. Linear operator Ψ : RN → RM provides a map between vector spaces that remain distinct when M = N ; u ∈ R(uv T )

u ∈ N (uv T ) ⇔ v Tu = 0 R(uv T ) ∩ N (uv T ) = ∅

B.1.0.1

¨

rank-one modification

If A ∈ RN ×N is any nonsingular matrix and 1 + v TA−1 u 6= 0, [206, App.6] [368, 2.3, prob.16] [139, 4.11.2] (Sherman-Morrison)

§

§

(A + uv T )−1 = A−1 − B.1.0.2

(1547)

A−1 uv T A−1 1 + v TA−1 u

then

(1548)

dyad symmetry

In the specific circumstance that v = u , then uuT ∈ RN ×N is symmetric, rank-one, and positive semidefinite having exactly N − 1 0-eigenvalues. In fact, (Theorem A.3.1.0.7) uv T º 0 ⇔ v = u

(1549)

and the remaining eigenvalue is almost always positive; λ = uT u = tr(uuT ) > 0 unless u = 0 The matrix

·

Ψ u uT 1

¸

(1550) (1551)

for example, is rank-1 positive semidefinite if and only if Ψ = uuT .

B.1.1

Dyad independence

Now we consider a sum of dyads like (1538) as encountered in diagonalization and singular value decomposition: ! Ã k k k X X ¢ X ¡ T T R(si ) ⇐ wi ∀ i are l.i. (1552) R s i wi = R s i wi = i=1

i=1

i=1

B.1. RANK-ONE MATRIX (DYAD)

613

range of summation is the vector sum of ranges.B.3 (Theorem B.1.1.1.1) Under the assumption the dyads are linearly independent (l.i.), then the vector sums are unique (p.742): for {wi } l.i. and {si } l.i. Ã k ! X ¡ ¢ ¡ ¢ R si wiT = R s1 w1T ⊕ . . . ⊕ R sk wkT = R(s1 ) ⊕ . . . ⊕ R(sk ) (1553) i=1

B.1.1.0.1 Definition. Linearly independent dyads. [196, p.29, thm.11] [316, p.2] The set of k dyads © ª si wiT | i = 1 . . . k (1554)

where si ∈ CM and wi ∈ CN , is said to be linearly independent iff ! Ã k X si wiT = k rank SW T ,

(1555)

i=1

where S , [s1 · · · sk ] ∈ CM ×k and W , [w1 · · · wk ] ∈ CN ×k .

△

Dyad independence does not preclude existence of a nullspace N (SW T ) , as defined, nor does it imply SW T were full-rank. In absence of assumption of independence, generally, rank SW T ≤ k . Conversely, any rank-k matrix can be written in the form SW T by singular value decomposition. ( A.6)

§

B.1.1.0.2 Theorem. Linearly independent (l.i.) dyads. Vectors {si ∈ CM , i = 1 . . . k} are l.i. and vectors {wi ∈ CN , i = 1 . . . k} are l.i. if and only if dyads {si wiT ∈ CM ×N , i = 1 . . . k} are l.i. ⋄ Proof. Linear independence of k dyads is identical to definition (1555). (⇒) Suppose {si } and {wi } are each linearly independent sets. Invoking Sylvester’s rank inequality, [188, 0.4] [368, 2.4]

§

§

rank S + rank W − k ≤ rank(SW T ) ≤ min{rank S , rank W } (≤ k) (1556) Then k ≤ rank(SW T ) ≤ k which implies the dyads are independent. (⇐) Conversely, suppose rank(SW T ) = k . Then k ≤ min{rank S , rank W } ≤ k implying the vector sets are each independent. B.3

(1557) ¨

Move of range R to inside the summation depends on linear independence of {wi }.

614

APPENDIX B. SIMPLE MATRICES

B.1.1.1

Biorthogonality condition, Range and Nullspace of Sum

Dyads characterized by a biorthogonality condition W TS = I are independent; id est, for S ∈ CM ×k and W ∈ CN ×k , if W TS = I then rank(SW T ) = k by the linearly independent dyads theorem because (confer E.1.1)

§

W TS = I ⇒ rank S = rank W = k ≤ M = N

(1558)

To see that, we need only show: N (S) = 0 ⇔ ∃ B Ä BS = I .B.4 (⇐) Assume BS = I . Then N (BS) = 0 = {x | BSx = 0} ⊇ N (S). (1539) (⇒) If N (S) = 0·then¸S must be full-rank skinny-or-square. B ∴ ∃ A, B, C Ä [ S A ] = I (id est, [ S A ] is invertible) ⇒ BS = I . C Left inverse B is given as W T here. Because of reciprocity with S , it immediately follows: N (W ) = 0 ⇔ ∃ S Ä S T W = I . ¨

§

Dyads produced by diagonalization, for example, are independent because of their inherent biorthogonality. ( A.5.0.3) The converse is generally false; id est, linearly independent dyads are not necessarily biorthogonal. B.1.1.1.1 Theorem. Nullspace and range of dyad sum. Given a sum of dyads represented by SW T where S ∈ CM ×k and W ∈ CN ×k N (SW T ) = N (W T ) ⇐ R(SW T ) = R(S) ⇐

∃ B Ä BS = I

∃ Z Ä W TZ = I

(1559) ⋄

Proof. (⇒) N (SW T ) ⊇ N (W T ) and R(SW T ) ⊆ R(S) are obvious. (⇐) Assume the existence of a left inverse B ∈ Rk×N and a right inverse Z ∈ RN ×k .B.5 N (SW T ) = {x | SW Tx = 0} ⊆ {x | BSW Tx = 0} = N (W T )

R(SW T ) = {SW Tx | x ∈ RN } ⊇ {SW TZ y | Zy ∈ RN } = R(S) B.4 B.5

(1560) (1561) ¨

Left inverse is not unique, in general. By counter-example, the theorem’s converse cannot be true; e.g., S = W = [ 1 0 ] .

B.2. DOUBLET R([ u v ])

615 R(Π) = R([u v]) 0r

r0

N (Π) = v ⊥ ∩ u⊥ N

R = R([ u v ]) ⊕ N

µ·

vT uT

¸¶

v ⊥ ∩ u⊥ N

µ·

vT uT

¸¶

§

⊕ R([ u v ]) = RN

Figure 149: Four fundamental subspaces [311, 3.6] of a doublet Π = uv T + vuT ∈ SN . Π(x) = (uv T + vuT )x is a linear bijective mapping from R([ u v ]) to R([ u v ]).

B.2

Doublet

Consider a sum of two linearly independent square dyads, one a transposition of the other: ¸ · [ u v ] vT T T = SW T ∈ SN (1562) Π = uv + vu = uT where u , v ∈ RN . Like the dyad, a doublet can be 0 only when u or v is 0; Π = uv T + vuT = 0 ⇔ u = 0 or v = 0

(1563)

By assumption of independence, a nonzero doublet has two nonzero eigenvalues λ1 , uTv + kuv T k , λ2 , uTv − kuv T k (1564) where λ1 > 0 > λ2 , with corresponding eigenvectors x1 ,

v u + , kuk kvk

x2 ,

u v − kuk kvk

(1565)

spanning the doublet range. Eigenvalue λ1 cannot be 0 unless u and v have opposing directions, but that is antithetical since then the dyads would no longer be independent. Eigenvalue λ2 is 0 if and only if u and v share the same direction, again antithetical. Generally we have λ1 > 0 and λ2 < 0, so Π is indefinite.

616

APPENDIX B. SIMPLE MATRICES N (uT )

r0

0r

R(E ) = N (v T )

R(v)

N (E ) = R(u) RN = N (uT ) ⊕ N (E )

R(v) ⊕ R(E ) = RN

§

Figure 150: v Tu = 1/ζ .

The four fundamentalµ subspaces ¶ [311, 3.6] of T uv elementary matrix E as a linear mapping E(x) = I − T x . v u By the nullspace and range of dyad sum theorem, doublet Π has N− remaining and corresponding eigenvectors spanning ¸¶ µ·2 zero-eigenvalues T v . We therefore have N uT R(Π) = R([ u v ]) ,

N (Π) = v ⊥ ∩ u⊥

(1566)

of respective dimension 2 and N − 2.

B.3

Elementary matrix

A matrix of the form E = I − ζ uv T ∈ RN ×N

(1567)

λ = 1 − ζ v Tu

(1568)

where ζ ∈ R is finite and u,v ∈ RN , is called an elementary matrix or a rank-one modification of the identity. [190] Any elementary matrix in RN ×N has N − 1 eigenvalues equal to 1 corresponding to real eigenvectors that span v ⊥ . The remaining eigenvalue

corresponds to eigenvector u .B.6 From [206, App.7.A.26] the determinant: ¡ ¢ det E = 1 − tr ζ uv T = λ (1569) B.6

Elementary matrix E is not always diagonalizable because eigenvector u need not be independent of the others; id est, u ∈ v ⊥ is possible.

B.3. ELEMENTARY MATRIX

617

§

If λ 6= 0 then E is invertible; [139] (confer B.1.0.1) E −1 = I +

ζ T uv λ

(1570)

Eigenvectors corresponding to 0 eigenvalues belong to N (E ) , and the number of 0 eigenvalues must be at least dim N (E ) which, here, can be at most one. ( A.7.3.0.1) The nullspace exists, therefore, when λ = 0 ; id est, when v Tu = 1/ζ ; rather, whenever u belongs to hyperplane {z ∈ RN | v Tz = 1/ζ}. Then (when λ = 0) elementary matrix E is a nonorthogonal projector projecting on its range (E 2 = E , E.1) and N (E ) = R(u) ; eigenvector u spans the nullspace when it exists. By conservation of dimension, dim R(E ) = N − dim N (E ). It is apparent from (1567) that v ⊥ ⊆ R(E ) , but dim v ⊥ = N − 1. Hence R(E ) ≡ v ⊥ when the nullspace exists, and the remaining eigenvectors span it. In summary, when a nontrivial nullspace of E exists,

§

§

R(E ) = N (v T ),

v Tu = 1/ζ

N (E ) = R(u),

(1571)

illustrated in Figure 150, which is opposite to the assignment of subspaces for a dyad (Figure 148). Otherwise, R(E ) = RN . When E = E T , the spectral norm is kEk2 = max{1 , |λ|}

B.3.1

(1572)

Householder matrix

§

§

§

§

An elementary matrix is called a Householder matrix when it has the defining form, for nonzero vector u [146, 5.1.2] [139, 4.10.1] [309, 7.3] [188, 2.2] H =I −2

uuT ∈ SN uT u

(1573)

which is a symmetric orthogonal (reflection) matrix (H −1 = H T = H ( B.5.2)). Vector u is normal to an N −1-dimensional subspace u⊥ through which this particular H effects pointwise reflection; e.g., Hu⊥ = u⊥ while Hu = −u . Matrix H has N − 1 orthonormal eigenvectors spanning that reflecting subspace u⊥ with corresponding eigenvalues equal to 1. The remaining eigenvector u has corresponding eigenvalue −1 ; so

§

det H = −1

(1574)

618

APPENDIX B. SIMPLE MATRICES

Due to symmetry of H , the matrix 2-norm (the spectral norm) is equal to the largest eigenvalue-magnitude. A Householder matrix is thus characterized, HT = H ,

H −1 = H T ,

H²0

kHk2 = 1 ,

For example, the permutation matrix   1 0 0 Ξ = 0 0 1  0 1 0

(1575)

(1576)

√ is a Householder matrix having u = [ 0 1 −1 ]T / 2 . Not all permutation matrices are Householder matrices, although all permutation matrices are orthogonal matrices ( B.5.1, ΞT Ξ = I ) [309, 3.4] because they are made by permuting rows and columns of the identity matrix. Neither are all symmetric  permutation matrices Householder matrices; e.g., 0 0 0 1  0 0 1 0   Ξ =  0 1 0 0  (1654) is not a Householder matrix. 1 0 0 0

§

B.4 B.4.1

§

Auxiliary V -matrices Auxiliary projector matrix V

§

It is convenient to define a matrix V that arises naturally as a consequence of translating the geometric center αc ( 5.5.1.0.1) of some list X to the origin. In place of X − αc 1T we may write XV as in (920) where V = I−

1 T 11 ∈ SN N

(859)

is an elementary matrix called the geometric centering matrix. Any elementary matrix in RN ×N has N − 1 eigenvalues equal to 1. For the particular elementary matrix V , the N th eigenvalue equals 0. The number of 0 eigenvalues must equal dim N (V ) = 1, by the 0 eigenvalues theorem ( A.7.3.0.1), because V = V T is diagonalizable. Because

§

V1=0

(1577)

B.4. AUXILIARY V -MATRICES

619

the nullspace N (V ) = R(1) is spanned by the eigenvector 1. The remaining eigenvectors span R(V ) ≡ 1⊥ = N (1T ) that has dimension N − 1. Because V2= V (1578)

§

and V T = V , elementary matrix V is also a projection matrix ( E.3) projecting orthogonally on its range N (1T ) which is a hyperplane containing the origin in RN V = I − 1(1T 1)−1 1T (1579)

§

The {0, 1} eigenvalues also indicate diagonalizable V is a projection matrix. [368, 4.1, thm.4.1] Symmetry of V denotes orthogonal projection; from (1839), V† = V ,

VT= V ,

kV k2 = 1 ,

V º0

(1580)

Matrix V is also circulant [154]. B.4.1.0.1 Example. Relationship of auxiliary to Householder matrix. Let H ∈ SN be a Householder matrix (1573) defined by   1 ..   .  ∈ RN  (1581) u= 1√  1+ N

§

Then we have [142, 2]

V =H Let D ∈ SN h and define

·

¸ I 0 H 0T 0

¸ · A b −HDH , − T b c

(1582)

(1583)

§

§

where b is a vector. Then because H is nonsingular ( A.3.1.0.5) [170, 3] ¸ · A 0 H º 0 ⇔ −A º 0 (1584) −V D V = −H 0T 0 and affine dimension is r = rank A when D is a Euclidean distance matrix. 2

620

APPENDIX B. SIMPLE MATRICES

B.4.2 1. VN

Schoenberg auxiliary matrix VN 1 = √ 2

·

−1T I

¸

∈ RN ×N −1

2. VNT 1 = 0

√ £ ¤ 3. I − e1 1T = 0 2VN √ ¤ £ 2VN VN = VN 4. 0 √ ¤ £ 2VN V = V 5. 0 √ √ ¤ £ ¤ £ 6. V 0 2VN = 0 2VN √ √ √ ¤£ ¤ £ ¤ £ 7. 0 2VN 0 2VN = 0 2VN · ¸ √ £ ¤† 0 0T 8. 0 2VN = V 0 I 9. 10. 11. 12. 13.

14. 15.

£

£

0

√

√

2VN

√ £ ¤† V = 0 2VN

¤£

16. VN† 1 = 0

17. VN† VN = I

0

√

2VN

¤†

=V ¸ · √ √ £ ¤† £ ¤ 0 0T 0 2VN 0 2VN = 0 I ¸ · √ √ £ £ ¤ 0 0T ¤ 0 = 0 2VN 2VN 0 I ¸ ¸ · · √ ¤ 0 0T £ 0 0T 0 2VN = 0 I 0 I " # † V N [ VN √12 1 ]−1 = √ 2 T 1 N √ £ ¤ VN† = 2 − N1 1 I − N1 11T ∈ RN −1×N , 0

2VN

¤†

¡

I − N1 11T ∈ SN −1

¢

B.4. AUXILIARY V -MATRICES 18. V T = V = VN VN† = I −

1 11T N

19. −VN† (11T − I )VN = I , 20. D = [dij ] ∈ SN h

621

¡

(861)

∈ SN

11T − I ∈ EDMN

tr(−V D V ) = tr(−V D) = tr(−VN† DVN ) =

¢

1 T 1 D1 N

=

1 N

tr(11TD) =

1 N

Any elementary matrix E ∈ SN of the particular form E = k1 I − k2 11T

(1585)

where k1 , k2 ∈ R ,B.7 will make tr(−ED) proportional to 21. D = [dij ] ∈ SN tr(−V D V ) =

22. D = [dij ] ∈ SN h

1 N

tr(−VNTDVN ) =

23. For Y ∈ SN

P

i,j i6=j

P

dij −

N −1 N

P i

P

dij .

dii = 1TD1 N1 − tr D

d1j

j

V (Y − δ(Y 1))V = Y − δ(Y 1)

B.4.3

Orthonormal auxiliary matrix VW

The skinny matrix  −1 √

N −1 √ N+ N

VW

  1+    ,  N +−1√N   ..  .  −1 √ N+ N

B.7

−1 √ N −1 √ N+ N

···

...

...

... −1 √ N+ N

···

−1 √ N −1 √ N+ N −1 √ N+ N

... ··· 1 +

.. . −1 √ N+ N



      ∈ RN ×N −1    

(1586)

If k1 is 1−ρ while k2 equals −ρ ∈ R , then all eigenvalues of E for −1/(N − 1) < ρ < 1 are guaranteed positive and therefore E is guaranteed positive definite. [279]

P i,j

dij

622

APPENDIX B. SIMPLE MATRICES

has R(VW ) = N (1T ) and orthonormal columns. [6] We defined three auxiliary V -matrices: V , VN (843), and VW sharing some attributes listed in Table B.4.4. For example, V can be expressed V = VW VWT = VN VN†

(1587)

but VWT VW = I means V is an orthogonal projector (1836) and VW† = VWT ,

B.4.4

R(V )

N (V T )

V TV

R(1)

V

rank V

N ×N

N−1

N (1T )

VN

N ×(N − 1)

N−1

N (1T )

R(1)

VW

N ×(N − 1)

N−1

N (1T )

R(1)

B.4.5

(1588)

Auxiliary V -matrix Table

dim V V

VWT 1 = 0

kVW k2 = 1 ,

1 (I 2

V

+ 11T )

1 2

·

I

More auxiliary matrices

§

VV†

VVT

V

N − 1 −1T −1 I

¸

V

§

Mathar shows [241, 2] that any elementary matrix ( B.3) of the form VM = I − b 1T ∈ RN ×N

(1589)

§

such that bT 1 = 1 (confer [148, 2]), is an auxiliary V -matrix having T R(VM ) = N (bT ) ,

N (VM ) = R(b) ,

R(VM ) = N (1T ) T N (VM ) = R(1)

(1590)

Given X ∈ Rn×N , the choice b = N1 1 (VM = V ) minimizes kX(I − b 1T )kF . [150, 3.2.1]

§

V V

B.5. ORTHOGONAL MATRIX

B.5 B.5.1

623

Orthogonal matrix Vector rotation

The property Q−1 = QT completely defines an orthogonal matrix Q ∈ Rn×n employed to effect vector rotation; [309, 2.6, 3.4] [311, 6.5] [188, 2.1] for any x ∈ Rn kQxk2 = kxk2 (1591)

§ §

§

§

In other words, the 2-norm is orthogonally invariant. A unitary matrix is a complex generalization of the orthogonal matrix. The conjugate transpose defines it: U −1 = U H . An orthogonal matrix is simply a real unitary matrix. Orthogonal matrix Q is a normal matrix further characterized: Q−1 = QT ,

kQk2 = 1

(1592)

Applying characterization (1592) to QT we see it too is an orthogonal matrix. Hence the rows and columns of Q respectively form an orthonormal set. Normalcy guarantees diagonalization ( A.5.1) so, for Q , SΛS H

§

SΛ−1 S H = S ∗ΛS T ,

kδ(Λ)k∞ = 1

(1593)

characterizes an orthogonal matrix in terms of eigenvalues and eigenvectors. All permutation matrices Ξ , for example, are orthogonal matrices. Any product of permutation matrices remains a permutator. Any product of a permutation matrix with an orthogonal matrix remains orthogonal. In fact, any product of orthogonal matrices AQ remains orthogonal by definition. Given any other dimensionally compatible orthogonal matrix U , the mapping g(A) = U TAQ is a linear bijection on the domain of orthogonal matrices (a nonconvex manifold of dimension 12 n(n−1) [48]). [222, 2.1] [223] The largest magnitude entry of an orthogonal matrix is 1; for each and every j ∈ 1 . . . n kQ(j , :)k∞ ≤ 1 (1594) kQ(: , j)k∞ ≤ 1

§

Each and every eigenvalue of a (real) orthogonal matrix has magnitude 1 λ(Q) ∈ C n ,

|λ(Q)| = 1

(1595)

while only the identity matrix can be simultaneously positive definite and orthogonal.

624

APPENDIX B. SIMPLE MATRICES

B.5.2

Reflection

A matrix for pointwise reflection is defined by imposing symmetry upon the orthogonal matrix; id est, a reflection matrix is completely defined by Q−1 = QT = Q . The reflection matrix is an orthogonal matrix, characterized: QT = Q ,

Q−1 = QT ,

§

kQk2 = 1

(1596)

The Householder matrix ( B.3.1) is an example of a symmetric orthogonal (reflection) matrix. Reflection matrices have eigenvalues equal to ±1 and so det Q = ±1. It is natural to expect a relationship between reflection and projection matrices because all projection matrices have eigenvalues belonging to {0, 1}. In fact, any reflection matrix Q is related to some orthogonal projector P by [190, 1, prob.44] Q = I − 2P (1597)

§

§

Yet P is, generally, neither orthogonal or invertible. ( E.3.2) λ(Q) ∈ Rn ,

|λ(Q)| = 1

(1598)

Reflection is with respect to R(P )⊥ . Matrix 2P −I represents antireflection. Every orthogonal matrix can be expressed as the product of a rotation and a reflection. The collection of all orthogonal matrices of particular dimension does not form a convex set.

B.5.3

Rotation of range and rowspace

Given orthogonal matrix Q , column vectors of a matrix X are simultaneously rotated about the origin via product QX . In three dimensions (X ∈ R3×N ), the precise meaning of rotation is best illustrated in Figure 151 where the gimbal aids visualization of what is achievable; mathematically, ( 5.5.2.0.1)

§



cos θ Q = 0 sin θ

 0 − sin θ 1 1 0  0 0 cos θ 0

  0 0 cos φ − sin φ 0 cos ψ − sin ψ  sin φ cos φ 0  (1599) sin ψ cos ψ 0 0 1

B.5. ORTHOGONAL MATRIX

625

Figure 151: Gimbal : a mechanism imparting three degrees of dimensional freedom to a Euclidean body suspended at the device’s center. Each ring is free to rotate about one axis. (Courtesy of The MathWorks Inc.)

B.5.3.0.1

Example. One axis of revolution. n+1

Partition an n + 1-dimensional Euclidean space R

,

an n-dimensional subspace

·

Rn R

R , {λ ∈ Rn+1 | 1T λ = 0}

¸

and define

(1600)

(a hyperplane through the origin). We want an orthogonal matrix that rotates a list in the columns of matrix X ∈ Rn+1×N through the dihedral angle between Rn and R ( 2.4.3) ¶ µ ¶ µ hen+1 , 1i 1 n radians (1601) Á(R , R) = arccos = arccos √ ken+1 k k1k n+1

§

The vertex-description of the nonnegative orthant in Rn+1 is {[ e1 e2 · · · en+1 ] a | a º 0} = {a º 0} = Rn+1 ⊂ Rn+1 +

(1602)

Consider rotation of these vertices via orthogonal matrix 1 Q , [ 1 √n+1

ΞVW ]Ξ ∈ Rn+1×n+1

(1603)

where permutation matrix Ξ ∈ S n+1 is defined in (1654), and VW ∈ Rn+1×n is the orthonormal auxiliary matrix defined in B.4.3. This particular orthogonal matrix is selected because it rotates any point in subspace Rn

§

626

APPENDIX B. SIMPLE MATRICES

about one axis of revolution onto R ; e.g., rotation Qen+1 aligns the last standard basis vector with subspace normal R⊥ = 1. The rotated standard basis vectors remaining are orthonormal spanning R . 2 Another interpretation of product QX is rotation/reflection of R(X). Rotation of X as in QXQT is the simultaneous rotation/reflection of range and rowspace.B.8 Proof. Any matrix can be expressed as a singular value decomposition X = U ΣW T (1495) where δ 2(Σ) = Σ , R(U ) ⊇ R(X) , and R(W ) ⊇ R(X T ). ¨

B.5.4

Matrix rotation

§

Orthogonal matrices are also employed to rotate/reflect other matrices like vectors: [146, 12.4.1] Given orthogonal matrix Q , the product QTA will 2 rotate A ∈ Rn×n in the Euclidean sense in Rn because the Frobenius norm is orthogonally invariant ( 2.2.1); p kQTAkF = tr(ATQQTA) = kAkF (1604)

§

(likewise for AQ). Were A symmetric, such a rotation would depart from S n . One remedy is to instead form the product QTA Q because q T (1605) kQ A QkF = tr(QTATQQTA Q) = kAkF

Matrix A is orthogonally equivalent to B if B = S TA S for some orthogonal matrix S . Every square matrix, for example, is orthogonally equivalent to a matrix having equal entries along the main diagonal. [188, 2.2, prob.3]

§

The product QTA Q can be regarded as a coordinate transformation; e.g., given linear map y = Ax : Rn → Rn and orthogonal Q , the transformation Qy = A Qx is a rotation/reflection of the range and rowspace (134) of matrix A where Qy ∈ R(A) and Qx ∈ R(AT ) (135). B.8

Appendix C Some analytical optimal results C.1

ˆ

Properties of infima inf ∅ , ∞ sup ∅ , −∞

(1606)

ˆ Given f (x) : X → R defined on arbitrary set X [185, §0.1.2] inf f (x) = − sup −f (x)

x∈X

x∈X

sup f (x) = − inf −f (x) x∈X

x∈X

arg inf f (x) = arg sup −f (x) x∈X

x∈X

arg sup f (x) = arg inf −f (x)

ˆ Given f (x) : X → R [185, §0.1.2]

(1607)

(1608)

x∈X

x∈X

and g(x) : X → R defined on arbitrary set X

inf (f (x) + g(x)) ≥ inf f (x) + inf g(x)

(1609)

X ⊂ Y ⇒ inf f (x) ≥ inf f (x)

(1610)

x∈X

x∈X

x∈X

ˆ Given f (x) : X ∪ Y → R and arbitrary sets X and Y [185, §0.1.2] x∈X

x∈Y

inf f (x) = min{ inf f (x) , inf f (x)}

(1611)

inf f (x) ≥ max{ inf f (x) , inf f (x)}

(1612)

x∈X ∪Y

x∈X

x∈X ∩Y

x∈X

x∈Y

x∈Y

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

627

628

C.2

APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

Trace, singular and eigen values

ˆ pFor A ∈ R

m×n

σ(A)1 =

and σ(A)1 connoting spectral norm,

λ(ATA)1 = kAk2 = sup kAxk2 = minimize t∈R

kxk=1

subject to

ˆ For A ∈ R

m×n

§

t ·

(603) tI A AT tI

¸

º0

and σ(A) denoting its singular values, the nuclear norm (Ky Fan norm) of matrix A (confer (39), A.1.1 no.15, [189, p.200]) is √ P tr(X TA) σ(A)i = tr ATA = sup tr(X TA) = maximize m×n X∈ R i kXk2 ≤1 · ¸ I X subject to º0 XT I (1613) = 12 minimize tr X + tr Y m n X∈ S , Y ∈ S · ¸ X A subject to º0 AT Y

§

This nuclear norm is convex (a supremum of functions linear in A) and dual to spectral norm. [189, p.214] [57, A.1.6] Given singular value decomposition A = SΣQT ∈ Rm×n (A.6), then X ⋆ = SQT ∈ Rm×n is an optimal solution to maximization ( 2.3.2.0.5) while X ⋆ = SΣS T ∈ S m and Y ⋆ = QΣQT ∈ S n is an optimal solution to the minimization [125]. Srebro [301] asserts P σ(A)i = 21 minimize kU k2F + kV k2F

§

U,V

i

subject to A = U V T

=

minimize kU kF kV kF

(1614)

U,V

subject to A = U V T

C.2.0.0.1 Exercise. Optimal matrix factorization. Prove (1614).C.1

H

Hint: Write A = S ΣQT ∈ Rm×n and · ¸ · ¸ T X A U [U V T ] = º0 T A Y V √ √ Show U ⋆ = S Σ ∈ Rm×min{m, n} and V ⋆ = Q Σ ∈ Rn×min{m, n} , hence kU ⋆ k2F = kV ⋆ k2F . C.1

C.2. TRACE, SINGULAR AND EIGEN VALUES

ˆ For

629

X ∈ Sm , Y ∈ S n , A ∈ C ⊆ Rm×n for set C convex, and σ(A) denoting the singular values of A [125, 3]

§

minimize A

P

σ(A)i

i

subject to A ∈ C

ˆ For A ∈ S

N +

1 2

≡

minimize tr X + tr Y A,X, Y · ¸ X A subject to º0 AT Y A∈C

(1615)

and β ∈ R β tr A = maximize X∈ SN

tr(XA)

(1616)

subject to X ¹ β I

But the following statement is numerically stable, preventing an unbounded solution in direction of a 0 eigenvalue: maximize sgn(β) tr(XA) X∈ SN

(1617)

subject to X ¹ |β| I X º −|β| I

where β tr A = tr(X ⋆ A). If β ≥ 0, then X º −|β|I ← X º 0.

ˆ For symmetric A ∈ S , its smallest and largest eigenvalue in λ(A) ∈ R is respectively [11, §4.1] [40, §I.6.15] [188, §4.2] [222, §2.1] [223] N

min{λ(A) i } = inf xTA x = minimize i

kxk=1

X∈ SN +

kxk=1

t∈R

subject to tr X = 1

max{λ(A) i } = sup xTA x = maximize i

tr(XA) = maximize

X∈ SN +

N

t

subject to A º t I (1618)

tr(XA) = minimize t∈R

t

subject to A ¹ t I (1619) The largest eigenvalue is always convex in A ∈ SN because, given particular x , xTA x is linear in A . Supremum of a family of linear functions is convex, as illustrated in Figure 71. Similarly, the smallest subject to tr X = 1

630

APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS eigenvalue of any symmetric matrix is a concave function of its entries. For v1 a normalized eigenvector of A corresponding to the largest eigenvalue, and vN a normalized eigenvector corresponding to the smallest eigenvalue, vN = arg inf xTA x (1620) kxk=1

v1 = arg sup xTA x

(1621)

kxk=1

ˆ For A ∈ S having eigenvalues λ(A) ∈ R , consider the unconstrained nonconvex optimization that is a projection on the rank-1 subset (§2.9.2.1, §3.7.0.0.1) of the boundary of positive semidefinite cone S : N

N

N +

Defining λ1 , max i {λ(A) i } and corresponding eigenvector v1

minimize kxxT − Ak2F = minimize tr(xxT(xTx) − 2AxxT + ATA) x x ( kλ(A)k2 , λ1 ≤ 0 (1622) = 2 2 kλ(A)k − λ1 , λ1 > 0 ( 0, λ1 ≤ 0 √ arg minimize kxxT − Ak2F = (1623) x v 1 λ1 , λ1 > 0

§

Proof. This is simply the Eckart & Young solution from 7.1.2: ( 0, λ1 ≤ 0 x⋆ x⋆T = (1624) λ1 v1 v1T , λ1 > 0

§

Given nonincreasingly ordered diagonalization A = QΛQT where Λ = δ(λ(A)) ( A.5), then (1622) has minimum value  kQΛQT k2F = kδ(Λ)k2 , λ1 ≤ 0       °2     °2 ° ° ° ° ° λ1 ° λ T 2 1 ° ° ° minimize kxx −AkF = ° 0 °   T ° ° 0  °  x   . . − δ(Λ) = Q − Λ Q ° , λ1 > 0 ° ° °  .      ..  . ° ° ° °   ° ° ° 0 ° 0 F (1625) ¨

C.2. TRACE, SINGULAR AND EIGEN VALUES

631

C.2.0.0.2 Exercise. Rank-1 approximation. Given symmetric matrix A ∈ SN , prove: v1 = arg minimize x

kxxT − Ak2F

(1626)

subject to kxk = 1

where v1 is a normalized eigenvector of A corresponding to its largest eigenvalue. What is the objective’s optimal value? H

ˆ (Ky Fan, 1949) For B ∈ S whose eigenvalues λ(B) ∈ R are arranged in nonincreasing order, and for 1 ≤ k ≤ N [11, §4.1] [200] [188, §4.3.18] [334, §2] [222, §2.1] [223] N

N P

N

λ(B)i = inf tr(U U TB) = minimize N

i=N −k+1

U ∈ RN ×k U TU = I

X∈ S+

tr(XB)

(a)

subject to X ¹ I tr X = k (k − N )µ + tr(B − Z ) = maximize N µ∈R , Z∈ S+

(b)

subject to µI + Z º B k P

i=1

tr(XB) λ(B)i = sup tr(U U TB) = maximize N U ∈ RN ×k U TU = I

X∈ S+

(c)

subject to X ¹ I tr X = k = minimizeN µ∈R , Z∈ S+

kµ + tr Z

subject to µI + Z º B

§

(d) (1627)

Given ordered diagonalization B = QΛQT , ( A.5.1) then an optimal U for the infimum is U ⋆ = Q(: , N − k+1 : N ) ∈ RN ×k whereas U ⋆ = Q(: , 1 : k) ∈ RN ×k for the supremum. In both cases, X ⋆ = U ⋆ U ⋆T . Optimization (a) searches the convex hull of outer product U U T of all N × k orthonormal matrices. ( 2.3.2.0.1)

§

632

APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

ˆ For B ∈ S

N

whose eigenvalues λ(B) ∈ RN are arranged in nonincreasing order, and for diagonal matrix Υ ∈ Sk whose diagonal entries are arranged in nonincreasing order where 1 ≤ k ≤ N , we utilize the main-diagonal δ operator’s self-adjointness property (1362): [12, 4.2]

§

k P

Υii λ(B)N −i+1 = inf tr(Υ U TBU ) = inf δ(Υ)T δ(U TBU )

i=1

U ∈ RN ×k

U ∈ RN ×k

U TU = I

=

U TU = I

(1628)

µ k ¶ P tr B (Υii −Υi+1,i+1 )Vi

minimize Vi ∈ SN

i=1

subject to tr Vi = i , I º Vi º 0 ,

i=1 . . . k i=1 . . . k

where Υk+1,k+1 , 0. We speculate, k P

Υii λ(B)i = sup tr(Υ U TBU ) = sup δ(Υ)T δ(U TBU )

i=1

U ∈ RN ×k

(1629)

U ∈ RN ×k

U TU = I

U TU = I

§

Alizadeh shows: [11, 4.2] k P

Υii λ(B)i =

i=1

minimize

µ∈Rk , Zi ∈ SN

k P

iµi + tr Zi

i=1

subject to µi I + Zi − (Υii −Υi+1,i+1 )B º 0 , Zi º 0 , ¶ µ k P = maximize tr B (Υii −Υi+1,i+1 )Vi Vi ∈ SN

i=1 . . . k i=1 . . . k

i=1

subject to tr Vi = i , I º Vi º 0 ,

i=1 . . . k i=1 . . . k

(1630)

where Υk+1,k+1 , 0.

ˆ The largest eigenvalue magnitude µ of A ∈ S

N

max { |λ(A)i | } i

=

minimize µ∈ R

µ

subject to −µI ¹ A ¹ µI

(1631)

C.2. TRACE, SINGULAR AND EIGEN VALUES

633

§

is minimized over convex set C by semidefinite program: (confer 7.1.5) minimize A

minimize

kAk2

µ,A

≡

subject to A ∈ C

µ

subject to −µI ¹ A ¹ µI A∈C

(1632)

id est, µ⋆ , max { |λ(A⋆ )i | , i = 1 . . . N } ∈ R+ i

ˆ For B ∈ S

(1633)

N

whose eigenvalues λ(B) ∈ RN are arranged in nonincreasing order, let Π λ(B) be a permutation of eigenvalues λ(B) such that their absolute value becomes arranged in nonincreasing order: |Π λ(B)|1 ≥ |Π λ(B)|2 ≥ · · · ≥ |Π λ(B)|N . Then, for 1 ≤ k ≤ N [11, 4.3]C.2

§

k P

i=1

|Π λ(B)|i = minimizeN µ∈R , Z∈ S+

kµ + tr Z

=

subject to µI + Z + B º 0 µI + Z − B º 0

maximize hB , V − W i V , W ∈ SN +

subject to I º V , W tr(V + W ) = k (1634)

k

For diagonal matrix Υ ∈ S whose diagonal entries are arranged in nonincreasing order where 1 ≤ k ≤ N k P

i=1

Υii |Π λ(B)|i =

minimize

µ∈Rk , Zi ∈ SN

k P

iµi + tr Zi

i=1

subject to µi I + Zi + (Υii −Υi+1,i+1 )B º 0 , i = 1 . . . k µi I + Zi − (Υii −Υi+1,i+1 )B º 0 , i = 1 . . . k Zi º 0 , i=1 . . . k µ k ¶ P = maximize tr B (Υii −Υi+1,i+1 )(Vi − Wi ) Vi , Wi ∈ SN

i=1

subject to tr(Vi + Wi ) = i , I º Vi º 0 , I º Wi º 0 ,

i=1 . . . k i=1 . . . k i=1 . . . k

(1635)

where Υk+1,k+1 , 0. C.2

We eliminate a redundant positive semidefinite variable from Alizadeh’s minimization. There exist typographical errors in [271, (6.49) (6.55)] for this minimization.

634

APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

ˆ For

A , B ∈ SN whose eigenvalues λ(A) , λ(B) ∈ RN are respectively arranged in nonincreasing order, and for nonincreasingly ordered diagonalizations A = WA ΥWAT and B = WB ΛWBT [186] [222, 2.1] [223]

§

λ(A)T λ(B) = sup tr(AT U TBU ) ≥ tr(ATB)

(1653)

U ∈ RN ×N U TU = I

(confer (1658)) where optimal U is U ⋆ = WB WAT ∈ RN ×N

(1650)

§

We can push that upper bound higher using a result in C.4.2.1: |λ(A)|T |λ(B)| = sup Re tr(AT U HBU )

(1636)

U ∈ CN ×N U HU = I

For step function ψ as defined in (1511), optimal U becomes p Hp U ⋆ = WB δ(ψ(δ(Λ))) δ(ψ(δ(Υ))) WAT ∈ CN ×N (1637)

C.3

Orthogonal Procrustes problem

Given matrices A , B ∈ Rn×N , their product having full singular value decomposition ( A.6.3)

§

AB T , U ΣQT ∈ Rn×n

(1638)

then an optimal solution R⋆ to the orthogonal Procrustes problem minimize R

kA − RTBkF

(1639)

subject to RT = R −1

§

maximizes tr(ATRTB) over the nonconvex manifold of orthogonal matrices: [188, 7.4.8] R⋆ = QU T ∈ Rn×n (1640) A necessary and sufficient condition for optimality AB TR⋆ º 0

§

holds whenever R⋆ is an orthogonal matrix. [150, 4]

(1641)

C.3. ORTHOGONAL PROCRUSTES PROBLEM

635

§

§

Optimal solution R⋆ can reveal rotation/reflection ( 5.5.2, B.5) of one list in the columns of matrix A with respect to another list in B . Solution is unique if rank BVN = n . [103, 2.4.1] In the case that A is a vector and permutation of B , solution R⋆ is not necessarily a permutation matrix ( 4.6.0.0.3) although the optimal objective will be zero. More generally, the optimal value for objective of minimization is ¡ ¢ tr ATA + B TB − 2AB TR⋆ = tr(ATA) + tr(B TB) − 2 tr(U ΣU T ) (1642) = kAk2F + kBk2F − 2δ(Σ)T 1

§

§

while the optimal value for corresponding trace maximization is sup tr(ATRTB) = tr(ATR⋆TB) = δ(Σ)T 1 ≥ tr(ATB)

(1643)

RT=R −1

The same optimal solution R⋆ solves maximize R

kA + RTBkF

(1644)

subject to RT = R −1

C.3.1

Effect of translation

Consider the impact on problem (1639) of dc offset in known lists A , B ∈ Rn×N . Rotation of B there is with respect to the origin, so better results may be obtained if offset is first accounted. Because the geometric centers of the lists AV and BV are the origin, instead we solve minimize R

kAV − RTBV kF

subject to RT = R −1

(1645)

§

where V ∈ SN is the geometric centering matrix ( B.4.1). Now we define the full singular value decomposition A V B T , U ΣQT ∈ Rn×n

(1646)

and an optimal rotation matrix

R⋆ = QU T ∈ Rn×n

(1640)

The desired result is an optimally rotated offset list R⋆TB V + A(I − V ) ≈ A

(1647)

which most closely matches the list in A . Equality is attained when the lists are precisely related by a rotation/reflection and an offset. When R⋆TB = A or B1 = A1 = 0, this result (1647) reduces to R⋆TB ≈ A .

636

APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

C.3.1.1

Translation of extended list

Suppose an optimal rotation matrix R⋆ ∈ Rn×n were derived as before from matrix B ∈ Rn×N , but B is part of a larger list in the columns of [ C B ] ∈ Rn×M +N where C ∈ Rn×M . In that event, we wish to apply the rotation/reflection and translation to the larger list. The expression supplanting the approximation in (1647) makes 1T of compatible dimension; R⋆T [ C −B 11TN1

B V ] + A11TN1

(1648)

id est, C −B 11TN1 ∈ Rn×M and A11TN1 ∈ Rn×M +N .

C.4

Two-sided orthogonal Procrustes

C.4.0.1

Minimization

§

Given symmetric A , B ∈ SN , each having diagonalization ( A.5.1) A , QA ΛA QAT ,

B , QB ΛB QBT

(1649)

where eigenvalues are arranged in their respective diagonal matrix Λ in nonincreasing order, then an optimal solution [120] R⋆ = QB QAT ∈ RN ×N

(1650)

to the two-sided orthogonal Procrustes problem ¡ ¢ minimize kA − RTBRkF minimize tr ATA − 2ATRTBR + B TB R R = subject to RT = R −1 subject to RT = R −1 (1651) maximizes tr(ATRTBR) over the nonconvex manifold of orthogonal matrices. Optimal product R⋆TBR⋆ has the eigenvectors of A but the eigenvalues of B . [150, 7.5.1] The optimal value for the objective of minimization is, by (43)

§

kQA ΛA QAT − R⋆TQB ΛB QBT R⋆ kF = kQA (ΛA − ΛB )QAT kF = kΛA − ΛB kF

(1652)

while the corresponding trace maximization has optimal value sup tr(ATRTBR) = tr(ATR⋆TBR⋆ ) = tr(ΛA ΛB ) ≥ tr(ATB)

RT=R −1

(1653)

C.4. TWO-SIDED ORTHOGONAL PROCRUSTES C.4.0.2

637

Maximization

§

Any permutation matrix is an orthogonal matrix. Defining a row- and column-swapping permutation matrix (a reflection matrix, B.5.2)   0 1   ·   T   · (1654) Ξ = Ξ =     1 1 0

then an optimal solution R⋆ to the maximization problem [sic] maximize R

kA − RTBRkF

subject to RT = R −1

(1655)

§

minimizes tr(ATRTBR) : [186] [222, 2.1] [223] R⋆ = QB ΞQAT ∈ RN ×N

(1656)

The optimal value for the objective of maximization is kQA ΛA QAT − R⋆TQB ΛB QBT R⋆ kF = kQA ΛA QAT − QA Ξ TΛB ΞQT A kF = kΛA − ΞΛB ΞkF

(1657)

while the corresponding trace minimization has optimal value inf tr(ATRTBR) = tr(ATR⋆TBR⋆ ) = tr(ΛA ΞΛB Ξ)

RT=R −1

C.4.1

(1658)

Procrustes’ relation to linear programming

Although these two-sided Procrustes problems are nonconvex, a connection with linear programming [89] was discovered by Anstreicher & Wolkowicz [12, 3] [222, 2.1] [223]: Given A , B ∈ SN , this semidefinite program in S and T

§

§

minimize R

tr(ATRTBR) = maximize

subject to RT = R −1

S , T ∈ SN

tr(S + T )

(1659)

subject to AT ⊗ B − I ⊗ S − T ⊗ I º 0

638

APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

§

§

(where ⊗ signifies Kronecker product ( D.1.2.1)) has optimal objective value (1658). These two problems are strong duals ( 2.13.1.0.3). Given ordered diagonalizations (1649), make the observation: ˆ TΛB R) ˆ inf tr(ATRTBR) = inf tr(ΛA R R

ˆ R

(1660)

ˆ , QTR Q on the set of orthogonal matrices (which includes the because R B A permutation matrices) is a bijection. This means, basically, diagonal matrices of eigenvalues ΛA and ΛB may be substituted for A and B , so only the main diagonals of S and T come into play; maximize S,T ∈ SN

1T δ(S + T )

subject to δ(ΛA ⊗ (ΞΛB Ξ) − I ⊗ S − T ⊗ I ) º 0

(1661)

a linear program in δ(S) and δ(T ) having the same optimal objective value as the semidefinite program (1659). We relate their results to Procrustes problem (1651) by manipulating signs (1607) and permuting eigenvalues: maximize R

tr(ATRTBR) = minimize

subject to RT = R −1

S , T ∈ SN

1T δ(S + T )

subject to δ(I ⊗ S + T ⊗ I − ΛA ⊗ ΛB ) º 0 = minimize S , T ∈ SN

tr(S + T )

(1662)

subject to I ⊗ S + T ⊗ I − AT ⊗ B º 0 This formulation has optimal objective value identical to that in (1653).

C.4.2

Two-sided orthogonal Procrustes via SVD

By making left- and right-side orthogonal matrices independent, we can push the upper bound on trace (1653) a little further: Given real matrices A , B each having full singular value decomposition ( A.6.3)

§

A , UA ΣA QAT ∈ Rm×n ,

B , UB ΣB QBT ∈ Rm×n

(1663)

then a well-known optimal solution R⋆ , S ⋆ to the problem minimize R,S

kA − SBRkF

subject to RH = R −1 S H = S −1

(1664)

C.4. TWO-SIDED ORTHOGONAL PROCRUSTES

639

maximizes Re tr(ATSBR) : [292] [268] [48] [181] optimal orthogonal matrices S ⋆ = UA UBH ∈ R m×m ,

R⋆ = QB QAH ∈ R n×n

§

(1665)

[sic] are not necessarily unique [188, 7.4.13] because the feasible set is not convex. The optimal value for the objective of minimization is, by (43) kUA ΣA QAH − S ⋆ UB ΣB QBH R⋆ kF = kUA (ΣA − ΣB )QAH kF = kΣA − ΣB kF

(1666)

§

while the corresponding trace maximization has optimal value [40, III.6.12] sup | tr(ATSBR)| = sup Re tr(ATSBR) = Re tr(ATS ⋆BR⋆ ) = tr(ΣAT ΣB ) ≥ tr(ATB) RH =R−1 S H =S −1

RH =R−1 S H =S −1

(1667)

for which it is necessary ATS ⋆BR⋆ º 0 ,

BR⋆ATS ⋆ º 0

(1668)

The lower bound on inner product of singular values in (1667) is due to von Neumann. Equality is attained if UAH UB = I and QBH QA = I . C.4.2.1

Symmetric matrices

§

Now optimizing over the complex manifold of unitary matrices ( B.5.1), the upper bound on trace (1653) is thereby raised: Suppose we are given diagonalizations for (real) symmetric A , B ( A.5)

§

A = WA ΥWAT ∈ S n ,

δ(Υ) ∈ KM

B = WB ΛWBT ∈ S n ,

δ(Λ) ∈ KM

(1669) (1670) n

having their respective eigenvalues in diagonal matrices Υ, Λ ∈ S arranged in nonincreasing order (membership to the monotone cone KM (408)). Then by splitting eigenvalue signs, we invent a symmetric SVD-like decomposition A , UA ΣA QAH ∈ S n ,

B , UB ΣB QBH ∈ S n

(1671)

§

where UA , UB , QA , QB ∈ C n×n are unitary matrices defined by (confer A.6.5) UA , WA

p δ(ψ(δ(Υ))) ,

UB , WB

p δ(ψ(δ(Λ))) ,

QA , WA

p

QB , WB

H

δ(ψ(δ(Υ))) ,

ΣA = |Υ|

(1672)

p H δ(ψ(δ(Λ))) ,

ΣB = |Λ|

(1673)

640

APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

where step function ψ is defined in (1511). In this circumstance, S ⋆ = UA UBH = R⋆T ∈ C n×n

(1674)

optimal matrices (1665) now unitary are related by transposition. optimal value of objective (1666) is kUA ΣA QAH − S ⋆ UB ΣB QBH R⋆ kF = k |Υ| − |Λ| kF

The (1675)

while the corresponding optimal value of trace maximization (1667) is sup Re tr(ATSBR) = tr(|Υ| |Λ|)

(1676)

RH =R−1 S H =S −1

C.4.2.2

Diagonal matrices

Now suppose A and B are diagonal matrices A = Υ = δ 2(Υ) ∈ S n ,

δ(Υ) ∈ KM

(1677)

B = Λ = δ 2(Λ) ∈ S n ,

δ(Λ) ∈ KM

(1678)

both having their respective main diagonal entries arranged in nonincreasing order: minimize kΥ − SΛRkF R,S (1679) subject to RH = R −1 S H = S −1 Then we have a symmetric decomposition from unitary matrices as in (1671) where p p H (1680) UA , δ(ψ(δ(Υ))) , QA , δ(ψ(δ(Υ))) , ΣA = |Υ| p p H UB , δ(ψ(δ(Λ))) , QB , δ(ψ(δ(Λ))) , ΣB = |Λ| (1681) Procrustes solution (1665) again sees the transposition relationship S ⋆ = UA UBH = R⋆T ∈ C n×n

(1674)

but both optimal unitary matrices are now themselves diagonal. So, S ⋆ΛR⋆ = δ(ψ(δ(Υ)))Λδ(ψ(δ(Λ))) = δ(ψ(δ(Υ)))|Λ|

(1682)

Appendix D Matrix calculus From too much study, and from extreme passion, cometh madnesse. − Isaac Newton [141, §5]

D.1 D.1.1

Directional derivative, Taylor series Gradients

Gradient of a differentiable real function f (x) : RK → R with respect to its vector argument is defined in terms of partial derivatives 

  ∇f (x) ,   

∂f (x) ∂x1 ∂f (x) ∂x2

.. .

∂f (x) ∂xK



   ∈ RK  

(1683)

while the second-order gradient of the twice differentiable real function with respect to its vector argument is traditionally called the Hessian ;  ∂ 2f (x)  ∂ 2f (x) ∂ 2f (x) · · · ∂x ∂x1 ∂x2 ∂x21 1 ∂xK  2   ∂ f (x) ∂ 2f (x) ∂ 2f (x)  · · ·   ∂x2 ∂xK  ∈ SK ∂x22 (1684) ∇ 2 f (x) ,  ∂x2.∂x1 . . ...   .. .. ..  2  ∂ f (x) ∂ 2f (x) ∂ 2f (x) ··· ∂xK ∂x1 ∂xK ∂x2 ∂x2 K

© 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

641

642

APPENDIX D. MATRIX CALCULUS

The gradient of vector-valued function v(x) : R → RN on real domain is a row-vector ∇v(x) ,

h

∂v1 (x) ∂x

∂v2 (x) ∂x

∂vN (x) ∂x

···

i

∈ RN

(1685)

i

(1686)

while the second-order gradient is ∇ 2 v(x) ,

h

∂ 2 v1 (x) ∂x2

∂ 2 v2 (x) ∂x2

···

∂ 2 vN (x) ∂x2

∈ RN

Gradient of vector-valued function h(x) : RK → RN on vector domain is 

   ∇h(x) ,   

∂h1 (x) ∂x1

∂h2 (x) ∂x1

∂h1 (x) ∂x2

∂h2 (x) ∂x2

∂h1 (x) ∂xK

∂h2 (x) ∂xK

.. .

.. .

···

∂hN (x) ∂x1

···

∂hN (x) ∂x2

···

∂hN (x) ∂xK

.. .

      

(1687)

= [ ∇h1 (x) ∇h2 (x) · · · ∇hN (x) ] ∈ RK×N

while the second-order gradient has a three-dimensional representation dubbed cubix ;D.1 

∇ ∂h∂x1 (x) ∇ ∂h∂x2 (x) ··· ∇ 1 1

∂hN (x) ∂x1

  ∂h1 (x) ∂hN (x) ∂h (x)  ∇ ∂x2 ∇ ∂x2 2 · · · ∇ ∂x 2 2 ∇ h(x) ,  .. .. ..  . . .  ∂hN (x) ∂h1 (x) ∂h2 (x) ∇ ∂xK ∇ ∂xK · · · ∇ ∂xK

      

(1688)

= [ ∇ 2 h1 (x) ∇ 2 h2 (x) · · · ∇ 2 hN (x) ] ∈ RK×N ×K

where the gradient of each real entry is with respect to vector x as in (1683). D.1

The word matrix comes from the Latin for womb ; related to the prefix matri- derived from mater meaning mother.

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES

643

The gradient of real function g(X) : RK×L → R on matrix domain is 

   ∇g(X) ,    =

£

∂g(X) ∂X11

∂g(X) ∂X12

···

∂g(X) ∂X1L

∂g(X) ∂X21

∂g(X) ∂X22

···

∂g(X) ∂X2L

∂g(X) ∂XK1

∂g(X) ∂XK2

···

∂g(X) ∂XKL

.. .

.. .

.. .



    ∈ RK×L  

(1689)

∇X(:,1) g(X)

∇X(:,2) g(X) ...

∈ RK×1×L ∇X(:,L) g(X)

¤

where the gradient ∇X(:,i) is with respect to the i th column of X . The strange appearance of (1689) in RK×1×L is meant to suggest a third dimension perpendicular to the page (not a diagonal matrix). The second-order gradient has representation



∇ ∂g(X) · · · ∇ ∂g(X) ∇ ∂g(X) ∂X11 ∂X12 ∂X1L

  ∂g(X)  ∇ ∂X21 ∇ ∂g(X) · · · ∇ ∂g(X) ∂X22 ∂X2L ∇ 2 g(X) ,  .. .. ..  . . .  ∂g(X) ∂g(X) ∂g(X) ∇ ∂XK1 ∇ ∂XK2 · · · ∇ ∂XKL =

£



    ∈ RK×L×K×L  

∇∇X(:,1) g(X)

∇∇X(:,2) g(X) ... ∇∇X(:,L) g(X)

∈ RK×1×L×K×L ¤

where the gradient ∇ is with respect to matrix X .

(1690)

644

APPENDIX D. MATRIX CALCULUS

Gradient of vector-valued function g(X) : RK×L → RN on matrix domain is a cubix £ ∇X(:,1) g1 (X) ∇X(:,1) g2 (X) · · · ∇X(:,1) gN (X)

∇g(X) ,

∇X(:,2) g1 (X) ∇X(:,2) g2 (X) · · · ∇X(:,2) gN (X) ... ... ...

∇X(:,L) g1 (X) ∇X(:,L) g2 (X) · · · ∇X(:,L) gN (X) = [ ∇g1 (X) ∇g2 (X) · · · ∇gN (X) ] ∈ RK×N ×L

¤

(1691)

while the second-order gradient has a five-dimensional representation; £ ∇∇X(:,1) g1 (X) ∇∇X(:,1) g2 (X) · · · ∇∇X(:,1) gN (X)

∇ 2 g(X) ,

∇∇X(:,2) g1 (X) ∇∇X(:,2) g2 (X) · · · ∇∇X(:,2) gN (X) ... ... ...

∇∇X(:,L) g1 (X) ∇∇X(:,L) g2 (X) · · · ∇∇X(:,L) gN (X) = [ ∇ 2 g1 (X) ∇ 2 g2 (X) · · · ∇ 2 gN (X) ] ∈ RK×N ×L×K×L

(1692)

The gradient of matrix-valued function g(X) : RK×L → RM ×N on matrix domain has a four-dimensional representation called quartix (fourth-order tensor )   ∇g11 (X) ∇g12 (X) · · · ∇g1N (X)    ∇g21 (X) ∇g22 (X) · · · ∇g2N (X)  ∇g(X) ,   ∈ RM ×N ×K×L (1693) . . . . . .   . . . ∇gM 1 (X) ∇gM 2 (X) · · · ∇gMN (X)

while the second-order gradient has six-dimensional representation  2  ∇ g11 (X) ∇ 2 g12 (X) · · · ∇ 2 g1N (X)  2   (X) ∇ 2 g22 (X) · · · ∇ 2 g2N (X)  ∈ RM ×N ×K×L×K×L ∇ 2 g(X) ,  ∇ g21  .. .. ..   . . . ∇ 2 gM 1 (X) ∇ 2 gM 2 (X) · · · ∇ 2 gMN (X) (1694) and so on.

¤

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES

D.1.2

645

Product rules for matrix-functions

Given dimensionally compatible matrix-valued functions of matrix variable f (X) and g(X) ¡ ¢ ∇X f (X)T g(X) = ∇X (f ) g + ∇X (g) f (1695)

§

while [49, 8.3] [293]

³ ¡ ¢ ¡ ¡ ¢ ¢´¯¯ T T T ∇X tr f (X) g(X) = ∇X tr f (X) g(Z ) + tr g(X) f (Z ) ¯

Z←X

(1696)

These expressions implicitly apply as well to scalar-, vector-, or matrix-valued functions of scalar, vector, or matrix arguments.

D.1.2.0.1 Example. Cubix. Suppose f (X) : R2×2 → R2 = X Ta and g(X) : R2×2 → R2 = Xb . We wish to find ¡ ¢ ∇X f (X)T g(X) = ∇X aTX 2 b (1697)

using the product rule. Formula (1695) calls for

∇X aTX 2 b = ∇X (X Ta) Xb + ∇X (Xb) X Ta

(1698)

Consider the first of the two terms: ∇X (f ) g = ∇X (X Ta) Xb £ ¤ = ∇(X Ta)1 ∇(X Ta)2 Xb

(1699)

The gradient of X Ta forms a cubix in R2×2×2 ; a.k.a, third-order tensor.   ∂(X Ta)2 ∂(X Ta)1 (1700) ∂X11  ∂X11II  I I   II II II II    T T  ∂(X a)1 ∂(X a)2  (Xb)   1 ∂X ∂X 12 12    2×1×2  ∇X (X Ta) Xb =  ∈R    ∂(X Ta)1  ∂(X Ta)2  ∂X  (Xb)2 ∂X 21I 21I   I I II II   I I   I I ∂(X Ta)1 ∂X22

∂(X Ta)2 ∂X22

646

APPENDIX D. MATRIX CALCULUS

Because gradient of the product (1697) requires total change with respect to change in each entry of matrix X , the Xb vector must make an inner product with each vector in the second dimension of the cubix (indicated by dotted line segments);   a1 0 ¸ ·  0 a1    b1 X11 + b2 X12 T ∇X (X a) Xb =  ∈ R2×1×2  b X + b X  a2  1 21 2 22 0 0 a2 (1701) ¸ · a1 (b1 X11 + b2 X12 ) a1 (b1 X21 + b2 X22 ) ∈ R2×2 = a2 (b1 X11 + b2 X12 ) a2 (b1 X21 + b2 X22 ) = abTX T

where the cubix appears as a complete 2 × 2 × 2 matrix. In like manner for the second term ∇X (g) f   b1 0 · ¸  b2 0    X11 a1 + X21 a2 T ∇X (Xb) X a =  ∈ R2×1×2   0  X12 a1 + X22 a2 b1 (1702) 0 b2 = X TabT ∈ R2×2

The solution ∇X aTX 2 b = abTX T + X TabT

(1703)

can be found from Table D.2.1 or verified using (1696). D.1.2.1

2

Kronecker product

A partial remedy for venturing into hyperdimensional representations, such as the cubix or quartix, is to first vectorize matrices as in (32). This device gives rise to the Kronecker product of matrices ⊗ ; a.k.a, direct product or tensor product. Although it sees reversal in the literature, [305, 2.1] we adopt the definition: for A ∈ Rm×n and B ∈ Rp×q   B11 A B12 A · · · B1q A  B21 A B22 A · · · B2q A    pm×qn B ⊗ A ,  .. (1704) .. ..  ∈ R  .  . . Bp1 A Bp2 A · · · Bpq A

§

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES

647

for which A ⊗ 1 = 1 ⊗ A = A (real unity acts like identity). One advantage to vectorization is existence of a traditional two-dimensional matrix representation for the second-order gradient of a real function with respect to a vectorized matrix. For example, from A.1.1 no.33 ( D.2.1) for square A , B ∈ Rn×n [152, 5.2] [12, 3]

§

§

§

§

2

2

2 2 n ×n T T T T T ∇vec X tr(AXBX ) = ∇vec X vec(X) (B ⊗A) vec X = B⊗A + B ⊗A ∈ R (1705) To disadvantage is a large new but known set of algebraic rules ( A.1.1) and the fact that its mere use does not generally guarantee two-dimensional matrix representation of gradients. Another application of the Kronecker product is to reverse order of appearance in a matrix product: Suppose we wish to weight the columns of a matrix S ∈ RM ×N , for example, by respective entries wi from the main diagonal in   w1 0 ...  ∈ SN W , (1706) 0T wN

§

A conventional means for accomplishing column weighting is to multiply S by diagonal matrix W on the right-hand side:   w1 0 £ ¤ ...  = S(: , 1)w1 · · · S(: , N )wN ∈ RM ×N (1707) S W = S 0T

wN

To reverse product order such that diagonal matrix W instead appears to the left of S : for I ∈ SM (Sze Wan)   S(: , 1) 0 0 .   0 S(: , 2) . .  ∈ RM ×N (1708) S W = (δ(W )T ⊗ I ) . .   .. .. 0 0 0 S(: , N )

To instead weight the rows of S via diagonal matrix W ∈ SM , for I ∈ SN   S(1 , :) 0 0 .   0 S(2 , :) . .  (δ(W ) ⊗ I ) ∈ RM ×N (1709) WS =  ... ...  0 0 0 S(M , :)

648

APPENDIX D. MATRIX CALCULUS

For any matrices of like size, S , Y ∈ RM ×N  S(: , 1) 0 0 .  £ ¤ 0 S(: , 2) . .   ∈ RM ×N S ◦Y = δ(Y (: , 1)) · · · δ(Y (: , N ))  ... ...  0 0 0 S(: , N ) (1710) which converts a Hadamard product into a standard matrix product. In the special case that S = s and Y = y are vectors in RM 

D.1.3

s ◦ y = δ(s)y

(1711)

sT ⊗ y = ysT s ⊗ y T = sy T

(1712)

Chain rules for composite matrix-functions

§

Given dimensionally compatible matrix-valued functions of matrix variable f (X) and g(X) [204, 15.7] ¡ ¢ ∇X g f (X)T = ∇X f T ∇f g (1713) ¡ ¢ ¡ ¢ ∇X2 g f (X)T = ∇X ∇X f T ∇f g = ∇X2 f ∇f g + ∇X f T ∇f2 g ∇X f (1714) D.1.3.1

D.1.3.1.1

Two arguments ¡ ¢ ∇X g f (X)T , h(X)T = ∇X f T ∇f g + ∇X hT ∇h g

Example. Chain rule for two arguments.

¡ ¢ g f (x)T , h(x)T = (f (x) + h(x))TA(f (x) + h(x)) ¸ ¸ · · εx1 x1 , h(x) = f (x) = x2 εx2 ¡ ¢ ∇x g f (x)T , h(x)T =

·

(1715)

§

[39, 1.1] (1716) (1717)

¸ ¸ · ε 0 1 0 T (A + AT )(f + h) (A + A )(f + h) + 0 1 0 ε (1718)

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES ¡ ¢ ∇x g f (x)T , h(x)T =

·

¸¶ ¸ · ¸ µ· εx1 1+ε 0 x1 T + (A + A ) x2 εx2 0 1+ε (1719)

¡ ¢ lim ∇x g f (x)T , h(x)T = (A + AT )x

(1720)

ε→0

from Table D.2.1.

649

2

These foregoing formulae remain correct when gradient produces hyperdimensional representation:

D.1.4

First directional derivative

Assume that a differentiable function g(X) : RK×L → RM ×N has continuous first- and second-order gradients ∇g and ∇ 2 g over dom g which is an open set. We seek simple expressions for the first and second directional derivatives →Y

→Y

in direction Y ∈ RK×L : respectively, dg ∈ RM ×N and dg 2 ∈ RM ×N . Assuming that the limit exists, we may state the partial derivative of the mn th entry of g with respect to the kl th entry of X ; gmn (X + ∆t ek eT ∂gmn (X) l ) − gmn (X) = lim ∈R ∆t→0 ∂Xkl ∆t

(1721)

where ek is the k th standard basis vector in RK while el is the l th standard basis vector in RL . The total number of partial derivatives equals KLM N while the gradient is defined in their terms; the mn th entry of the gradient is   ∂g (X) ∂g (X) mn mn (X) mn · · · ∂g∂X ∂X11 ∂X12 1L    ∂gmn (X) ∂gmn (X) ∂gmn (X)  · · ·   ∂X21 ∂X22 ∂X2L (1722) ∇gmn (X) =   ∈ RK×L .. .. ..   . . .   ∂gmn (X) ∂gmn (X) ∂gmn (X) · · · ∂XKL ∂XK1 ∂XK2 while the gradient is a quartix   ∇g11 (X) ∇g12 (X) · · · ∇g1N (X)  ∇g (X) ∇g (X) · · · ∇g (X)    21 22 2N ∇g(X) =   ∈ RM ×N ×K×L (1723) . . . .. .. ..   ∇gM 1 (X) ∇gM 2 (X) · · · ∇gMN (X)

650

APPENDIX D. MATRIX CALCULUS

By simply rotating our perspective of the four-dimensional representation of gradient matrix, we find one of three useful transpositions of this quartix (connoted T1 ): 

  ∇g(X) =    T1

∂g(X) ∂X11 ∂g(X) ∂X21

∂g(X) ∂X12 ∂g(X) ∂X22

···

∂g(X) ∂XK1

∂g(X) ∂XK2

···

.. .

···

.. .

∂g(X) ∂X1L ∂g(X) ∂X2L

.. .

∂g(X) ∂XKL



   ∈ RK×L×M ×N  

(1724)

When the limit for ∆t ∈ R exists, it is easy to show by substitution of variables in (1721) gmn (X + ∆t Ykl ek eT ∂gmn (X) l ) − gmn (X) Ykl = lim ∈R ∆t→0 ∂Xkl ∆t

(1725)

which may be interpreted as the change in gmn at X when the change in Xkl is equal to Ykl , the kl th entry of any Y ∈ RK×L . Because the total change in gmn (X) due to Y is the sum of change with respect to each and every Xkl , the mn th entry of the directional derivative is the corresponding total differential [204, 15.8]

§

dgmn (X)|dX→Y =

X ∂gmn (X) ∂Xkl

k,l

=

X k,l

¡ ¢ Ykl = tr ∇gmn (X)T Y

(1726)

gmn (X + ∆t Ykl ek eT l ) − gmn (X) (1727) ∆t→0 ∆t lim

gmn (X + ∆t Y ) − gmn (X) = lim ∆t→0 ∆t ¯ d ¯¯ gmn (X + t Y ) = dt ¯t=0

§

§

(1728) (1729)

where t ∈ R . Assuming finite Y , equation (1728) is called the Gˆ ateaux differential [38, App.A.5] [185, D.2.1] [333, 5.28] whose existence is implied by existence of the Fr´echet differential (the sum in (1726)). [232, 7.2] Each may be understood as the change in gmn at X when the change in X is equal

§

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES in magnitude and direction to Y .D.2 Hence the  dg11 (X) dg12 (X) · · · dg1N (X)  →Y  dg21 (X) dg22 (X) · · · dg2N (X) dg(X) ,  .. .. ..  . . . dgM 1 (X) dgM 2 (X) · · · dgMN (X) 

  = 

¡ ¢ tr ∇g11 (X)T Y ¡ ¢ tr ∇g21 (X)T Y .. . ¡ ¢ tr ∇gM 1 (X)T Y

 P ∂g11 (X)

Ykl

directional derivative, ¯ ¯ ¯ ¯ ¯  ¯ ∈ RM ×N ¯ ¯ ¯ dX→Y

¡ ¢ ¡ ¢ tr ∇g12 (X)T Y · · · tr ∇g1N (X)T Y ¡ ¢ ¡ ¢ tr ∇g22 (X)T Y · · · tr ∇g2N (X)T Y .. .. . . ¡ ¢ ¡ ¢ tr ∇gM 2 (X)T Y · · · tr ∇gMN (X)T Y

P ∂g12 (X)

Ykl

···

P ∂g1N (X)

Ykl

∂Xkl ∂Xkl  k,l ∂Xkl k,l k,l  P P ∂g2N (X) P ∂g22 (X) ∂g21 (X)  Ykl Ykl · · · Ykl  ∂Xkl ∂Xkl ∂Xkl =  k,l k,l k,l .. .. ..  . . .  P ∂gM 2 (X) P ∂gMN (X)  P ∂gM 1 (X) Ykl Ykl · · · Ykl ∂Xkl ∂Xkl ∂Xkl k,l

k,l

from which it follows

→Y

651

dg(X) =

k,l

X ∂g(X) k,l

∂Xkl

Ykl

    

        

(1730)

(1731)

Yet for all X ∈ dom g , any Y ∈ RK×L , and some open interval of t ∈ R →Y

g(X + t Y ) = g(X) + t dg(X) + o(t2 )

§

(1732)

§

which is the first-order Taylor series expansion about X . [204, 18.4] [140, 2.3.4] Differentiation with respect to t and subsequent t-zeroing isolates the second term of expansion. Thus differentiating and zeroing g(X + t Y ) in t is an operation equivalent to individually differentiating and zeroing every entry gmn (X + t Y ) as in (1729). So the directional derivative of g(X) : RK×L → RM ×N in any direction Y ∈ RK×L evaluated at X ∈ dom g becomes ¯ →Y d ¯¯ (1733) dg(X) = ¯ g(X + t Y ) ∈ RM ×N dt t=0 D.2

Although Y is a matrix, we may regard it as a vector in RKL .

652

APPENDIX D. MATRIX CALCULUS ­T ­ ­ ­ ­ (f (α), α)­ ­ ­ ­ ­

υ

f (α + t y) 

 υ,

∇x f (α) →∇x f (α) 1 df (α) 2

  

f (x)

∂H

Figure 152: Strictly convex quadratic bowl in R2 × R ; f (x) = xTx : R2 → R versus x on some open disc in R2 . Plane slice ∂H is perpendicular to function domain. Slice intersection with domain connotes bidirectional vector y . Slope of tangent line T at point (α , f (α)) is value of ∇x f (α)Ty directional derivative (1758) at α in slice direction y . Negative gradient −∇x f (x) ∈ R2 is direction of steepest descent. [350] [204, 15.6] [140] When vector υ ∈ ·R3 entry ¸ υ3 is half directional derivative in gradient direction at α υ1 = ∇x f (α) , then −υ points directly toward bowl bottom. and when υ2

§

§ §

§

[257, 2.1, 5.4.5] [32, 6.3.1] which is simplest. In case of a real function g(X) : RK×L → R →Y ¡ ¢ dg(X) = tr ∇g(X)T Y

(1755)

In case g(X) : RK → R

→Y

dg(X) = ∇g(X)T Y

(1758)

Unlike gradient, directional derivative does not expand dimension; directional derivative (1733) retains the dimensions of g . The derivative with respect to t makes the directional derivative resemble ordinary calculus

§

→Y

§

( D.2); e.g., when g(X) is linear, dg(X) = g(Y ). [232, 7.2]

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES D.1.4.1

653

Interpretation of directional derivative

In the case of any differentiable real function g(X) : RK×L → R , the directional derivative of g(X) at X in any direction Y yields the slope of g along the line {X + t Y | t ∈ R} through its domain evaluated at t = 0. For higher-dimensional functions, by (1730), this slope interpretation can be applied to each entry of the directional derivative. Figure 152, for example, shows a plane slice of a real convex bowl-shaped function f (x) along a line {α + t y | t ∈ R} through its domain. The slice reveals a one-dimensional real function of t ; f (α + t y). The directional derivative at x = α in direction y is the slope of f (α + t y) with respect to t at t = 0. In the case of a real function having vector argument h(X) : RK → R , its directional derivative in the normalized direction of its gradient is the gradient magnitude. (1758) For a real function of real variable, the directional derivative evaluated at any point in the function domain is just the slope of that function there scaled by the real direction. (confer 3.7) Directional derivative generalizes our one-dimensional notion of derivative to a multidimensional domain. When direction Y coincides with a member of the standard Cartesian basis ek eT l (55), then a single partial derivative ∂g(X)/∂Xkl is obtained from directional derivative (1731); such is each entry of gradient ∇g(X) in equalities (1755) and (1758), for example.

§

§

D.1.4.1.1 Theorem. Directional derivative optimality condition. [232, 7.4] Suppose f (X) : RK×L → R is minimized on convex set C ⊆ Rp×k by X ⋆ , and the directional derivative of f exists there. Then for all X ∈ C →X−X ⋆

df (X) ≥ 0

(1734) ⋄

D.1.4.1.2 Example. Simple bowl. Bowl function (Figure 152) f (x) : RK → R , (x − a)T (x − a) − b

(1735)

has function offset −b ∈ R , axis of revolution at x = a , and positive definite Hessian (1684) everywhere in its domain (an open hyperdisc in RK ); id est, strictly convex quadratic f (x) has unique global minimum equal to −b at

654

APPENDIX D. MATRIX CALCULUS

x = a . A vector −υ based anywhere in dom f × R pointing toward the unique bowl-bottom is specified: ¸ · x−a ∈ RK × R (1736) υ ∝ f (x) + b Such a vector is

since the gradient is



 υ=

∇x f (x) →∇x f (x) 1 df (x) 2

  

∇x f (x) = 2(x − a)

(1737)

(1738)

and the directional derivative in the direction of the gradient is (1758) →∇x f (x)

df (x) = ∇x f (x)T ∇x f (x) = 4(x − a)T (x − a) = 4(f (x) + b)

D.1.5

(1739) 2

Second directional derivative

By similar argument, it so happens: the second directional derivative is equally simple. Given g(X) : RK×L → RM ×N on open domain,   ∂ 2gmn (X) ∂ 2gmn (X) ∂ 2gmn (X) · · · ∂X ∂Xkl ∂X11 ∂Xkl ∂X12 ∂X 1L kl   2  ∂ gmn (X) ∂ 2gmn (X) ∂ 2gmn (X)  ∂∇gmn (X) ∂gmn (X) · · · ∂Xkl ∂X2L   = =  ∂Xkl ∂X21 ∂Xkl ∂X22 ∇  ∈ RK×L (1740) .. .. ..   ∂Xkl ∂Xkl . . .   ∂ 2gmn (X) ∂ 2gmn (X) ∂ 2gmn (X) · · · ∂Xkl ∂XKL ∂Xkl ∂XK1 ∂Xkl ∂XK2  ∂g (X)  ∂gmn (X) ∂gmn (X) mn ∇ ∂X ∇ · · · ∇ ∂X ∂X 11 12 1L    ∂gmn (X) ∂gmn (X) ∂gmn (X)  ∇ ∇ · · · ∇   ∂X21 ∂X22 ∂X2L ∇ 2 gmn (X) =   ∈ RK×L×K×L .. .. ..   . . .   ∂gmn (X) ∂gmn (X) ∂gmn (X) ∇ ∂XK1 ∇ ∂XK2 · · · ∇ ∂XKL (1741)  ∂∇gmn (X) ∂∇gmn (X)  ∂∇gmn (X) ··· ∂X11 ∂X12 ∂X1L  ∂∇gmn (X) ∂∇gmn (X)  ∂∇gmn (X)   · · ·   ∂X ∂X ∂X = .. 21 .. 22 ..2L  . . .   ∂∇gmn (X) ∂∇gmn (X) ∂∇gmn (X) ··· ∂XK1 ∂XK2 ∂XKL

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES

655

Rotating our perspective, we get several views of the second-order gradient: 

∇ 2 g11 (X)

∇ 2 g12 (X)

· · · ∇ 2 g1N (X)

 ∇ 2 g (X) ∇ 2 g (X) · · · ∇ 2 g (X)  21 22 2N ∇ g(X) =  . . .. .. ..  . ∇ 2 gM 1 (X) ∇ 2 gM 2 (X) · · · ∇ 2 gMN (X) 2

2

T1

∇ g(X)

2

∇ ∂g(X) · · · ∇ ∂g(X) ∇ ∂g(X) ∂X11 ∂X12 ∂X1L

 ∂g(X)  ∇ ∇ ∂g(X) · · · ∇ ∂g(X)  ∂X ∂X ∂X2L 21 22 = .. .. .. . . .  ∂g(X) ∂g(X) ∂g(X) ∇ ∂XK1 ∇ ∂XK2 · · · ∇ ∂XKL

T2

∇ g(X)





  =  

∂∇g(X) ∂X11 ∂∇g(X) ∂X21

∂∇g(X) ∂X12 ∂∇g(X) ∂X22

···

∂∇g(X) ∂XK1

∂∇g(X) ∂XK2

···

.. .

.. .

···

∂∇g(X) ∂X1L ∂∇g(X) ∂X2L

.. .

∂∇g(X) ∂XKL



   ∈ RM ×N ×K×L×K×L (1742)  

   ∈ RK×L×M ×N ×K×L (1743)   

   ∈ RK×L×K×L×M ×N  

(1744)

Assuming the limits exist, we may state the partial derivative of the mn th entry of g with respect to the kl th and ij th entries of X ; ∂ 2gmn (X) ∂Xkl ∂Xij

=

T T T gmn (X+∆t ek eT l +∆τ ei ej )−gmn (X+∆t ek el )−(gmn (X+∆τ ei ej )−gmn (X)) ∆τ ∆t ∆τ,∆t→0

lim

(1745) Differentiating (1725) and then scaling by Yij

∂ 2gmn (X) Y Y ∂Xkl ∂Xij kl ij

=

∂gmn (X+∆t Ykl ek eT l )−∂gmn (X) Yij ∂X ∆t ij ∆t→0

= lim

(1746)

T T T gmn (X+∆t Ykl ek eT l +∆τ Yij ei ej )−gmn (X+∆t Ykl ek el )−(gmn (X+∆τ Yij ei ej )−gmn (X)) ∆τ ∆t ∆τ,∆t→0

lim

656

APPENDIX D. MATRIX CALCULUS

which can be proved by substitution of variables in (1745). second-order total differential due to any Y ∈ RK×L is 2

d gmn (X)|dX→Y =

X X ∂ 2gmn (X) i,j

=

X i,j

k,l

∂Xkl ∂Xij

The mn th

³ ¡ ¢T ´ T Ykl Yij = tr ∇X tr ∇gmn (X) Y Y (1747)

∂gmn (X + ∆t Y ) − ∂gmn (X) Yij ∆t→0 ∂Xij ∆t

(1748)

lim

gmn (X + 2∆t Y ) − 2gmn (X + ∆t Y ) + gmn (X) = lim ∆t→0 ∆t2 ¯ 2 ¯ d ¯ = gmn (X + t Y ) dt2 ¯t=0

(1749) (1750)

Hence the second directional derivative, 

 →Y  dg 2(X) ,  

d 2g11 (X) d 2g21 (X) .. .

d 2g12 (X) d 2g22 (X) .. .

· · · d 2g1N (X) · · · d 2g2N (X) .. .

¯ ¯ ¯ ¯ ¯  ¯ ∈ RM ×N ¯ ¯ 2 2 2 d gM 1 (X) d gM 2 (X) · · · d gMN (X) ¯dX→Y

³ ³  ³ ¡ ¢T ´  ¡ ¢T ´ ¡ ¢T ´ T T T tr ∇tr ∇g12 (X) Y Y · · · tr ∇tr ∇g1N (X) Y Y tr ∇tr ∇g11 (X) Y Y   ³ ³ ´ ³ ¡ ¢T ´  ¡ ¢T ´ ¡ ¢  T T  tr ∇tr ∇g21 (X)T Y T Y tr ∇tr ∇g22 (X) Y Y · · · tr ∇tr ∇g2N (X) Y Y   =   . . . . . .   . . .  ³ ´ ³ ´ ³ ´ ¡ ¢ ¡ ¢ ¡ ¢ T T T tr ∇tr ∇gM 1 (X)T Y Y tr ∇tr ∇gM 2 (X)T Y Y · · · tr ∇tr ∇gMN (X)T Y Y  PP

∂ 2g11 (X) Y Y ∂Xkl ∂Xij kl ij

PP

∂ 2g12 (X) Y Y ∂Xkl ∂Xij kl ij

···

PP

∂ 2g1N (X) Y Y ∂Xkl ∂Xij kl ij

 i,j k,l i,j k,l i,j k,l  PP 2 P P P P ∂ 2g2N (X) 2g (X) ∂ g (X) ∂  21 22 Y Y Y Y ··· Y Y  ∂Xkl ∂Xij kl ij ∂Xkl ∂Xij kl ij ∂Xkl ∂Xij kl ij =  i,j k,l i,j k,l i,j k,l .. .. ..  . . .  P P ∂ 2gMN (X) P P ∂ 2gM 2 (X)  P P ∂ 2gM 1 (X) Y Y Y Y ··· Y Y ∂Xkl ∂Xij kl ij ∂Xkl ∂Xij kl ij ∂Xkl ∂Xij kl ij i,j k,l

i,j k,l

i,j k,l

        

(1751)

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES

657

from which it follows →Y

dg 2(X) =

X ∂ →Y X X ∂ 2g(X) Ykl Yij = dg(X) Yij ∂Xkl ∂Xij ∂Xij i,j i,j k,l

(1752)

Yet for all X ∈ dom g , any Y ∈ RK×L , and some open interval of t ∈ R →Y

g(X + t Y ) = g(X) + t dg(X) +

1 2 →Y2 t dg (X) + o(t3 ) 2!

(1753)

§

§

which is the second-order Taylor series expansion about X . [204, 18.4] [140, 2.3.4] Differentiating twice with respect to t and subsequent t-zeroing isolates the third term of the expansion. Thus differentiating and zeroing g(X + t Y ) in t is an operation equivalent to individually differentiating and zeroing every entry gmn (X + t Y ) as in (1750). So the second directional derivative of g(X) : RK×L → RM ×N becomes [257, 2.1, 5.4.5] [32, 6.3.1]

§ §

→Y 2

¯ d 2 ¯¯ dg (X) = 2 ¯ g(X + t Y ) ∈ RM ×N dt t=0

§

(1754)

which is again simplest. (confer (1733)) Directional derivative retains the dimensions of g .

D.1.6

directional derivative expressions

In the case of a real function g(X) : RK×L → R , all its directional derivatives are in R : →Y ¡ ¢ dg(X) = tr ∇g(X)T Y (1755) →Y 2

¶ µ ³ →Y ¡ ¢T ´ T T dg (X) = tr ∇X tr ∇g(X) Y Y = tr ∇X dg(X) Y

(1756)

! Ã µ ³ ´T ¶ →Y ¡ ¢ T (1757) dg (X) = tr ∇X tr ∇X tr ∇g(X)T Y Y Y = tr ∇X dg 2(X)T Y →Y 3

658

APPENDIX D. MATRIX CALCULUS

In the case g(X) : RK → R has vector argument, they further simplify: →Y

dg(X) = ∇g(X)T Y

(1758)

dg 2(X) = Y T ∇ 2 g(X)Y →Y ¡ ¢T dg 3(X) = ∇X Y T ∇ 2 g(X)Y Y

(1759)

→Y

and so on.

D.1.7

(1760)

Taylor series

Series expansions of the differentiable matrix-valued function g(X) , of matrix argument, were given earlier in (1732) and (1753). Assuming g(X) has continuous first-, second-, and third-order gradients over the open set dom g , then for X ∈ dom g and any Y ∈ RK×L the complete Taylor series is expressed on some open interval of µ ∈ R →Y

g(X+µY ) = g(X) + µ dg(X) +

1 2 →Y2 1 →Y µ dg (X) + µ3 dg 3(X) + o(µ4 ) (1761) 2! 3!

or on some open interval of kY k2

1 →Y2 −X 1 →Y3 −X dg (X) + dg (X) + o(kY k4 ) (1762) 2! 3! which are third-order expansions about X . The mean value theorem from calculus is what insures finite order of the series. [204] [39, 1.1] [38, App.A.5] [185, 0.4] These somewhat unbelievable formulae imply that a function can be determined over the whole of its domain by knowing its value and all its directional derivatives at a single point X . →Y −X

g(Y ) = g(X) + dg(X) +

§

§

D.1.7.0.1 Example. Inverse matrix function. Say g(Y ) = Y −1 . Continuing the table on page 664, ¯ →Y d ¯¯ dg(X) = ¯ g(X + t Y ) = −X −1 Y X −1 dt t=0 ¯ →Y d 2 ¯¯ 2 dg (X) = 2 ¯ g(X + t Y ) = 2X −1 Y X −1 Y X −1 dt t=0 ¯ →Y 3 ¯ d dg 3(X) = 3 ¯¯ g(X + t Y ) = −6X −1 Y X −1 Y X −1 Y X −1 dt t=0

(1763) (1764) (1765)

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES

659

Let’s find the Taylor series expansion of g about X = I : Since g(I ) = I , for kY k2 < 1 (µ = 1 in (1761)) g(I + Y ) = (I + Y )−1 = I − Y + Y 2 − Y 3 + . . .

(1766)

If Y is small, (I + Y )−1 ≈ I − Y . D.3 Now we find Taylor series expansion about X : g(X + Y ) = (X + Y )−1 = X −1 − X −1 Y X −1 + 2X −1 Y X −1 Y X −1 − . . . (1767) −1 −1 −1 −1 If Y is small, (X + Y ) ≈ X − X Y X . 2 D.1.7.0.2 Exercise. log det . (confer [57, p.644]) Find the first three terms of a Taylor series expansion for log det Y . Specify an open interval over which the expansion holds in vicinity of X . H

D.1.8

Correspondence of gradient to derivative

From the foregoing expressions for directional derivative, we derive a relationship between the gradient with respect to matrix X and the derivative with respect to real variable t : D.1.8.1

first-order

Removing evaluation at t = 0 from (1733),D.4 we find an expression for the directional derivative of g(X) in direction Y evaluated anywhere along a line {X + t Y | t ∈ R} intersecting dom g →Y

dg(X + t Y ) =

d g(X + t Y ) dt

(1768)

In the general case g(X) : RK×L → RM ×N , from (1726) and (1729) we find ¡ ¢ d tr ∇X gmn (X + t Y )T Y = gmn (X + t Y ) dt

(1769)

Had we instead set g(Y ) = (I + Y )−1 , then the equivalent expansion would have been about X = 0. D.4 Justified by replacing X with X + t Y in (1726)-(1728); beginning, D.3

dgmn (X + t Y )|dX→Y =

X ∂gmn (X + t Y ) k, l

∂Xkl

Ykl

660

APPENDIX D. MATRIX CALCULUS

which is valid at t = 0, of course, when X ∈ dom g . In the important case of a real function g(X) : RK×L → R , from (1755) we have simply ¡ ¢ d (1770) tr ∇X g(X + t Y )T Y = g(X + t Y ) dt When, additionally, g(X) : RK → R has vector argument, ∇X g(X + t Y )T Y =

d g(X + t Y ) dt

(1771)

§

D.1.8.1.1 Example. Gradient. g(X) = wTX TXw , X ∈ RK×L , w ∈ RL . Using the tables in D.2, ¡ ¢ ¡ ¢ tr ∇X g(X + t Y )T Y = tr 2wwT(X T + t Y T )Y T

T

T

= 2w (X Y + t Y Y )w

(1772) (1773)

Applying the equivalence (1770), d d T g(X + t Y ) = w (X + t Y )T (X + t Y )w dt dt ¡ ¢ = wT X T Y + Y TX + 2t Y T Y w T

T

T

= 2w (X Y + t Y Y )w

which is the same as (1773); hence, equivalence is demonstrated. It is easy to extract ∇g(X) from (1776) knowing only (1770): ¡ ¢ tr ∇X g(X + t Y )T Y = 2wT¡(X T Y + t Y T Y )w ¢ = 2 tr wwT(X T + t Y T )Y ¡ ¢ ¡ ¢ tr ∇X g(X)T Y = 2 tr wwTX T Y ⇔ ∇X g(X) = 2XwwT D.1.8.2

(1774) (1775) (1776)

(1777)

2

second-order

Likewise removing the evaluation at t = 0 from (1754), →Y

d2 g(X + t Y ) (1778) dt2 we can find a similar relationship between the second-order gradient and the second derivative: In the general case g(X) : RK×L → RM ×N from (1747) and (1750), dg 2(X + t Y ) =

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES ³ ¡ ¢T ´ d2 T tr ∇X tr ∇X gmn (X + t Y ) Y Y = 2 gmn (X + t Y ) dt

In the case of a real function g(X) : RK×L → R we have, of course, ³ ¡ ¢T ´ d2 tr ∇X tr ∇X g(X + t Y )T Y Y = 2 g(X + t Y ) dt

661 (1779)

(1780)

From (1759), the simpler case, where the real function g(X) : RK → R has vector argument, Y T ∇X2 g(X + t Y )Y =

d2 g(X + t Y ) dt2

(1781)

D.1.8.2.1 Example. Second-order gradient. Given real function g(X) = log det X having domain int SK + , we want to find ∇ 2 g(X) ∈ RK×K×K×K . From the tables in D.2,

§

h(X) , ∇g(X) = X −1 ∈ int SK +

so ∇ 2 g(X) = ∇h(X). By (1769) and (1732), for Y ∈ SK ¯ ¡ ¢ d ¯¯ T hmn (X + t Y ) tr ∇hmn (X) Y = dt ¯t=0 µ ¯ ¶ d ¯¯ = h(X + t Y ) dt ¯t=0 mn µ ¯ ¶ d ¯¯ = (X + t Y )−1 dt ¯t=0 mn ¡ −1 ¢ −1 = − X Y X mn

(1782)

(1783) (1784) (1785) (1786)

K×K Setting Y to a member of {ek eT | k, l = 1 . . . K } , and employing a l ∈ R property (34) of the trace function we find ¡ ¢ ¡ ¢ −1 ∇ 2 g(X)mnkl = tr ∇hmn (X)T ek eT = ∇hmn (X)kl = − X −1 ek eT l l X mn

(1787)

∇ 2 g(X)kl = ∇h(X)kl

¡ ¢ −1 = − X −1 ek eT ∈ RK×K l X

(1788) 2

From all these first- and second-order expressions, we may generate new ones by evaluating both sides at arbitrary t (in some open interval) but only after the differentiation.

662

APPENDIX D. MATRIX CALCULUS

D.2

Tables of gradients and derivatives

ˆ [152] [61] When proving results for symmetric matrices algebraically, it

is critical to take gradients ignoring symmetry and to then substitute symmetric entries afterward.

ˆ a, b∈R , n

x, y ∈ Rk , A , B ∈ Rm×n , X , Y ∈ RK×L , t, µ∈R , i, j, k, ℓ, K, L , m , n , M , N are integers, unless otherwise noted.

ˆx

means δ(δ(x)µ ) for µ ∈ R ; id est, entrywise vector exponentiation. δ is the main-diagonal linear operator (1359). x0 , 1, X 0 , I if square.

ˆ

µ

d dx



d dx1



§

→y  .  →y ,  .. , dg(x) , dg 2(x) (directional derivatives D.1), log x , e x , d dxk

√ |x| , sgn x , x/y (Hadamard quotient), x (entrywise square root), etcetera, are maps f : Rk → Rk that maintain dimension; e.g., ( A.1.1)

§

d −1 x , ∇x 1T δ(x)−1 1 dx

(1789)

ˆ For A a scalar or square matrix, we have the Taylor series [73, §3.6] ∞ X 1 k e , A k! k=0 A

§

(1790)

Further, [309, 5.4] eA ≻ 0

∀ A ∈ Sm

(1791)

ˆ For all square A and integer k detk A = det Ak

(1792)

D.2. TABLES OF GRADIENTS AND DERIVATIVES

D.2.1

663

algebraic

∇x x = ∇x xT = I ∈ Rk×k

∇X X = ∇X X T , I ∈ RK×L×K×L

(identity)

∇x (Ax − b) = AT ¡ ¢ ∇x xTA − bT = A

∇x (Ax − b)T(Ax − b) = 2AT(Ax − b) ∇x2 (Ax − b)T(Ax − b) = 2ATA ∇x kAx − bk = AT(Ax − b)/kAx − bk ∇x z T |Ax − b| = AT δ(z) sgn(Ax − b) , z¶ i 6= 0 ⇒ (Ax − b)i 6= 0 µ ¯ ¯ df (y) ∇x 1T f (|Ax − b|) = AT δ dy ¯ sgn(Ax − b) y=|Ax−b|

¡ ¢ ¡ ¢ ∇x xTAx + 2xTB y + y TC y = A + AT x + 2B y ∇x (x + y)TA(x + y) = (A +¢AT )(x + y) ¡ T 2 ∇x x Ax + 2xTB y + y TC y = A + AT

∇X aTXb = ∇X bTX Ta = abT ∇X aTX 2 b = X T abT + abT X T ∇X aTX −1 b = −X −T abT X −T

∇x aTxTxb = 2xaT b

confer ∂X −1 −1 (1724) = −X −1 ek eT X , l ∂Xkl (1788) ∇X aTX TXb = X(abT + baT )

∇x aTxxT b = (abT + baT )x

∇X aTXX T b = (abT + baT )X

∇x aTxTxa = 2xaTa

∇X aTX TXa = 2XaaT

∇x aTxxTa = 2aaTx

∇X aTXX T a = 2aaT X

∇x aTyxT b = b aTy

∇X aT Y X T b = baT Y

∇x aTy Tx b = y bTa

∇X aT Y TXb = Y abT

∇x aTxy T b = a bTy

∇X aTXY T b = abT Y

∇x aTxTy b = y aT b

∇X aTX T Y b = Y baT

∇X (X −1 )kl =

664

APPENDIX D. MATRIX CALCULUS

algebraic continued d (X + dt

tY ) = Y

d B T (X + dt d B T (X + dt d B T (X + dt

t Y )−1 A = −B T (X + t Y )−1 Y (X + t Y )−1 A t Y )−TA = −B T (X + t Y )−T Y T (X + t Y )−TA t Y )µ A = ... ,

d2 B T (X + dt2 d dt

¡

d2 dt2

−1 ≤ µ ≤ 1, X , Y ∈ SM +

t Y )−1 A = 2B T (X + t Y )−1 Y (X + t Y )−1 Y (X + t Y )−1 A

¢ (X + t Y )TA(X + t Y ) = Y TAX + X TAY + 2 t Y TAY

¡

¢ (X + t Y )TA(X + t Y ) = 2 Y TAY

d ((X + dt

t Y )A(X + t Y )) = YAX + XAY + 2 t YAY

d2 ((X + dt2

t Y )A(X + t Y )) = 2 YAY

§

D.2.1.0.1 Exercise. Expand these tables. Provide unfinished table entries indicated by . . . throughout D.2.

H

§

D.2.1.0.2 Exercise. log . ( D.1.7) Find the first four terms of the Taylor series expansion for log x about x = 1. Prove that log x ≤ x − 1; ·alternatively, ¸ · ¸plot the supporting hyperplane to 1 x . H = the hypograph of log x at 0 log x

D.2.2

trace Kronecker

∇vec X tr(AXBX T ) = ∇vec X vec(X)T (B T ⊗ A) vec X = (B ⊗ AT + B T ⊗ A) vec X 2 2 T T T T T ∇vec X tr(AXBX ) = ∇vec X vec(X) (B ⊗ A) vec X = B ⊗ A + B ⊗ A

D.2. TABLES OF GRADIENTS AND DERIVATIVES

D.2.3

665

trace

∇x µ x = µI

∇X tr µX = ∇X µ tr X = µI

d −1 x = −x−2 ∇x 1T δ(x)−1 1 = dx T −1 ∇x 1 δ(x) y = −δ(x)−2 y

∇X tr X −1 = −X −2T ∇X tr(X −1 Y ) = ∇X tr(Y X −1 ) = −X −T Y TX −T

d µ dx x

∇X tr X µ = µX µ−1 ,

= µx µ−1

X ∈ SM

∇X tr X j = jX (j−1)T ∇x (b − aTx)−1 = (b − aTx)−2 a ∇x (b − aTx)µ = −µ(b − aTx)µ−1 a ∇x xTy = ∇x y Tx = y

¡ ¢ ¡ ¢T ∇X tr (B − AX)−1 = (B − AX)−2 A ∇X tr(X T Y ) = ∇X tr(Y X T ) = ∇X tr(Y TX) = ∇X tr(XY T ) = Y ∇X tr(AXBX T ) = ∇X tr(XBX TA) = ATXB T + AXB ∇X tr(AXBX) = ∇X tr(XBXA) = ATX TB T + B TX TAT ∇X tr(AXAXAX) = ∇X tr(XAXAXA) = 3(AXAXA )T ∇X tr(Y X k ) = ∇X tr(X k Y ) =

k−1 P¡ i=0

X i Y X k−1−i

¢T

∇X tr(Y TXX T Y ) = ∇X tr(X T Y Y TX) = 2 Y Y TX ∇X tr(Y TX TXY ) = ∇X tr(XY Y TX T ) = 2XY Y T

¡ ¢ ∇X tr (X + Y )T (X + Y ) = 2(X + Y ) = ∇X kX + Y k2F ∇X tr((X + Y )(X + Y )) = 2(X + Y )T ∇X tr(ATXB) = ∇X tr(X TAB T ) = AB T T −1 −T T −T ∇X tr(A X B) = ∇X tr(X AB ) = −X AB T X −T ∇X aTXb = ∇X tr(baTX) = ∇X tr(XbaT ) = abT ∇X bTX Ta = ∇X tr(X TabT ) = ∇X tr(abTX T ) = abT ∇X aTX −1 b = ∇X tr(X −T abT ) = −X −T abT X −T µ ∇X aTX b = ...

666

APPENDIX D. MATRIX CALCULUS

trace continued d dt

tr g(X + t Y ) = tr dtd g(X + t Y )

d dt

tr(X + t Y ) = tr Y

d dt

tr j(X + t Y ) = j tr j−1(X + t Y ) tr Y

d dt

tr(X + t Y )j = j tr((X + t Y )j−1 Y )

d dt

tr((X + t Y )Y ) = tr Y 2

d dt

¡ ¢ tr (X + t Y )k Y =

d dt

¡ ¢ tr(Y (X + t Y )k ) = k tr (X + t Y )k−1 Y 2 ,

d dt

¡ ¢ tr (X + t Y )k Y =

d dt

tr(Y (X + t Y )k ) = tr

d dt d dt d dt d dt d dt

(X + t Y )i Y (X + t Y )k−1−i Y

i=0

¡ ¢ ¡ ¢ tr B T (X + t Y )−1 A = 2 tr B T (X + t Y )−1 Y (X + t Y )−1 Y (X + t Y )−1 A ¡ ¢ ¡ ¢ tr (X + t Y )TA(X + t Y ) = 2 tr Y TAY

tr((X + t Y )A(X + t Y )) = tr(YAX + XAY + 2 t YAY )

d2 dt2

k ∈ {0, 1, 2}

k−1 P

¡ ¢ ¡ ¢ tr (X + t Y )TA(X + t Y ) = tr Y TAX + X TAY + 2 t Y TAY

d2 dt2 d dt

(∀ j)

tr((X + t Y )−1 Y ) = − tr((X + t Y )−1 Y (X + t Y )−1 Y ) ¡ T ¢ ¡ ¢ tr B (X + t Y )−1 A = − tr B T (X + t Y )−1 Y (X + t Y )−1 A ¡ ¢ ¡ ¢ tr B T (X + t Y )−TA = − tr B T (X + t Y )−T Y T (X + t Y )−TA ¡ ¢ tr B T (X + t Y )−k A = ... , k > 0 ¡ ¢ tr B T (X + t Y )µ A = ... , −1 ≤ µ ≤ 1, X , Y ∈ SM +

d2 dt2 d dt

[189, p.491]

tr((X + t Y )A(X + t Y )) = 2 tr(YAY )

D.2. TABLES OF GRADIENTS AND DERIVATIVES

D.2.4

667

logarithmic determinant

x ≻ 0, det X > 0 on some neighborhood of X , and det(X + t Y ) > 0 on some open interval of t ; otherwise, log( ) would be discontinuous. [78, p.75] d dx

log x = x−1

∇X log det X = X −T ∇X2 log det(X)kl =

¡ ¢ ∂X −T −1 T = − X −1 ek eT X , confer (1741)(1788) l ∂Xkl

d dx

log x−1 = −x−1

∇X log det X −1 = −X −T

d dx

log x µ = µx−1

∇X log detµ X = µX −T µ

∇X log det X = µX −T ∇X log det X k = ∇X log detk X = kX −T ∇X log detµ (X + t Y ) = µ(X + t Y )−T 1 ∇x log(aTx + b) = a aTx+b

∇X log det(AX + B) = AT(AX + B)−T ∇X log det(I ± ATXA) = ±A(I ± ATXA)−TAT ∇X log det(X + t Y )k = ∇X log detk (X + t Y ) = k(X + t Y )−T d dt

log det(X + t Y ) = tr ((X + t Y )−1 Y )

d2 dt2 d dt

log det(X + t Y )−1 = − tr ((X + t Y )−1 Y )

d2 dt2 d dt

log det(X + t Y ) = − tr ((X + t Y )−1 Y (X + t Y )−1 Y )

log det(X + t Y )−1 = tr ((X + t Y )−1 Y (X + t Y )−1 Y )

log det(δ(A(x + t y) + a)2 + µI) ¡ ¢ = tr (δ(A(x + t y) + a)2 + µI)−1 2δ(A(x + t y) + a)δ(Ay)

668

D.2.5

APPENDIX D. MATRIX CALCULUS

determinant

∇X det X = ∇X det X T = det(X)X −T ∇X det X −1 = − det(X −1 )X −T = − det(X)−1 X −T ∇X detµ X = µ detµ (X)X −T µ

µ

∇X det X = µ det(X )X −T ¡ ¢ ∇X det X k = k detk−1(X) tr(X)I − X T ,

X ∈ R2×2

∇X det X k = ∇X detk X = k det(X k )X −T = k detk (X)X −T ∇X detµ (X + t Y ) = µ detµ (X + t Y )(X + t Y )−T ∇X det(X + t Y )k = ∇X detk (X + t Y ) = k detk (X + t Y )(X + t Y )−T d dt

det(X + t Y ) = det(X + t Y ) tr((X + t Y )−1 Y )

d2 dt2 d dt

det(X + t Y )−1 = − det(X + t Y )−1 tr((X + t Y )−1 Y )

d2 dt2 d dt

det(X + t Y ) = det(X + t Y )(tr 2 ((X + t Y )−1 Y ) − tr((X + t Y )−1 Y (X + t Y )−1 Y ))

det(X + t Y )−1 = det(X + t Y )−1 (tr 2 ((X + t Y )−1 Y ) + tr((X + t Y )−1 Y (X + t Y )−1 Y ))

detµ (X + t Y ) = µ detµ (X + t Y ) tr((X + t Y )−1 Y )

D.2.6

logarithmic

Matrix logarithm. d log(X + dt d log(I − dt

t Y )µ = µY (X + t Y )−1 = µ(X + t Y )−1 Y ,

XY = Y X

t Y )µ = −µY (I − t Y )−1 = −µ(I − t Y )−1 Y

[189, p.493]

D.2. TABLES OF GRADIENTS AND DERIVATIVES

D.2.7

exponential

§ §

669

§

Matrix exponential. [73, 3.6, 4.5] [309, 5.4] ∇X etr(Y

TX)

= ∇X det eY

∇X tr eY X = eY

TX T

TX

= etr(Y

Y T = Y T eX

TX)

Y

(∀ X, Y )

TY T

∇x 1T eAx = ATeAx ∇x 1T e|Ax| = AT δ(sgn(Ax))e|Ax| ∇x log(1T e x ) = ∇x2

1 1T e x

(Ax)i 6= 0

ex

µ ¶ 1 1 x x xT log(1 e ) = T x δ(e ) − T x e e 1 e 1 e T x

µ k 1¶ 1 Q ∇x xi = xik 1/x k i=1 i=1 k Q

∇x2

1 k

µ k 1¶µ ¶ 1 Q 1 T −2 k xi = − x δ(x) − (1/x)(1/x) k i=1 i k i=1 k Q

d tY e dt

1 k

= etY Y = Y etY

d X+ t Y e dt d 2 X+ t Y e dt2

= eX+ t Y Y = Y eX+ t Y , = eX+ t Y Y 2 = Y eX+ t Y Y = Y 2 eX+ t Y ,

d j tr(X+ t Y ) e dt j

= etr(X+ t Y ) tr j(Y )

XY = Y X XY = Y X

670

APPENDIX D. MATRIX CALCULUS

Appendix E Projection

§

§

For any A ∈ Rm×n , the pseudoinverse [188, 7.3, prob.9] [232, 6.12, prob.19] [146, 5.5.4] [309, App.A]

§

A† , lim+(ATA + t I )−1AT = lim+AT(AAT + t I )−1 ∈ Rn×m t→0

t→0

(1793)

is a unique matrix from the optimal solution set to minimize kAX − Ik2F X ( 3.7.0.0.2). For any t > 0

§

I − A(ATA + t I )−1AT = t(AAT + t I )−1

(1794)

Equivalently, pseudoinverse A† is that unique matrix satisfying the Moore-Penrose conditions : [190, 1.3] [348]

§

1. AA†A = A 2. A†AA† = A†

3. (AA† )T = AA† 4. (A†A)T = A†A

which are necessary and sufficient to establish the pseudoinverse whose principal action is to injectively map R(A) onto R(AT ) (Figure 153). Conditions 1 and 3 are necessary and sufficient for AA† to be the orthogonal projector on R(A) , while conditions 2 and 4 hold iff A†A is the orthogonal projector on R(AT ). Range and nullspace of the pseudoinverse [248] [306, III.1, exer.1]

§

R(A† ) = R(AT ) , N (A† ) = N (AT ) ,

R(A†T ) = R(A) N (A†T ) = N (A)

§

(1795) (1796)

can be derived by singular value decomposition ( A.6). © 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

671

672

APPENDIX E. PROJECTION p = AA† b

Rn x R(AT )

x = A† p

Rm Ax = p

R(A) p

x = A† b

0

{x}

†

0 = A (b − p) N (A)

0

b

{b}

b−p

N (AT )

Figure 153: (confer Figure 16) Pseudoinverse A† ∈ Rn×m action: [309, p.449] Component of vector b in N (AT ) maps to origin, while component of b in R(A) maps to rowspace R(AT ). For any A ∈ Rm×n , inversion is bijective ∀ p ∈ R(A). The following relations reliably hold without qualification: a. b. c. d. e. f.

AT† = A†T A†† = A (AAT )† = A†TA† (ATA)† = A†A†T (AA† )† = AA† (A†A)† = A†A

Yet for arbitrary A,B it is generally true that (AB)† 6= B †A† : E.0.0.0.1 Theorem. Pseudoinverse of product. [155] [53] [224, exer.7.23] For A ∈ Rm×n and B ∈ Rn×k (AB)† = B †A†

(1797)

R(ATAB) ⊆ R(B) and R(BB TAT ) ⊆ R(AT )

(1798) ⋄

if and only if

673 U T = U † for orthonormal (including the orthogonal) matrices U . So, for orthonormal matrices U, Q and arbitrary A (UAQT )† = QA† U T E.0.0.0.2 Prove:

Exercise. Kronecker inverse. (A ⊗ B)† = A† ⊗ B †

E.0.1

(1800) H

Logical deductions

When A is invertible, A† = A−1 ; so A†A = AA† = I . A ∈ Rm×n [139, 5.3.3.1] [224, 7] [274]

§

g. h. i. j. k. l.

(1799)

A†A = I , AA† = I , A†Aω = ω , AA† υ = υ , A†A = AA† , Ak† = A†k ,

§

Otherwise, for

A† = (ATA)−1 AT , rank A = n † T T −1 A = A (AA ) , rank A = m T ω ∈ R(A ) υ ∈ R(A) A normal A normal, k an integer

Equivalent to the corresponding Moore-Penrose condition: 1. AT = ATAA†

or

AT = A†AAT

2. A†T = A†TA†A

or

A†T = AA†A†T

§

When A is symmetric, A† is symmetric and ( A.6) A º 0 ⇔ A† º 0

§

(1801)

E.0.1.0.1 Example. Solution to classical linear equation Ax = b . In 2.5.1.1, the solution set to matrix equation Ax = b was represented as an intersection of hyperplanes. Regardless of rank of A or its shape (fat or skinny), interpretation as a hyperplane intersection describing a possibly empty affine set generally holds. If matrix A is rank deficient or fat, there is an infinity of solutions x when b ∈ R(A). A unique solution occurs when the hyperplanes intersect at a single point.

674

APPENDIX E. PROJECTION

Given arbitrary matrix A (of any rank and dimension) and vector b not necessarily in R(A) , we wish to find a best solution x⋆ to Ax ≈ b

(1802)

in a Euclidean sense by solving an algebraic expression for orthogonal projection of b on R(A) minimize kAx − bk2 x

(1803)

Necessary and sufficient condition for optimal solution to this unconstrained optimization is the so-called normal equation that results from zeroing the convex objective’s gradient: ( D.2.1)

§

ATA x = AT b

(1804)

normal because error vector b −Ax is perpendicular to R(A) ; id est, AT (b −Ax) = 0. Given any matrix A and any vector b , the normal equation is solvable exactly; always so, because R(ATA) = R(AT ) and AT b ∈ R(AT ). When A is skinny-or-square full-rank, normal equation (1804) can be solved exactly by inversion: x⋆ = (ATA)−1 AT b ≡ A† b

(1805)

For matrix A of arbitrary rank and shape, on the other hand, ATA might not be invertible. Yet the normal equation can always be solved exactly by: (1793) x⋆ = lim+(ATA + t I )−1 AT b = A† b (1806) t→0

invertible for any positive value of t by (1395). The exact inversion (1805) and this pseudoinverse solution (1806) each solve a limited regularization °· ¸ · ¸°2 ° A b ° 2 2 ° √ lim+ minimize kAx − bk + t kxk ≡ lim+ minimize ° x − ° 0 ° x x tI t→0 t→0 (1807) simultaneously providing least squares solution to (1803) and the classical least norm solutionE.1 [309, App.A.4] (confer 3.3.0.0.1)

§

arg minimize kxk2 x

subject to Ax = AA† b

E.1

(1808)

This means: optimal solutions of lesser norm than the so-called least norm solution (1808) can be obtained (at expense of approximation Ax ≈ b hence, of perpendicularity) by ignoring the limiting operation and introducing finite positive values of t into (1807).

E.1. IDEMPOTENT MATRICES

675

where AA† b is the orthogonal projection of vector b on R(A). Least norm solution can be interpreted as orthogonal projection of the origin 0 on affine subset A = {x | Ax = AA† b} ; ( E.5.0.0.5, E.5.0.0.6)

§

§

arg minimize kx − 0k2

(1809)

x

subject to x ∈ A

equivalently, maximization of the Euclidean ball until it kisses A ; rather, arg dist(0, A). 2

E.1

Idempotent matrices

§ §

§

Projection matrices are square and defined by idempotence, P 2 = P ; [309, 2.6] [190, 1.3] equivalent to the condition, P be diagonalizable [188, 3.3, prob.3] with eigenvalues φi ∈ {0, 1}. [368, 4.1, thm.4.1] Idempotent matrices are not necessarily symmetric. The transpose of an idempotent matrix remains idempotent; P TP T = P T . Solely excepting P = I , all projection matrices are neither orthogonal ( B.5) or invertible. [309, 3.4] The collection of all projection matrices of particular dimension does not form a convex set. Suppose we wish to project nonorthogonally (obliquely ) on the range of any particular matrix A ∈ Rm×n . All idempotent matrices projecting nonorthogonally on R(A) may be expressed:

§

§

§

P = A(A† + BZ T ) ∈ Rm×m

(1810)

ATZ = A† Z = 0

(1811)

where R(P ) = R(A) ,E.2 B ∈ Rn×k for k ∈ {1 . . . m} is otherwise arbitrary, and Z ∈ Rm×k is any matrix whose range is in N (AT ) ; id est, Evidently, the collection of nonorthogonal projectors projecting on R(A) is an affine subset © ª Pk = A(A† + BZ T ) | B ∈ Rn×k (1812) E.2

Proof. R(P ) ⊆ R(A) is obvious [309, §3.6]. By (134) and (135),

R(A† + BZ T ) = {(A† + BZ T )y | y ∈ Rm } ⊇ {(A† + BZ T )y | y ∈ R(A)} = R(AT ) R(P ) = {A(A† + BZ T )y | y ∈ Rm } ⊇ {A(A† + BZ T )y | (A† + BZ T )y ∈ R(AT )} = R(A)

¨

676

APPENDIX E. PROJECTION

When matrix A in (1810) is skinny full-rank (A† A = I ) or has orthonormal columns (ATA = I ), either property leads to a biorthogonal characterization of nonorthogonal projection:

E.1.1

Biorthogonal characterization

Any nonorthogonal projector P 2 = P ∈ Rm×m projecting on nontrivial R(U ) can be defined by a biorthogonality condition QT U = I ; the biorthogonal decomposition of P being (confer (1810)) P = U QT ,

§

QT U = I

(1813)

whereE.3 ( B.1.1.1) N (P ) = N (QT )

R(P ) = R(U ) ,

(1814)

and where generally (confer (1839))E.4 P † 6= P ,

P T 6= P ,

P ²0

kP k2 6= 1 ,

(1815)

and P is not nonexpansive (1840).

§

§

(⇐) To verify assertion (1813) we observe: because idempotent matrices are diagonalizable ( A.5), [188, 3.3, prob.3] they must have the form (1480) P = SΦS

−1

=

m X

φi si wiT

i=1

§

=

kX ≤m

si wiT

(1816)

i=1

§

that is a sum of k = rank P independent projector dyads (idempotent dyads, B.1.1, E.6.2.1) where φi ∈ {0, 1} are the eigenvalues of P [368, 4.1, thm.4.1] in diagonal matrix Φ ∈ Rm×m arranged in nonincreasing

§

E.3

Proof. Obviously, R(P ) ⊆ R(U ). Because QT U = I R(P ) = {U QT x | x ∈ Rm }

⊇ {U QT U y | y ∈ Rk } = R(U )

E.4

¨

Orthonormal decomposition (1836) (confer §E.3.4) is a special case of biorthogonal decomposition (1813) characterized by (1839). So, these characteristics (1815) are not necessary conditions for biorthogonality.

E.1. IDEMPOTENT MATRICES

677

order, and where si , wi ∈ Rm are the right- and left-eigenvectors of P , respectively, which are independent and real.E.5 Therefore £ ¤ U , S(: , 1 : k) = s1 · · · sk ∈ Rm×k (1817)

is the full-rank matrix S ∈ Rm×m having m − k columns truncated (corresponding to 0 eigenvalues), while  T  w1 . T −1 (1818) Q , S (1 : k , :) =  ..  ∈ Rk×m T wk

is matrix S −1 having the corresponding m − k rows truncated. By the 0 eigenvalues theorem ( A.7.3.0.1), R(U ) = R(P ) , R(Q) = R(P T ) , and

§

R(P ) N (P ) R(P T ) N (P T )

= = = =

span {si span {si span {wi span {wi

| | | |

φi φi φi φi

=1 =0 =1 =0

∀ i} ∀ i} ∀ i} ∀ i}

(1819)

Thus biorthogonality QT U = I is a necessary condition for idempotence, and so the collection of nonorthogonal projectors projecting on R(U ) is the affine m×k T subset Pk = U QT }. k where Qk = {Q | Q U = I , Q ∈ R (⇒) Biorthogonality is a sufficient condition for idempotence; 2

P =

k X

si wiT

i=1

k X

sj wjT = P

id est, if the cross-products are annihilated, then P 2 = P . Nonorthogonal projection of biorthogonal expansion,

(1820)

j=1

¨

x on R(P ) has expression like a

P x = U QT x =

k X

wiTx si

(1821)

i=1

When the domain is restricted to range of P , say x = U ξ for ξ ∈ Rk , then x = P x = U QT U ξ = U ξ and expansion is unique because the eigenvectors E.5

Eigenvectors of a real matrix corresponding to real eigenvalues must be real. (§A.5.0.0.1)

678

APPENDIX E. PROJECTION

T x

T ⊥ R(Q) cone(Q)

cone(U )

Px

Figure 154: Nonorthogonal projection of x ∈ R3 on R(U ) = R2 under biorthogonality condition; id est, P x = U QT x such that QT U = I . Any point along imaginary line T connecting x to P x will be projected nonorthogonally on P x with respect to horizontal plane constituting R2 = aff cone(U ) in this example. Extreme directions of cone(U ) correspond to two columns of U ; likewise for cone(Q). For purpose of illustration, we truncate each conic hull by truncating coefficients of conic combination at unity. Conic hull cone(Q) is headed upward at an angle, out of plane of page. Nonorthogonal projection would fail were N (QT ) in R(U ) (were T parallel to a line in R(U )).

E.1. IDEMPOTENT MATRICES

679

are linearly independent. Otherwise, any component of x in N (P ) = N (QT ) will be annihilated. Direction of nonorthogonal projection is orthogonal to R(Q) ⇔ QT U = I ; id est, for P x ∈ R(U ) P x − x ⊥ R(Q) in Rm E.1.1.0.1 Example. Illustration of nonorthogonal projector. Figure 154 shows cone(U ) , the conic hull of the columns of   1 1 U =  −0.5 0.3  0 0

(1822)

(1823)

from nonorthogonal projector P = U QT . Matrix U has a limitless number of left inverses because N (U T ) is nontrivial. Similarly depicted is left inverse QT from (1810)     0 0.3750 0.6250 Q = U †T + ZB T =  −1.2500 1.2500  +  0  [ 0.5 0.5 ] 1 0 0 (1824)   0.3750 0.6250 =  −1.2500 1.2500  0.5000 0.5000

where Z ∈ N (U T ) and matrix B is selected arbitrarily; id est, QT U = I because U is full-rank. Direction of projection on R(U ) is orthogonal to R(Q). Any point along line T in the figure, for example, will have the same projection. Were matrix Z instead equal to 0, then cone(Q) would become the relative dual to cone(U ) (sharing the same affine hull; 2.13.8, confer Figure 54(a)). In that case, projection P x = U U † x of x on R(U ) becomes orthogonal projection (and unique minimum-distance). 2

§

E.1.2

Idempotence summary

Nonorthogonal subspace-projector P is a (convex) linear operator defined by idempotence or biorthogonal decomposition (1813), but characterized not by symmetry or positive semidefiniteness or nonexpansivity (1840).

680

APPENDIX E. PROJECTION

E.2 I−P , Projection on algebraic complement It follows from the diagonalizability of idempotent matrices that I − P must also be a projection matrix because it too is idempotent, and because it may be expressed I − P = S(I − Φ)S

−1

=

m X i=1

(1 − φi )si wiT

(1825)

where (1 − φi ) ∈ {1, 0} are the eigenvalues of I − P (1396) whose eigenvectors si , wi are identical to those of P in (1816). A consequence of that complementary relationship of eigenvalues is the fact, [320, 2] [315, 2] for subspace projector P = P 2 ∈ Rm×m

§

R(P ) N (P ) R(P T ) N (P T )

= = = =

span {si span {si span {wi span {wi

| | | |

φi φi φi φi

=1 =0 =1 =0

∀ i} ∀ i} ∀ i} ∀ i}

= = = =

span {si span {si span {wi span {wi

| | | |

(1 − φi ) = 0 (1 − φi ) = 1 (1 − φi ) = 0 (1 − φi ) = 1

∀ i} ∀ i} ∀ i} ∀ i}

= = = =

§

N (I − P ) R(I − P ) N (I − P T ) R(I − P T ) (1826)

that is easy to see from (1816) and (1825). Idempotent I − P therefore projects vectors on its range, N (P ). Because all eigenvectors of a real idempotent matrix are real and independent, the algebraic complement of R(P ) [211, 3.3] is equivalent to N (P ) ;E.6 id est,

§

R(P )⊕N (P ) = R(P T )⊕N (P T ) = R(P T )⊕N (P ) = R(P )⊕N (P T ) = Rm (1827) because R(P ) ⊕ R(I − P ) = Rm . For idempotent P ∈ Rm×m , consequently, rank P + rank(I − P ) = m E.2.0.0.1

Theorem. Rank/Trace.

rank P = tr P E.6

(1828)

§

[368, 4.1, prob.9] (confer (1844))

P2 = P ⇔ and rank(I − P ) = tr(I − P )

(1829) ⋄

The same phenomenon occurs with symmetric (nonidempotent) matrices, for example. When the summands in A ⊕ B = Rm are orthogonal vector spaces, the algebraic complement is the orthogonal complement.

E.3. SYMMETRIC IDEMPOTENT MATRICES

E.2.1

681

Universal projector characteristic

Although projection is not necessarily orthogonal and R(P ) 6⊥ R(I − P ) in general, still for any projector P and any x ∈ Rm P x + (I − P )x = x

(1830)

must hold where R(I − P ) = N (P ) is the algebraic complement of R(P ). The algebraic complement of closed convex cone K , for example, is the ∗ negative dual cone −K . (1946)

E.3

Symmetric idempotent matrices

When idempotent matrix P is symmetric, P is an orthogonal projector. In other words, the direction of projection of point x ∈ Rm on subspace R(P ) is orthogonal to R(P ) ; id est, for P 2 = P ∈ Sm and projection P x ∈ R(P ) P x − x ⊥ R(P ) in Rm

§

(1831)

(confer (1822)) Perpendicularity is a necessary and sufficient condition for orthogonal projection on a subspace. [99, 4.9] A condition equivalent to (1831) is: Norm of direction x − P x is the infimum over all nonorthogonal projections of x on R(P ) ; [232, 3.3] for P 2 = P ∈ Sm , R(P ) = R(A) , matrices A , B , Z and positive integer k as defined for (1810), and given x ∈ Rm

§

kx − P xk2 = kx − AA† xk2 = inf kx − A(A† + BZ T )xk2 = dist(x , R(P )) n×k B∈R (1832) The infimum is attained for R(B) ⊆ N (A) over any affine subset of nonorthogonal projectors (1812) indexed by k .

§

Proof is straightforward: The vector 2-norm is a convex function. Setting gradient of the norm-square to 0, applying D.2, ¡ T ¢ A ABZ T − AT(I − AA† ) xxTA = 0 ⇔ (1833) T T T A ABZ xx A = 0 because AT = ATAA† . Projector P = AA† is therefore unique; the minimum-distance projector is the orthogonal projector, and vice versa. ¨

682

APPENDIX E. PROJECTION

We get P = AA† so this projection matrix must be symmetric. Then for any matrix A ∈ Rm×n , symmetric idempotent P projects a given vector x in Rm orthogonally on R(A). Under either condition (1831) or (1832), the projection P x is unique minimum-distance; for subspaces, perpendicularity and minimum-distance conditions are equivalent.

E.3.1

Four subspaces

We summarize the orthogonal projectors projecting on the four fundamental subspaces: for A ∈ Rm×n A†A AA† I −A†A I −AA†

: : : :

Rn on R(A†A) Rm on R(AA† ) Rn on R(I −A†A) Rm on R(I −AA† )

= = = =

R(AT ) R(A) N (A) N (AT )

(1834)

For completeness:E.7 (1826) N (A†A) N (AA† ) N (I −A†A) N (I −AA† )

E.3.2

= = = =

N (A) N (AT ) R(AT ) R(A)

(1835)

Orthogonal characterization

Any symmetric projector P 2 = P ∈ Sm projecting on nontrivial R(Q) can be defined by the orthonormality condition QTQ = I . When skinny matrix Q ∈ Rm×k has orthonormal columns, then Q† = QT by the Moore-Penrose conditions. Hence, any P having an orthonormal decomposition ( E.3.4)

§

§

P = QQT ,

QTQ = I

(1836)

where [309, 3.3] (1539) R(P ) = R(Q) ,

N (P ) = N (QT )

(1837)

Proof is by singular value decomposition (§A.6.2): N (A†A) ⊆ N (A) is obvious. Conversely, suppose A†Ax = 0. Then xTA†Ax = xTQQTx = kQTxk2 = 0 where A = U ΣQT is the subcompact singular value decomposition. Because R(Q) = R(AT ) , then x ∈ N (A) which implies N (A†A) ⊇ N (A). ∴ N (A†A) = N (A). ¨ E.7

E.3. SYMMETRIC IDEMPOTENT MATRICES

683

is an orthogonal projector projecting on R(Q) having, for P x ∈ R(Q) (confer (1822)) P x − x ⊥ R(Q) in Rm (1838)

§

From (1836), orthogonal projector P is obviously positive semidefinite ( A.3.1.0.6); necessarily, PT = P ,

P† = P ,

kP k2 = 1 ,

P º0

(1839)

and kP xk = kQQTxk = kQTxk because kQyk = kyk ∀ y ∈ Rk . All orthogonal projectors are therefore nonexpansive because p hP x , xi = kP xk = kQTxk ≤ kxk ∀ x ∈ Rm (1840)

the Bessel inequality, [99] [211] with equality when x ∈ R(Q). From the diagonalization of idempotent matrices (1816) on page 676 P = SΦS

T

=

m X

φi si sT i

=

i=1

§

kX ≤m

si sT i

(1841)

i=1

orthogonal projection of point x on R(P ) has expression like an orthogonal expansion [99, 4.10] P x = QQT x =

k X

sT i x si

(1842)

¤ s1 · · · sk ∈ Rm×k

(1843)

i=1

where Q = S(: , 1 : k) =

£

and where the si [sic] are orthonormal eigenvectors of symmetric idempotent P . When the domain is restricted to range of P , say x = Qξ for ξ ∈ Rk , then x = P x = QQTQξ = Qξ and expansion is unique. Otherwise, any component of x in N (QT ) will be annihilated. E.3.2.0.1

Theorem. Symmetric rank/trace.

rank P = tr P = kP k2F

(confer (1829) (1400))

PT = P , P2 = P ⇔ and rank(I − P ) = tr(I − P ) = kI − P k2F ⋄

(1844)

684

APPENDIX E. PROJECTION

§

Proof. We take as given Theorem E.2.0.0.1 establishing idempotence. We have left only to show tr P = kP k2F ⇒ P T = P , established in [368, 7.1]. ¨

E.3.3

§

Summary, symmetric idempotent

§

(confer E.1.2) Orthogonal projector P is a (convex) linear operator defined [185, A.3.1] by idempotence and symmetry, and characterized by positive semidefiniteness and nonexpansivity. The algebraic complement ( E.2) to R(P ) becomes the orthogonal complement R(I − P ) ; id est, R(P ) ⊥ R(I − P ).

§

E.3.4

Orthonormal decomposition

When Z = 0 in the general nonorthogonal projector A(A† + BZ T ) (1810), an orthogonal projector results (for any matrix A) characterized principally by idempotence and symmetry. Any real orthogonal projector may, in fact, be represented by an orthonormal decomposition such as (1836). [190, 1, prob.42]

§

To verify that assertion for the four fundamental subspaces (1834), we need only to express A by subcompact singular value decomposition ( A.6.2): From pseudoinverse (1508) of A = U ΣQT ∈ Rm×n

§

AA† = U ΣΣ† U T = U U T , I − AA† = I − U U T = U ⊥ U ⊥T ,

A†A = QΣ† ΣQT = QQT I − A†A = I − QQT = Q⊥ Q⊥T (1845)

where U ⊥ ∈ Rm×m−rank A holds columnar an orthonormal basis for the orthogonal complement of R(U ) , and likewise for Q⊥ ∈ Rn×n−rank A . Existence of an orthonormal decomposition is sufficient to establish idempotence and symmetry of an orthogonal projector (1836). ¨

E.3.5

Unifying trait of all projectors: direction

§

Relation (1845) shows: orthogonal projectors simultaneously possess a biorthogonal decomposition (confer E.1.1) (for example, AA† for skinny-or-square A full-rank) and an orthonormal decomposition (U U T whence P x = U U Tx).

E.3. SYMMETRIC IDEMPOTENT MATRICES E.3.5.1

685

orthogonal projector, orthonormal decomposition

Consider orthogonal expansion of x ∈ R(U ) : T

x = UU x =

n X

ui uT i x

i=1

(1846)

§

a sum of one-dimensional orthogonal projections ( E.6.3), where U , [ u 1 · · · un ]

and

U TU = I

(1847)

and where the subspace projector has two expressions, (1845) AA† , U U T

(1848)

where A ∈ Rm×n has rank n . The direction of projection of x on uj for some j ∈ {1 . . . n} , for example, is orthogonal to uj but parallel to a vector in the span of all the remaining vectors constituting the columns of U ; T uT j (uj uj x − x) = 0

T T uj uT j x − x = uj uj x − U U x ∈ R({ui | i = 1 . . . n , i 6= j})

E.3.5.2

(1849)

orthogonal projector, biorthogonal decomposition

We get a similar result for the biorthogonal expansion of x ∈ R(A). Define A , [ a1 a2 · · · an ] ∈ Rm×n and the rows of the pseudoinverseE.8  ∗T a1  ∗T  a A† ,  2.  .. a∗T n



   ∈ Rn×m 

(1850)

(1851)

under biorthogonality condition A† A = I . In biorthogonal expansion ( 2.13.8) n X † ai a∗T (1852) x = AA x = i x

§

i=1

E.8

Notation * in this context connotes extreme direction of a dual cone; e.g., (379) or Example E.5.0.0.3.

686

APPENDIX E. PROJECTION

the direction of projection of x on aj for some particular j ∈ {1 . . . n} , for example, is orthogonal to a∗j and parallel to a vector in the span of all the remaining vectors constituting the columns of A ; ∗T a∗T j (aj aj x − x) = 0

∗T † aj a∗T j x − x = aj aj x − AA x ∈ R({ai | i = 1 . . . n , i 6= j})

E.3.5.3

(1853)

nonorthogonal projector, biorthogonal decomposition

§

Because the result in E.3.5.2 is independent of matrix symmetry AA† = (AA† )T , we must get the same result for any nonorthogonal projector characterized by a biorthogonality condition; namely, for nonorthogonal projector P = U QT (1813) under biorthogonality condition QT U = I , in the biorthogonal expansion of x ∈ R(U ) T

x = UQ x =

k X

ui qiT x

(1854)

i=1

where

£ ¤ U , u1 · · · uk ∈ Rm×k  T  q1 . QT ,  ..  ∈ Rk×m qkT

(1855)

the direction of projection of x on uj is orthogonal to qj and parallel to a vector in the span of the remaining ui : qjT(uj qjT x − x) = 0

uj qjT x − x = uj qjT x − U QT x ∈ R({ui | i = 1 . . . k , i 6= j})

E.4

(1856)

Algebra of projection on affine subsets

Let PA x denote projection of x on affine subset A , R + α where R is a subspace and α ∈ A . Then, because R is parallel to A , it holds: PA x = PR+α x = (I − PR )(α) + PR x = PR (x − α) + α

(1857)

Subspace projector PR is a linear operator (PA is not), and PR (x + y) = PR x whenever y ⊥ R and PR is an orthogonal projector.

E.5. PROJECTION EXAMPLES

687

§

E.4.0.0.1 Theorem. Orthogonal projection on affine subset. [99, 9.26] Let A = R + α be an affine subset where α ∈ A , and let R⊥ be the orthogonal complement of subspace R . Then PA x is the orthogonal projection of x ∈ Rn on A if and only if PA x ∈ A ,

hPA x − x , a − αi = 0 ∀ a ∈ A

(1858)

PA x − x ∈ R⊥

(1859) ⋄

or if and only if PA x ∈ A ,

E.5

Projection examples

E.5.0.0.1 Example. Orthogonal projection on orthogonal basis. Orthogonal projection on a subspace can instead be accomplished by orthogonally projecting on the individual members of an orthogonal basis for that subspace. Suppose, for example, matrix A ∈ Rm×n holds an orthonormal basis for R(A) in its columns; A , [ a1 a2 · · · an ] . Then orthogonal projection of vector x ∈ Rn on R(A) is a sum of one-dimensional orthogonal projections P x = AA† x = A(ATA)−1ATx = AATx =

n X

ai aT i x

(1860)

i=1

§

where each symmetric dyad ai aT i is an orthogonal projector projecting on R(ai ). ( E.6.3) Because kx − P xk is minimized by orthogonal projection, P x is considered to be the best approximation (in the Euclidean sense) to x from the set R(A) . [99, 4.9] 2

§

E.5.0.0.2 Example. Orthogonal projection on span of nonorthogonal basis. Orthogonal projection on a subspace can also be accomplished by projecting nonorthogonally on the individual members of any nonorthogonal basis for that subspace. This interpretation is in fact the principal application of the pseudoinverse we discussed. Now suppose matrix A holds a nonorthogonal basis for R(A) in its columns, A = [ a1 a2 · · · an ] ∈ Rm×n

(1850)

688

APPENDIX E. PROJECTION

and define the rows a∗T of its pseudoinverse A† as in (1851). Then i orthogonal projection of vector x ∈ Rn on R(A) is a sum of one-dimensional nonorthogonal projections P x = AA† x =

n X

ai a∗T i x

(1861)

i=1

where each nonsymmetric dyad ai a∗T i is a nonorthogonal projector projecting on R(ai ) , ( E.6.1) idempotent because of biorthogonality condition A† A = I . The projection P x is regarded as the best approximation to x from the set R(A) , as it was in Example E.5.0.0.1. 2

§

E.5.0.0.3 Example. Biorthogonal expansion as nonorthogonal projection. Biorthogonal expansion can be viewed as a sum of components, each a nonorthogonal projection on the range of an extreme direction of a pointed polyhedral cone K ; e.g., Figure 155. Suppose matrix A ∈ Rm×n holds a nonorthogonal basis for R(A) in its columns as in (1850), and the rows of pseudoinverse A† are defined as in (1851). Assuming the most general biorthogonality condition † T (A + BZ )A = I with BZ T defined as for (1810), then biorthogonal expansion of vector x is a sum of one-dimensional nonorthogonal projections; for x ∈ R(A) †

T

†

x = A(A + BZ )x = AA x =

n X

ai a∗T i x

(1862)

i=1

where each dyad ai a∗T is a nonorthogonal projector projecting on R(ai ). i ( E.6.1) The extreme directions of K = cone(A) are {a1 , . . . , an } the linearly independent columns of A , while the extreme directions {a∗1 , . . . , a∗n } ∗ of relative dual cone K ∩ aff K = cone(A†T ) ( 2.13.9.4) correspond to the linearly independent ( B.1.1.1) rows of A† . Directions of nonorthogonal projection are determined by the pseudoinverse; id est, direction of ∗ E.9 projection ai a∗T i x−x on R(ai ) is orthogonal to ai .

§

§

E.9

§

This remains true in high dimension although only a little more difficult to visualize in R3 ; confer , Figure 61.

E.5. PROJECTION EXAMPLES

689

a∗2

K

a2

∗

x z

K a1

y 0 a1 ⊥ a∗2

a2 ⊥ a∗1

x = y + z = Pa1 x + Pa2 x

K

∗

a∗1 Figure 155: (confer Figure 60) Biorthogonal expansion of point x ∈ aff K is found by projecting x nonorthogonally on extreme directions of polyhedral cone K ⊂ R2 . (Dotted lines of projection bound this translated negated cone.) Direction of projection on extreme direction a1 is orthogonal to extreme ∗ direction a∗1 of dual cone K and parallel to a2 ( E.3.5); similarly, direction of projection on a2 is orthogonal to a∗2 and parallel to a1 . Point x is sum of nonorthogonal projections: x on R(a1 ) and x on R(a2 ). Expansion is unique because extreme directions of K are linearly independent. Were ∗ a1 orthogonal to a2 , then K would be identical to K and nonorthogonal projections would become orthogonal.

§

690

APPENDIX E. PROJECTION

Because the extreme directions of this cone K are linearly independent, the component projections are unique in the sense:

ˆ there is only one linear combination of extreme directions of K that yields a particular point x ∈ R(A) whenever

R(A) = aff K = R(a1 ) ⊕ R(a2 ) ⊕ . . . ⊕ R(an )

(1863) 2

E.5.0.0.4 Example. Nonorthogonal projection on elementary matrix. Suppose PY is a linear nonorthogonal projector projecting on subspace Y , and suppose the range of a vector u is linearly independent of Y ; id est, for some other subspace M containing Y suppose M = R(u) ⊕ Y

(1864)

Assuming PM x = Pu x + PY x holds, then it follows for vector x ∈ M Pu x = x − PY x ,

PY x = x − Pu x

(1865)

nonorthogonal projection of x on R(u) can be determined from nonorthogonal projection of x on Y , and vice versa. Such a scenario is realizable were there some arbitrary basis for Y populating a full-rank skinny-or-square matrix A u ] ∈ Rn+1

A , [ basis Y

(1866)

Then PM = AA† fulfills the requirements, with Pu = A(: , n + 1)A† (n + 1 , :) and PY = A(: , 1 : n)A† (1 : n , :). Observe, PM is an orthogonal projector whereas PY and Pu are nonorthogonal projectors. Now suppose, for example, PY is an elementary matrix ( B.3); in particular, i h √ 2VN ∈ RN ×N (1867) PY = I − e1 1T = 0 √ where Y = N (1T ) . We have M = RN , A = [ 2VN e1 ] , and u = e1 . Thus Pu = e1 1T is a nonorthogonal projector projecting on R(u) in a direction parallel to a vector in Y ( E.3.5), and PY x = x − e1 1T x is a nonorthogonal projection of x on Y in a direction parallel to u . 2

§

§

E.5. PROJECTION EXAMPLES

691

§

E.5.0.0.5 Example. Projecting the origin on a hyperplane. (confer 2.4.2.0.2) Given the hyperplane representation having b ∈ R and nonzero normal a ∈ Rm ∂H = {y | aTy = b} ⊂ Rm

(108)

orthogonal projection of the origin P 0 on that hyperplane is the unique optimal solution to a minimization problem: (1832) kP 0 − 0k2 = inf

y∈∂H

ky − 0k2

= inf kZ ξ + xk2 m−1

(1868)

ξ∈ R

where x is any solution to aTy = b , and where the columns of Z ∈ Rm×m−1 constitute a basis for N (aT ) so that y = Z ξ + x ∈ ∂H for all ξ ∈ Rm−1 . The infimum can be found by setting the gradient (with respect to ξ) of the strictly convex norm-square to 0. We find the minimizing argument ξ ⋆ = −(Z TZ )−1 Z T x

(1869)

¡ ¢ y ⋆ = I − Z(Z TZ )−1 Z T x

(1870)

so and from (1834)

P 0 = y ⋆ = a(aTa)−1 aT x =

a T b a aT † x= a x , A Ax = a kak kak kak2 kak2

(1871)

In words, any point x in the hyperplane ∂H projected on its normal a (confer (1895)) yields that point y ⋆ in the hyperplane closest to the origin. 2 E.5.0.0.6 Example. Projection on affine subset. The technique of Example E.5.0.0.5 is extensible. Given an intersection of hyperplanes A = {y | Ay = b} ⊂ Rm (1872)

where each row of A ∈ Rm×n is nonzero and b ∈ R(A) , then the orthogonal projection P x of any point x ∈ Rn on A is the solution to a minimization problem: kP x − xk2 = inf ky − xk2 y∈A (1873) = inf kZ ξ + yp − xk2 ξ∈ R

n−rank A

692

APPENDIX E. PROJECTION

where yp is any solution to Ay = b , and where the columns of Z ∈ Rn×n−rank A constitute a basis for N (A) so that y = Z ξ + yp ∈ A for all ξ ∈ Rn−rank A . The infimum is found by setting the gradient of the strictly convex norm-square to 0. The minimizing argument is ξ ⋆ = −(Z TZ )−1 Z T (yp − x)

(1874)

¡ ¢ y ⋆ = I − Z(Z TZ )−1 Z T (yp − x) + x

(1875)

so and from (1834),

P x = y ⋆ = x − A† (Ax − b)

(1876)

= (I − A†A)x + A†Ayp

which is a projection of x on N (A) then translated perpendicularly with respect to the nullspace until it meets the affine subset A . 2 E.5.0.0.7 Example. Projection on affine subset, vertex-description. Suppose now we instead describe the affine subset A in terms of some given minimal set of generators arranged columnar in X ∈ Rn×N (71); id est, A = aff X = {Xa | aT 1 = 1} ⊆ Rn

(73)

√ Here minimal set means XVN = [ x2 − x1 x3 − x1 · · · xN − x1 ]/ 2 (906) is full-rank ( 2.4.2.2) where VN ∈ RN ×N −1 is the Schoenberg auxiliary matrix ( B.4.2). Then the orthogonal projection P x of any point x ∈ Rn on A is the solution to a minimization problem:

§

§

kP x − xk2 = inf

aT 1=1

= inf ξ∈ R

N −1

kXa − xk2 kX(VN ξ + ap ) − xk2

(1877)

where ap is any solution to aT 1 = 1. We find the minimizing argument ξ ⋆ = −(VNTX TXVN )−1 VNTX T(Xap − x)

§

(1878)

and so the orthogonal projection is [191, 3] P x = Xa⋆ = (I − XVN (XVN )† )Xap + XVN (XVN )† x

(1879)

a projection of point x on R(XVN ) then translated perpendicularly with respect to that range until it meets the affine subset A . 2

E.6. VECTORIZATION INTERPRETATION,

693

E.5.0.0.8 Example. Projecting on hyperplane, halfspace, slab. Given the hyperplane representation having b ∈ R and nonzero normal a ∈ Rm ∂H = {y | aTy = b} ⊂ Rm

(108)

the orthogonal projection of any point x ∈ Rm on that hyperplane is P x = x − a(aTa)−1 (aTx − b)

(1880)

Orthogonal projection of x on the halfspace parametrized by b ∈ R and nonzero normal a ∈ Rm H− = {y | aTy ≤ b} ⊂ Rm

(100)

is the point P x = x − a(aTa)−1 max{0 , aTx − b}

(1881)

B , {y | c ≤ aTy ≤ b} ⊂ Rm

(1882)

Orthogonal projection of x on the convex slab (Figure 11), for c < b

§

is the point [135, 5.1] ¡ ¢ P x = x − a(aTa)−1 max{0 , aTx − b} − max{0 , c − aTx}

E.6 E.6.1

(1883) 2

Vectorization interpretation, projection on a matrix Nonorthogonal projection on a vector

§

Nonorthogonal projection of vector x on the range of vector y is accomplished using a normalized dyad P0 ( B.1); videlicet, hz, xi z Tx yz T y = T y = T x , P0 x hz, yi z y z y

(1884)

where hz, xi/hz, yi is the coefficient of projection on y . Because P02 = P0 and R(P0 ) = R(y) , rank-one matrix P0 is a nonorthogonal projector dyad projecting on R(y). Direction of nonorthogonal projection is orthogonal to z ; id est, P0 x − x ⊥ R(P0T ) (1885)

694

APPENDIX E. PROJECTION

E.6.2

Nonorthogonal projection on vectorized matrix

Formula (1884) is extensible. Given X,Y, Z ∈ Rm×n , we have the one-dimensional nonorthogonal projection of X in isomorphic Rmn on the range of vectorized Y : ( 2.2)

§

hZ , X i Y , hZ , Y i

hZ , Y i = 6 0

(1886)

where hZ , X i/hZ , Y i is the coefficient of projection. The inequality accounts for the fact: projection on R(vec Y ) is in a direction orthogonal to vec Z . E.6.2.1

Nonorthogonal projection on dyad

Now suppose we have nonorthogonal projector dyad P0 =

yz T ∈ Rm×m z Ty

(1887)

Analogous to (1884), for X ∈ Rm×m P0 XP0 =

z TXy T hzy T , X i yz T yz T X = yz T yz = z Ty z Ty (z Ty)2 hzy T , yz T i

(1888)

is a nonorthogonal projection of matrix X on the range of vectorized dyad P0 ; from which it follows: ¿ T À T z TXy yz T zy yz hP0T , X i T P0 XP0 = T , X i P = = , X = hP P0 0 0 z y z Ty z Ty z Ty hP0T , P0 i (1889) Yet this relationship between matrix product and vector inner-product only holds for a dyad projector. When nonsymmetric projector P0 is rank-one as in (1887), therefore, 2

R(vec P0 XP0 ) = R(vec P0 ) in Rm

(1890)

and 2

P0 XP0 − X ⊥ P0T in Rm

(1891)

E.6. VECTORIZATION INTERPRETATION,

695

§

E.6.2.1.1 Example. λ as coefficients of nonorthogonal projection. Any diagonalization ( A.5)

X = SΛS

−1

=

m X i=1

λ i si wiT ∈ Rm×m

(1480)

may be expressed as a sum of one-dimensional nonorthogonal projections of X , each on the range of a vectorized eigenmatrix Pj , sj wjT ; X = =

m P

j=1

= , =

−1 T −1 h(Sei eT ) , X i Sei eT jS jS

i, j=1 m P m P

j=1 m P

j=1 m P

j=1

h(sj wjT )T , X i sj wjT +

h(sj wjT )T , X i sj wjT hPjT , X i Pj = λj sj wjT

m P

j=1

m P

−1 T −1 h(Sei eT ) , SΛS −1 i Sei eT jS jS

i, j=1 j 6= i

sj wjT Xsj wjT =

(1892) m P

Pj XPj

j=1

This biorthogonal expansion of matrix X is a sum of nonorthogonal projections because the term outside the projection coefficient h i is not identical to the inside-term. ( E.6.4) The eigenvalues λj are coefficients of nonorthogonal projection of X , while the remaining M (M − 1)/2 coefficients (for i 6= j) are zeroed by projection. When Pj is rank-one as in (1892),

§

2

R(vec Pj XPj ) = R(vec sj wjT ) = R(vec Pj ) in Rm

(1893)

and 2

Pj XPj − X ⊥ PjT in Rm

(1894)

§

Were matrix X symmetric, then its eigenmatrices would also be. So the one-dimensional projections would become orthogonal. ( E.6.4.1.1) 2

696

APPENDIX E. PROJECTION

E.6.3

Orthogonal projection on a vector

§

§

The formula for orthogonal projection of vector x on the range of vector y (one-dimensional projection) is basic analytic geometry; [13, 3.3] [309, 3.2] [340, 2.2] [355, 1-8]

§

§

hy, xi y Tx yy T y = T y = T x , P1 x hy, yi y y y y

(1895)

where hy, xi/hy, yi is the coefficient of projection on R(y) . An equivalent description is: Vector P1 x is the orthogonal projection of vector x on R(P1 ) = R(y). Rank-one matrix P1 is a projection matrix because P12 = P1 . The direction of projection is orthogonal P1 x − x ⊥ R(P1 )

(1896)

because P1T = P1 .

E.6.4

Orthogonal projection on a vectorized matrix

From (1895), given instead X , Y ∈ Rm×n , we have the one-dimensional orthogonal projection of matrix X in isomorphic Rmn on the range of vectorized Y : ( 2.2) hY , X i Y (1897) hY , Y i

§

where hY , X i/hY , Y i is the coefficient of projection. For orthogonal projection, the term outside the vector inner-products h i must be identical to the terms inside in three places. E.6.4.1

Orthogonal projection on dyad

§

There is opportunity for insight when Y is a dyad yz T ( B.1): Instead given X ∈ Rm×n , y ∈ Rm , and z ∈ Rn y TXz hyz T , X i T yz = yz T hyz T , yz T i y Ty z Tz

(1898)

is the one-dimensional orthogonal projection of X in isomorphic Rmn on the range of vectorized yz T . To reveal the obscured symmetric projection

E.6. VECTORIZATION INTERPRETATION,

697

matrices P1 and P2 we rewrite (1898): y TXz yy T zz T T yz = X T , P1 XP2 y Ty z Tz y Ty z z

(1899)

So for projector dyads, projection (1899) is the orthogonal projection in Rmn if and only if projectors P1 and P2 are symmetric;E.10 in other words,

ˆ for orthogonal projection on the range of a vectorized dyad yz , the T

term outside the vector inner-products h i in (1898) must be identical to the terms inside in three places.

When P1 and P2 are rank-one symmetric projectors as in (1899), (32) R(vec P1 XP2 ) = R(vec yz T ) in Rmn

(1900)

P1 XP2 − X ⊥ yz T in Rmn

(1901)

and When y = z then P1 = P2 = P2T and P1 XP1 = hP1 , X i P1 =

hP1 , X i P1 hP1 , P1 i

(1902)

meaning, P1 XP1 is equivalent to orthogonal projection of matrix X on the range of vectorized projector dyad P1 . Yet this relationship between matrix product and vector inner-product does not hold for general symmetric projector matrices. 2

For diagonalizable X ∈ Rm×m (§A.5), its orthogonal projection in isomorphic Rm on the range of vectorized yz T ∈ Rm×m becomes:

E.10

P1 XP2 =

m X

λ i P1 si wiT P2

i=1

When R(P1 ) = R(wj ) and R(P2 ) = R(sj ) , the j th dyad term from the diagonalization is isolated but only, in general, to within a scale factor because neither set of left or right eigenvectors is necessarily orthonormal unless X is normal [368, §3.2]. Yet when R(P2 ) = R(sk ) , k 6= j ∈ {1 . . . m} , then P1 XP2 = 0.

698

APPENDIX E. PROJECTION

E.6.4.1.1 Example. Eigenvalues λ as coefficients of orthogonal projection. Let C represent any convex subset of subspace SM , and let C1 be any element of C . Then C1 can be expressed as the orthogonal expansion C1 =

M X M X i=1

j=1 j≥i

hEij , C1 i Eij ∈ C ⊂ SM

(1903)

where Eij ∈ SM is a member of the standard orthonormal basis for SM (54). This expansion is a sum of one-dimensional orthogonal projections of C1 ; each projection on the range of a vectorized standard basis matrix. Vector inner-product hEij , C1 i is the coefficient of projection of svec C1 on R(svec Eij ). When C1 is any member of a convex set C whose dimension is L , Carath´eodory’s theorem [103] [285] [185] [38] [39] guarantees that no more than L + 1 affinely independent members from C are required to faithfully represent C1 by their linear combination.E.11 Dimension of SM is L = M (M + 1)/2 in isometrically isomorphic RM (M +1)/2 . Yet because any symmetric matrix can be diagonalized, ( A.5.1) C1 ∈ SM is a linear combination of its M eigenmatrices qi qiT ( A.5.0.3) weighted by its eigenvalues λ i ;

§

T

C1 = QΛQ =

M X

λ i qi qiT

§

(1904)

i=1

where Λ ∈ SM is a diagonal matrix having δ(Λ)i = λ i , and Q = [ q1 · · · qM ] is an orthogonal matrix in RM ×M containing corresponding eigenvectors. To derive eigen decomposition (1904) from expansion (1903), M standard basis matrices Eij are rotated ( B.5) into alignment with the M eigenmatrices qi qiT of C1 by applying a similarity transformation; [309, 5.6]

§

{QEij QT } = E.11

(

§

qi qiT , i = j = 1...M ¡ ¢ √1 q q T + q q T , 1 ≤ i < j ≤ M j i 2 i j

)

(1905)

Carath´eodory’s theorem guarantees existence of a biorthogonal expansion for any element in aff C when C is any pointed closed convex cone.

E.6. VECTORIZATION INTERPRETATION,

699

which remains an orthonormal basis for SM . Then remarkably C1 = =

M P

hQEij QT , C1 i QEij QT

M P

hqi qiT , C1 i qi qiT +

i, j=1 j≥i

i=1

=

M P

i=1

, =

M P

i=1 M P

i=1

hqi qiT , C1 i qi qiT hPi , C1 i Pi = λ i qi qiT

M P

i=1

M P

hQEij QT , QΛQT i QEij QT

i, j=1 j>i

qi qiT C1 qi qiT =

(1906) M P

i=1

Pi C1 Pi

this orthogonal expansion becomes the diagonalization; still a sum of one-dimensional orthogonal projections. The eigenvalues λ i = hqi qiT , C1 i

(1907)

§

are clearly coefficients of projection of C1 on the range of each vectorized eigenmatrix. (confer E.6.2.1.1) The remaining M (M − 1)/2 coefficients (i 6= j) are zeroed by projection. When Pi is rank-one symmetric as in (1906), R(svec Pi C1 Pi ) = R(svec qi qiT ) = R(svec Pi ) in RM (M +1)/2

(1908)

and Pi C1 Pi − C1 ⊥ Pi in RM (M +1)/2 E.6.4.2

(1909) 2

Positive semidefiniteness test as orthogonal projection

For any given X ∈ Rm×m the familiar quadratic construct y TXy ≥ 0, over broad domain, is a fundamental test for positive semidefiniteness. ( A.2) It is a fact that y TXy is always proportional to a coefficient of orthogonal projection; letting z in formula (1898) become y ∈ Rm , then P2 = P1 = yy T/y Ty = yy T/kyy T k2 (confer (1542)) and formula (1899) becomes

§

hyy T , X i y TXy yy T yy T yy T T yy = = X T , P1 XP1 hyy T , yy T i y Ty y Ty y Ty y y

(1910)

700

APPENDIX E. PROJECTION

By (1897), product P1 XP1 is the one-dimensional orthogonal projection of 2 X in isomorphic Rm on the range of vectorized P1 because, for rank P1 = 1 and P12 = P1 ∈ Sm (confer (1889)) y TXy yy T P1 XP1 = T = y y y Ty

¿

À T yy T yy hP1 , X i , X = hP , X i P = P1 1 1 y Ty y Ty hP1 , P1 i (1911)

The coefficient of orthogonal projection hP1 , X i= y TXy/(y Ty) is also known as Rayleigh’s quotient.E.12 When P1 is rank-one symmetric as in (1910), 2

R(vec P1 XP1 ) = R(vec P1 ) in Rm

(1912)

and 2

P1 XP1 − X ⊥ P1 in Rm

(1913)

The test for positive semidefiniteness, then, is a test for nonnegativity of the coefficient of orthogonal projection of X on the range of each and every vectorized extreme direction yy T ( 2.8.1) from the positive semidefinite cone in the ambient space of symmetric matrices.

§

When y becomes the j th eigenvector sj of diagonalizable X , for example, hP1 , X i becomes the j th eigenvalue: [182, §III]

E.12

µ

sT j hP1 , X i|y=sj =

m P

i=1

λ i si wiT sT j sj

¶

sj = λj

Similarly for y = wj , the j th left-eigenvector,

hP1 , X i|y=wj =

µm ¶ P wjT λ i si wiT wj i=1

wjTwj

= λj

A quandary may arise regarding the potential annihilation of the antisymmetric part of X when sT j Xsj is formed. Were annihilation to occur, it would imply the eigenvalue thus found came instead from the symmetric part of X . The quandary is resolved recognizing that diagonalization of real X admits complex eigenvectors; hence, annihilation could only T H T come about by forming Re(sH j Xsj ) = sj (X + X )sj /2 [188, §7.1] where (X + X )/2 is H the symmetric part of X , and sj denotes conjugate transpose.

E.7. PROJECTION ON MATRIX SUBSPACES E.6.4.3

701

P XP º 0

In some circumstances, it may be desirable to limit the domain of test y TXy ≥ 0 for positive semidefiniteness; e.g., {kyk = 1}. Another example of limiting domain-of-test is central to Euclidean distance geometry: For R(V ) = N (1T ) , the test −V D V º 0 determines whether D ∈ SN h is a Euclidean distance matrix. The same test may be stated: For D ∈ SN h (and optionally kyk = 1) D ∈ EDMN ⇔ −y TDy = hyy T , −Di ≥ 0 ∀ y ∈ R(V )

(1914)

The test −V D V º 0 is therefore equivalent to a test for nonnegativity of the coefficient of orthogonal projection of −D on the range of each and every vectorized extreme direction yy T from the positive semidefinite cone SN + such that R(yy T ) = R(y) ⊆ R(V ). (The validity of this result is independent of whether V is itself a projection matrix.)

E.7 E.7.1

Projection on matrix subspaces P XP misinterpretation for higher-rank P

For a projection matrix P of rank greater than 1, P XP is generally not ,Xi P as is the case for projector dyads (1911). Yet commensurate with hP hP , P i for a symmetric idempotent matrix P of any rank we are tempted to say “ P XP is the orthogonal projection of X ∈ Sm on R(vec P ) ”. The fallacy is: vec P XP does not necessarily belong to the range of vectorized P ; the most basic requirement for projection on R(vec P ).

E.7.2

Orthogonal projection on matrix subspaces

With A1 ∈ Rm×n , B1 ∈ Rn×k , Z1 ∈ Rm×k , A2 ∈ Rp×n , B2 ∈ Rn×k , Z2 ∈ Rp×k as defined for nonorthogonal projector (1810), and defining P1 , A1 A†1 ∈ Sm ,

P2 , A2 A†2 ∈ S p

(1915)

then, given compatible X kX − P1 XP2 kF =

inf

B1 , B2 ∈ R

n×k

T T kX − A1 (A†1 +B1 Z1T )X(A†T 2 +Z2 B2 )A2 kF (1916)

702

APPENDIX E. PROJECTION

As for all subspace projectors, range of the projector is the subspace on which projection is made: {P1 Y P2 | Y ∈ Rm×p }. For projectors P1 and P2 of any rank, altogether, this means projection P1 XP2 is unique minimum-distance, orthogonal P1 XP2 − X ⊥ {P1 Y P2 | Y ∈ Rm×p } in Rmp (1917) and P1 and P2 must each be symmetric (confer (1899)) to attain the infimum. E.7.2.0.1 Proof. Minimum Frobenius norm (1916). Defining P , A1 (A†1 + B1 Z1T ) , T T 2 inf kX − A1 (A†1 + B1 Z1T )X(A†T 2 + Z2 B2 )A2 kF

B1 , B2

T T 2 = inf kX − P X(A†T 2 + Z2 B2 )A2 kF B1 , B2 ³ ´ T T = inf tr (X T − A2 (A†2 + B2 Z2T )X TP T )(X − P X(A†T + Z B )A ) 2 2 2 2 B1 , B2 ³ † T T T T T = inf tr X TX −X TP X(A†T 2 +Z2 B2 )A2 −A2 (A2 +B2 Z2 )X P X B1 , B2 ´ T T +Z B )A +A2 (A†2 +B2 Z2T )X TP TP X(A†T 2 2 2 2

(1918)

§

Necessary conditions for a global minimum are ∇B1 = 0 and ∇B2 = 0. Terms not containing B2 in (1918) will vanish from gradient ∇B2 ; ( D.2.3) ³ † T T T T T T T ∇B2 tr −X TP XZ2 B2TAT 2 −A2 B2 Z2 X P X +A2 A2 X P P XZ2 B2 A2 =

´

T T T T T T +A2 B2 Z2TX TP TP XA†T 2 A2 +A2 B2 Z2 X P P XZ2 B2 A2 † T T T T T T T T T −2A ³ 2 X P XZ2 + 2A2 A2 A2 X P P XZ2 + ´ 2A2 A2 B2 Z2 X P P XZ2 † T T T T T T AT P XZ2 2 −X + A2 A2 X P + A2 B2 Z2 X P (1919)

= = 0

⇔ R(B1 ) ⊆ N (A1 ) and R(B2 ) ⊆ N (A2 )

(or Z2 = 0) because AT = ATAA† . Symmetry requirement (1915) is implicit. T T Were instead P T , (A†T 2 + Z2 B2 )A2 and the gradient with respect to B1 observed, then similar results are obtained. The projector is unique. Perpendicularity (1917) establishes uniqueness [99, 4.9] of projection P1 XP2 on a matrix subspace. The minimum-distance projector is the orthogonal projector, and vice versa. ¨

§

E.7. PROJECTION ON MATRIX SUBSPACES

703

E.7.2.0.2 Example. P XP redux & N (V). Suppose we define a subspace of m × n matrices, each elemental matrix having columns constituting a list whose geometric center ( 5.5.1.0.1) is the origin in Rm :

§

Rm×n , {Y ∈ Rm×n | Y 1 = 0} c

= {Y ∈ Rm×n | N (Y ) ⊇ 1} = {Y ∈ Rm×n | R(Y T ) ⊆ N (1T )}

= {XV | X ∈ Rm×n } ⊂ Rm×n

(1920)

the nonsymmetric geometric center subspace. Further suppose V ∈ S n is a projection matrix having N (V ) = R(1) and R(V ) = N (1T ). Then linear mapping T (X) = XV is the orthogonal projection of any X ∈ Rm×n on Rm×n c in the Euclidean (vectorization) sense because V is symmetric, N (XV ) ⊇ 1, and R(VX T ) ⊆ N (1T ). Now suppose we define a subspace of symmetric n × n matrices each of whose columns constitute a list having the origin in Rn as geometric center, Scn , {Y ∈ S n | Y 1 = 0}

= {Y ∈ S n | N (Y ) ⊇ 1} = {Y ∈ S n | R(Y ) ⊆ N (1T )}

(1921)

the geometric center subspace. Further suppose V ∈ S n is a projection matrix, the same as before. Then V X V is the orthogonal projection of any X ∈ S n on Scn in the Euclidean sense (1917) because V is symmetric, V X V 1 = 0, and R(V X V ) ⊆ N (1T ). Two-sided projection is necessary only to remain in the ambient symmetric matrix subspace. Then Scn = {V X V | X ∈ S n } ⊂ S n

(1922)

has dim Scn = n(n−1)/2 in isomorphic Rn(n+1)/2 . We find its orthogonal complement as the aggregate of all negative directions of orthogonal projection on Scn : the translation-invariant subspace ( 5.5.1.1)

§

Scn⊥ , {X − V X V | X ∈ S n } ⊂ S n

(1923)

= {u1T + 1uT | u ∈ Rn }

§

Defining the characterized by the doublet u1T + 1uT ( B.2).E.13 1 geometric center mapping V(X) = −V X V 2 consistently with (942), then E.13

Proof. {X − V X V | X ∈ S n } = {X − (I − n1 11T )X(I − 11T n1 ) | X ∈ S n } = { n1 11T X + X11T n1 −

1 T T1 n 11 X 11 n

| X ∈ Sn}

704

APPENDIX E. PROJECTION

§

N (V) = R(I − V) on domain S n analogously to vector projectors ( E.2); id est, N (V) = Scn⊥

(1924)

a subspace of S n whose dimension is dim Scn⊥ = n in isomorphic Rn(n+1)/2 . Intuitively, operator V is an orthogonal projector; any argument duplicitously in its range is a fixed point. So, this symmetric operator’s nullspace must be orthogonal to its range. Now compare the subspace of symmetric matrices having all zeros in the first row and column S1n , {Y ∈ S n | Y e1 = 0} ½· ¸ · ¸ ¾ 0 0T 0 0T n = X | X∈ S 0 I 0 I o n£ √ √ ¤T £ ¤ = 0 2VN Z 0 2VN | Z ∈ SN

(1925)

¸ 0 0T where P = is an orthogonal projector. Then, similarly, P XP is 0 I the orthogonal projection of any X ∈ S n on S1n in the Euclidean sense (1917), and ·

S1n⊥

0 0T 0 I

0 0T , X 0 I © T ª = ue1 + e1 uT | u ∈ Rn ½·

¸

·

¸

−X | X∈ S

n

¾

⊂ Sn

(1926)

Obviously, S1n ⊕ S1n⊥ = S n .

2

Because {X1 | X ∈ S n } = Rn , {X − V X V | X ∈ S n } = {1ζ T + ζ 1T − 11T (1T ζ n1 ) | ζ ∈ Rn } 1 = {1ζ T(I − 11T 2n ) + (I −

1 11T is invertible. where I − 2n

1 T T 2n 11 )ζ 1

| ζ ∈ Rn } ¨

E.8. RANGE/ROWSPACE INTERPRETATION

E.8

705

Range/Rowspace interpretation

For idempotent matrices P1 and P2 of any rank, P1 XP2T is a projection of R(X) on R(P1 ) and a projection of R(X T ) on R(P2 ) : For any given η P X = U ΣQT = σi ui qiT ∈ Rm×p , as in compact SVD (1495), i=1

P1 XP2T

=

η X

σi P1 ui qiT P2T

i=1

=

η X

σi P1 ui (P2 qi )T

(1927)

i=1

§

where η , min{m , p}. Recall ui ∈ R(X) and qi ∈ R(X T ) when the corresponding singular value σi is nonzero. ( A.6.1) So P1 projects ui on R(P1 ) while P2 projects qi on R(P2 ) ; id est, the range and rowspace of any X are respectively projected on the ranges of P1 and P2 .E.14

E.9

Projection on convex set

Thus far we have discussed only projection on subspaces. Now we generalize, considering projection on arbitrary convex sets in Euclidean space; convex because projection is, then, unique minimum-distance and a convex optimization problem: For projection PC x of point x on any closed set C ⊆ Rn it is obvious: C ≡ {PC x | x ∈ Rn }

(1928)

where PC is a projection operator that is convex when C is convex. [57, p.88] If C ⊆ Rn is a closed convex set, then for each and every x ∈ Rn there exists a unique point PC x belonging to C that is closest to x in the Euclidean sense. Like (1832), unique projection P x (or PC x) of a point x on convex set C is that point in C closest to x ; [232, 3.12]

§

kx − P xk2 = inf kx − yk2 = dist(x , C) y∈C

(1929)

There exists a converse (in finite-dimensional Euclidean space): When P1 and P2 are symmetric and R(P1 ) = R(uj ) and R(P2 ) = R(qj ) , then the j th dyad term from the singular value decomposition of X is isolated by the projection. Yet if R(P2 ) = R(qℓ ) , ℓ 6= j ∈ {1 . . . η} , then P1 XP2 = 0.

E.14

706

APPENDIX E. PROJECTION

§

E.9.0.0.1 Theorem. (Bunt-Motzkin) Convex set if projections unique. [346, 7.5] [183] If C ⊆ Rn is a nonempty closed set and if for each and every x in Rn there is a unique Euclidean projection P x of x on C belonging to C , then C is convex. ⋄

§

Borwein & Lewis propose, for closed convex set C [51, 3.3 exer.12(d)] ∇kx − P xk22 = 2(x − P x)

(1930)

for any point x whereas, for x ∈ /C

∇kx − P xk2 = (x − P x) kx − P xk−1 2

E.9.0.0.2 Exercise. Norm gradient. Prove (1930) and (1931). (Not proved in [51].)

(1931)

H

A well-known equivalent characterization of projection on a convex set is a generalization of the perpendicularity condition (1831) for projection on a subspace:

E.9.1

Dual interpretation of projection on convex set

E.9.1.0.1 Definition. Normal vector. Vector z is normal to convex set C at point P x ∈ C if hz , y−P xi ≤ 0

∀y ∈ C

[285, p.15] (1932) △

A convex set has at least one nonzero normal at each of its boundary points. [285, p.100] (Figure 64) Hence, the normal or dual interpretation of projection:

§

§

§

E.9.1.0.2 Theorem. Unique minimum-distance projection. [185, A.3.1] [232, 3.12] [99, 4.1] [74] (Figure 160b p.723) Given a closed convex set C ⊆ Rn , point P x is the unique projection of a given point x ∈ Rn on C (P x is that point in C nearest x) if and only if Px ∈ C ,

hx − P x , y − P xi ≤ 0 ∀ y ∈ C

(1933) ⋄

E.9. PROJECTION ON CONVEX SET

707

As for subspace projection, convex operator P is idempotent in the sense: for each and every x ∈ Rn , P (P x) = P x . Yet operator P is nonlinear;

ˆ Projector P is a linear operator if and only if convex set C (on which projection is made) is a subspace. (§E.4) [99, §4.3] E.9.1.0.3 Theorem. Unique projection via normal cone. E.15

n

Given closed convex set C ⊆ R , point P x is the unique projection of a given point x ∈ Rn on C if and only if Px ∈ C ,

P x − x ∈ (C − P x)∗

(1934)

In other words, P x is that point in C nearest x if and only if P x − x belongs to that cone dual to translate C − P x . ⋄ E.9.1.1

Dual interpretation as optimization

§

Deutsch [101, thm.2.3] [102, 2] and Luenberger [232, p.134] carry forward Nirenberg’s dual interpretation of projection [258] as solution to a maximization problem: Minimum distance from a point x ∈ Rn to a convex set C ⊂ Rn can be found by maximizing distance from x to hyperplane ∂H over the set of all hyperplanes separating x from C . Existence of a separating hyperplane ( 2.4.2.7) presumes point x lies on the boundary or exterior to set C . The optimal separating hyperplane is characterized by the fact it also supports C . Any hyperplane supporting C (Figure 29(a)) has form © ª ∂H− = y ∈ Rn | aTy = σC (a) (123)

§

where the support function is convex, defined σC (a) = sup aTz

(519)

z∈C

When point x is finite and set C contains finite points, under this projection interpretation, if the supporting hyperplane is a separating hyperplane then the support function is finite. From Example E.5.0.0.8, projection P∂H− x of x on any given supporting hyperplane ∂H− is ¡ ¢ (1935) P∂H− x = x − a(aTa)−1 aTx − σC (a)

E.15

−(C − P x)∗ is the normal cone to set C at point P x . (§E.10.3.2)

708

APPENDIX E. PROJECTION

∂H−

C

PC x

(a)

H−

H+ P∂H− x κa x

(b)

Figure 156: Dual interpretation of projection of point x on convex set C ¡ T ¢ 2 T −1 in R . (a) κ = (a a) a x − σC (a) (b) Minimum distance from x to C is found by maximizing distance to all hyperplanes supporting C and separating it from x . A convex problem for any convex set, distance of maximization is unique.

E.9. PROJECTION ON CONVEX SET

709

With reference to Figure 156, identifying H+ = {y ∈ Rn | aTy ≥ σC (a)}

(101)

then kx − PC xk = sup

kx − P∂H− xk = sup ka(aTa)−1 (aTx − σC (a))k a | x∈H+

∂H− | x∈H+

= sup a | x∈H+

1 |aTx kak

− σC (a)|

(1936)

which can be expressed as a convex optimization, for arbitrary positive constant τ 1 kx − PC xk = maximize aTx − σC (a) a (1937) τ subject to kak ≤ τ

The unique minimum-distance projection on convex set C is therefore ¡ ¢1 (1938) PC x = x − a⋆ a⋆Tx − σC (a⋆ ) 2 τ

where optimally ka⋆ k = τ .

E.9.1.1.1 Exercise. Dual projection technique on polyhedron. Test that projection paradigm from Figure 156 on any convex polyhedral set. H E.9.1.2

Dual interpretation of projection on cone

In the circumstance set C is a closed convex cone K and there exists a hyperplane separating given point x from K , then optimal σK (a⋆ ) takes value 0 [185, C.2.3.1]. So problem (1937) for projection of x on K becomes

§

kx − PK xk =

1 maximize aTx a τ subject to kak ≤ τ a∈K

◦

(1939)

§

The norm inequality in (1939) can be handled by Schur complement ( 3.6.2). Normals a to all hyperplanes supporting K belong to the polar cone ◦ ∗ K = −K by definition: (292) a∈K

◦

⇔ ha , xi ≤ 0 for all x ∈ K

(1940)

710

APPENDIX E. PROJECTION

Projection on cone K is PK x = (I −

1 ⋆ ⋆T a a )x τ2

(1941)

§

∗

whereas projection on the polar cone −K is ( E.9.2.2.1) PK◦ x = x − PK x =

1 ⋆ ⋆T aa x τ2

(1942)

Negating vector a , this maximization problem (1939) becomes a minimization (the same problem) and the polar cone becomes the dual cone: 1 kx − PK xk = − minimize aTx a τ subject to kak ≤ τ a∈K

E.9.2

(1943)

∗

Projection on cone

When convex set C is a cone, there is a finer statement of optimality conditions:

§

§

E.9.2.0.1 Theorem. Unique projection on cone. [185, A.3.2] ∗ n Let K ⊆ R be a closed convex cone, and K its dual ( 2.13.1). Then P x is the unique minimum-distance projection of x ∈ Rn on K if and only if Px ∈ K ,

hP x − x , P xi = 0 ,

Px − x ∈ K

∗

(1944) ⋄

In words, P x is the unique minimum-distance projection of x on K if and only if 1) projection P x lies in K 2) direction P x−x is orthogonal to the projection P x ∗

3) direction P x−x lies in the dual cone K . As the theorem is stated, it admits projection on K having empty interior; id est, on closed convex cones in a proper subspace of Rn .

E.9. PROJECTION ON CONVEX SET

711 ∗

Projection on K of any point x ∈ −K , belonging to the negative dual cone, is the origin. By (1944): the set of all points reaching the origin, when projecting on K , constitutes the negative dual cone; a.k.a, the polar cone ◦

∗

K = −K = {x ∈ Rn | P x = 0} E.9.2.1

(1945)

Relationship to subspace projection

Conditions 1 and 2 of the theorem are common with orthogonal projection on a subspace R(P ) : Condition 1 corresponds to the most basic requirement; namely, the projection P x ∈ R(P ) belongs to the subspace. Recall the perpendicularity requirement for projection on a subspace; P x − x ⊥ R(P ) or P x − x ∈ R(P )⊥

(1831)

which corresponds to condition 2. Yet condition 3 is a generalization of subspace projection; id est, for unique minimum-distance projection on a closed convex cone K , polar cone ∗ −K plays the role R(P )⊥ plays for subspace projection (PR x = x − PR⊥ x). ∗ Indeed, −K is the algebraic complement in the orthogonal vector sum (p.742) [250] [185, A.3.2.5]

§

∗

K ⊞ −K = Rn ⇔ cone K is closed and convex

(1946)

Also, given unique minimum-distance projection P x on K satisfying Theorem E.9.2.0.1, then by projection on the algebraic complement via I − P in E.2 we have

§

∗

−K = {x − P x | x ∈ Rn } = {x ∈ Rn | P x = 0}

(1947)

consequent to Moreau (1950). Recalling any subspace is a closed convex coneE.16 ∗ K = R(P ) ⇔ −K = R(P )⊥ (1948)

§

meaning, when a cone is a subspace R(P ) , then the dual cone becomes its orthogonal complement R(P )⊥ . [57, 2.6.1] In this circumstance, condition 3 becomes coincident with condition 2. The properties of projection on cones following in E.9.2.2 further generalize to subspaces by: (4) K = R(P ) ⇔ −K = R(P ) E.16

but a proper subspace is not a proper cone (§2.7.2.2.1).

§

(1949)

712

APPENDIX E. PROJECTION

E.9.2.2 Salient properties: Projection P x on closed convex cone K

§

§

[185, A.3.2] [99, 5.6] For x , x1 , x2 ∈ Rn 1.

PK (αx) = α PK x ∀ α ≥ 0

2.

kPK xk ≤ kxk

3.

PK x = 0 ⇔ x ∈ −K

4.

PK (−x) = −P−K x

5.

(Jean-Jacques Moreau (1962)) [250]

(nonnegative homogeneity)

∗

∗

x1 ∈ K , x2 ∈ −K , ⇔ x1 = PK x , x2 = P−K∗ x

x = x1 + x2 ,

x1 ⊥ x2

6.

K = {x − P−K∗ x | x ∈ Rn } = {x ∈ Rn | P−K∗ x = 0}

7.

−K = {x − PK x | x ∈ Rn } = {x ∈ Rn | PK x = 0}

∗

(1950)

(1947)

§

E.9.2.2.1 Corollary. I − P for cones. (confer E.2) ∗ n Denote by K ⊆ R a closed convex cone, and call K its dual. Then x − P−K∗ x is the unique minimum-distance projection of x ∈ Rn on K if and ∗ only if P−K∗ x is the unique minimum-distance projection of x on −K the polar cone. ⋄ Proof. Assume x1 = PK x . Then by Theorem E.9.2.0.1 we have x1 ∈ K ,

x 1 − x ⊥ x1 ,

x1 − x ∈ K

∗

(1951)

Now assume x − x1 = P−K∗ x . Then we have ∗

x − x1 ∈ −K ,

−x1 ⊥ x − x1 ,

−x1 ∈ −K

(1952)

But these two assumptions are apparently identical. We must therefore have x − P−K∗ x = x1 = PK x

(1953) ¨

E.9. PROJECTION ON CONVEX SET

§

713

§

E.9.2.2.2 Corollary. Unique projection via dual or normal cone. [99, 4.7] ( E.10.3.2, confer Theorem E.9.1.0.3) Given point x ∈ Rn and closed convex cone K ⊆ Rn , the following are equivalent statements: 1.

point P x is the unique minimum-distance projection of x on K

2.

Px ∈ K ,

x − P x ∈ −(K − P x)∗ = −K ∩ (P x)⊥

3.

Px ∈ K ,

hx − P x , P xi = 0 ,

∗

hx − P x , yi ≤ 0 ∀ y ∈ K ⋄

E.9.2.2.3 Example. Unique projection on nonnegative orthant. (confer (1266)) From Theorem E.9.2.0.1, to project matrix H ∈ Rm×n on the self-dual orthant ( 2.13.5.1) of nonnegative matrices Rm×n in isomorphic + mn R , the necessary and sufficient conditions are:

§

H ¡⋆ ≥ 0 ¢ tr (H ⋆ − H)T H ⋆ = 0 H⋆ − H ≥ 0

(1954)

where the inequalities denote entrywise comparison. The optimal solution H ⋆ is simply H having all its negative entries zeroed; Hij⋆ = max{Hij , 0} ,

i, j ∈ {1 . . . m} × {1 . . . n}

(1955)

Now suppose the nonnegative orthant is translated by T ∈ Rm×n ; id est, consider Rm×n + T . Then projection on the translated orthant is [99, 4.8] +

§

Hij⋆ = max{Hij , Tij }

(1956) 2

E.9.2.2.4 Example. Unique projection on truncated convex cone. Consider the problem of projecting a point x on a closed convex cone that is artificially bounded; really, a bounded convex polyhedron having a vertex at the origin: minimize kx − Ayk2 y∈RN

subject to y º 0

kyk∞ ≤ 1

(1957)

714

APPENDIX E. PROJECTION

§

where the convex cone has vertex-description ( 2.12.2.0.1), for A ∈ Rn×N K = {Ay | y º 0}

(1958)

and where kyk∞ ≤ 1 is the artificial bound. This is a convex optimization problem having no known closed-form solution, in general. It arises, for example, in the fitting of hearing aids designed around a programmable graphic equalizer (a filter bank whose only adjustable parameters are gain per band each bounded above by unity). [91] The problem is equivalent to a Schur-form semidefinite program ( 3.6.2)

§

minimize y∈RN , t∈R

subject to

t ·

tI x − Ay T (x − Ay) t

0¹y¹1

¸

(1959)

º0

2

E.9.3

nonexpansivity

§

§

E.9.3.0.1 Theorem. Nonexpansivity. [161, 2] [99, 5.3] n When C ⊂ R is an arbitrary closed convex set, projector P projecting on C is nonexpansive in the sense: for any vectors x, y ∈ Rn kP x − P yk ≤ kx − yk with equality when x −P x = y −P y .E.17

(1960) ⋄

Proof. [50] kx − yk2 = kP x − P yk2 + k(I − P )x − (I − P )yk2

+ 2hx − P x , P x − P yi + 2hy − P y , P y − P xi

(1961)

§

Nonnegativity of the last two terms follows directly from the unique ¨ minimum-distance projection theorem ( E.9.1.0.2). E.17

This condition for equality corrects an error in [74] (where the norm is applied to each side of the condition given here) easily revealed by counter-example.

E.9. PROJECTION ON CONVEX SET

715

The foregoing proof reveals another flavor of nonexpansivity; for each and every x, y ∈ Rn kP x − P yk2 + k(I − P )x − (I − P )yk2 ≤ kx − yk2

§

(1962)

Deutsch shows yet another: [99, 5.5] kP x − P yk2 ≤ hx − y , P x − P yi

E.9.4

(1963)

Easy projections

ˆ To project any matrix H ∈ R orthogonally in Euclidean/Frobenius sense on subspace of symmetric matrices S in isomorphic R , take symmetric part of H ; (§2.2.2.0.1) id est, (H + H )/2 is the projection. ˆ To project any H ∈ R orthogonally on symmetric hollow subspace S in isomorphic R (§2.2.3.0.1, §7.0.1), take symmetric part then zero n×n

n2

n

T

n×n

n2

n h

all entries along main diagonal or vice versa (because this is projection on intersection of two subspaces); id est, (H + H T )/2 − δ 2(H).

ˆ To project a matrix on nonnegative orthant R

m×n , +

simply clip all negative entries to 0. Likewise, projection on nonpositive orthant Rm×n − sees all positive entries clipped to 0. Projection on other orthants is equally simple with appropriate clipping.

ˆ Projecting on hyperplane, halfspace, slab: §E.5.0.0.8. ˆ Projection of y ∈ R on Euclidean ball B = {x ∈ R n

for y 6= a , PB y =

c (y ky−ak

n

− a) + a .

| kx − ak ≤ c} :

ˆ Clipping in excess of |1| each entry of point x ∈ R is equivalent to unique minimum-distance projection of x on a hypercube centered at the origin. (confer §E.10.3.2) ˆ Projection of x ∈ R on a (rectangular) hyperbox : [57, §8.1.1] n

n

C = {y ∈ Rn | l ¹ y   lk , xk , P (x)k=0... n =  uk ,

¹ u , l ≺ u}

(1964)

xk ≤ lk lk ≤ xk ≤ uk x k ≥ uk

(1965)

716

APPENDIX E. PROJECTION

ˆ Projection of x on set of all cardinality-k vectors {y | card y ≤ k} keeps k entries of greatest magnitude and clips to 0 those remaining.

ˆ Unique minimum-distance projection of H ∈ S

n

on positive semidefinite cone in Euclidean/Frobenius sense is accomplished by eigen decomposition (diagonalization) followed by clipping all negative eigenvalues to 0. Sn+

ˆ Unique minimum-distance projection on generally nonconvex subset of all matrices belonging to S having rank not exceeding ρ (§2.9.2.1) is accomplished by clipping all negative eigenvalues to 0 and zeroing smallest nonnegative eigenvalues keeping only ρ largest. (§7.1.2) ˆ Unique minimum-distance projection, in Euclidean/Frobenius sense, of H ∈ R on generally nonconvex subset of all m × n matrices of rank no greater than k is singular value decomposition (§A.6) of H n +

m×n

having all singular values beyond k th zeroed. This is also a solution to projection in sense of spectral norm. [306, p.79, p.208]

ˆ Projection on monotone nonnegative cone K ⊂ R in less than one cycle (in sense of alternating projections §E.10): [Wıκımization]. ˆ Projection on closed convex cone K of any point x ∈ −K , belonging to polar cone, is equivalent to projection on origin. (§E.9.2) ˆP =P P (1210) ˆP = P P (§7.0.1.1) ˆP = P P (§E.9.5) M+

n

∗

N SN + ∩ Sc

SN +

SN c

×N RN ∩ SN + h

×N RN +

SN h

×N RN ∩ SN +

×N RN +

SN

E.9.4.0.1 Exercise. Projection on spectral norm ball. Find the unique minimum-distance projection on the convex set of all m × n matrices whose largest singular value does not exceed 1 ; id est, on H {X ∈ Rm×n | kXk2 ≤ 1} the spectral norm ball ( 2.3.2.0.5).

§

E.9. PROJECTION ON CONVEX SET

" b

717

B B "b xs " b BS " b Rn⊥ BB " b BS " b B S B " b b B ""BBs S b b B" z Q S "B b QS y " b Q Q Ss n " b R " b b

b

b

x−z ⊥ z−y

C b

b

b

" b " Bb " b B " b " b B " b B b" B

"

"

" "

"

"

b "

Figure 157: Closed convex set C belongs to subspace Rn (shown bounded in sketch and drawn without proper perspective). Point y is unique minimum-distance projection of x on C ; equivalent to product of orthogonal projection of x on Rn and minimum-distance projection of result z on C . E.9.4.1

notes

Projection on Lorentz (second-order) cone: [57, exer.8.3(c)]. Deutsch [102] provides an algorithm for projection on polyhedral cones. Youla [367, 2.5] lists eleven “useful projections”, of square-integrable uni- and bivariate real functions on various convex sets, in closed form. Unique minimum-distance projection on an ellipsoid: Example 4.6.0.0.1. Numerical algorithms for projection on the nonnegative simplex and 1-norm ball are in [116].

§

E.9.5

Projection on convex set in subspace

Suppose a convex set C is contained in some subspace Rn . Then unique minimum-distance projection of any point in Rn ⊕ Rn⊥ on C can be accomplished by first projecting orthogonally on that subspace, and then projecting the result on C ; [99, 5.14] id est, the ordered product of two individual projections that is not commutable.

§

718

APPENDIX E. PROJECTION

Proof. (⇐) To show that, suppose unique minimum-distance projection PC x on C ⊂ Rn is y as illustrated in Figure 157; kx − yk ≤ kx − qk ∀ q ∈ C

(1966)

Further suppose PRn x equals z . By the Pythagorean theorem kx − yk2 = kx − zk2 + kz − yk2

(1967)

§

because x − z ⊥ z − y . (1831) [232, 3.3] Then point y = PC x is the same as PC z because kz − yk2 = kx − yk2 − kx − zk2 ≤ kz − qk2 = kx − qk2 − kx − zk2

∀q ∈ C (1968)

which holds by assumption (1966). (⇒) Now suppose z = PRn x and kz − yk ≤ kz − qk ∀ q ∈ C

(1969)

meaning y = PC z . Then point y is identical to PC x because kx − yk2 = kx − zk2 + kz − yk2 ≤ kx − qk2 = kx − zk2 + kz − qk2 by assumption (1969).

§

∀q ∈ C (1970) ¨

This proof is extensible via translation argument. ( E.4) Unique minimum-distance projection on a convex set contained in an affine subset is, therefore, similarly accomplished. Projecting matrix H ∈ Rn×n on convex cone K = S n ∩ Rn×n in isomorphic + 2 n n R can be accomplished, for example, by first projecting on S and only then projecting the result on Rn×n (confer 7.0.1). This is because that projection + product is equivalent to projection on the subset of the nonnegative orthant in the symmetric matrix subspace.

§

E.10. ALTERNATING PROJECTION

E.10

719

Alternating projection

Alternating projection is an iterative technique for finding a point in the intersection of a number of arbitrary closed convex sets Ck , or for finding the distance between two nonintersecting closed convex sets. Because it can sometimes be difficult or inefficient to compute the intersection or express it analytically, one naturally asks whether it is possible to instead project (unique minimum-distance) alternately on the individual Ck ; often easier and what motivates adoption of this technique. Once a cycle of alternating projections (an iteration) is complete, we then iterate (repeat the cycle) until convergence. If the intersection of two closed convex sets is empty, then by convergence we mean the iterate (the result after a cycle of alternating projections) settles to a point of minimum distance separating the sets. While alternating projection can find the point in the nonempty intersection closest to a given point b , it does not necessarily find the closest point. Finding that closest point is made dependable by an elegantly simple enhancement via correction to the alternating projection technique: this Dykstra algorithm (2008) for projection on the intersection is one of the most beautiful projection algorithms ever discovered. It is accurately interpreted as the discovery of what alternating projection originally sought to accomplish: unique minimum-distance projection on the nonempty intersection of a number of arbitrary closed convex sets Ck . Alternating projection is, in fact, a special case of the Dykstra algorithm whose discussion we defer until E.10.3.

§

E.10.0.1

commutative projectors

Given two arbitrary convex sets C1 and C2 and their respective minimum-distance projection operators P1 and P2 , if projectors commute for each and every x ∈ Rn then it is easy to show P1 P2 x ∈ C1 ∩ C2 and P2 P1 x ∈ C1 ∩ C2 . When projectors commute (P1 P2 = P2 P1 ), a point in the intersection can be found in a finite number of steps; while commutativity is a sufficient condition, it is not necessary ( 6.8.1.1.1 for example). When C1 and C2 are subspaces, in particular, projectors P1 and P2 commute if and only if P1 P2 = PC1 ∩ C2 or iff P2 P1 = PC1 ∩ C2 or iff P1 P2 is the orthogonal projection on a Euclidean subspace. [99, lem.9.2] Subspace projectors will commute, for example, when P1 (C2 ) ⊂ C2 or P2 (C1 ) ⊂ C1 or C1 ⊂ C2 or C2 ⊂ C1 or C1 ⊥ C2 . When subspace projectors commute, this

§

720

APPENDIX E. PROJECTION b

C2

∂KC⊥1 ∩ C2(P b) + P b C1

Figure 158: First several alternating projections in von Neumann-style projection (1981), of point b , converging on closest point P b in intersection of two closed convex sets in R2 : C1 and C2 are partially drawn in vicinity of their intersection. Pointed normal cone K⊥ (2010) is translated to P b , the unique minimum-distance projection of b on intersection. For this particular example, it is possible to start anywhere in a large neighborhood of b and still converge to P b . Alternating projections are themselves robust with respect to significant noise because they belong to translated normal cone.

means we can find a point in the intersection of those subspaces in a finite number of steps; we find, in fact, the closest point.

§

E.10.0.1.1 Theorem. Kronecker projector. [305, 2.7] Given any projection matrices P1 and P2 (subspace projectors), then P1 ⊗ P2

and

P1 ⊗ I

(1971)

are projection matrices. The product preserves symmetry if present. ⋄ E.10.0.2

noncommutative projectors

Typically, one considers the method of alternating projection when projectors do not commute; id est, when P1 P2 6= P2 P1 . The iconic example for noncommutative projectors illustrated in Figure 158 shows the iterates converging to the closest point in the intersection of two arbitrary convex sets. Yet simple examples like Figure 159 reveal that noncommutative alternating projection does not

E.10. ALTERNATING PROJECTION

721

b H1 P2 b Pb H1 ∩ H2

H2

P1 P2 b

Figure 159: The sets {Ck } in this example comprise two halfspaces H1 and H2 . The von Neumann-style alternating projection in R2 quickly converges to P1 P2 b (feasibility). The unique minimum-distance projection on the intersection is, of course, P b .

always yield the closest point, although we shall show it always yields some point in the intersection or a point that attains the distance between two convex sets. Alternating projection is also known as successive projection [164] [161] [59], cyclic projection [135] [242, 3.2], successive approximation [74], or projection on convex sets [303] [304, 6.4]. It is traced back to von Neumann (1933) [344] and later Wiener [351] who showed that higher iterates of a product of two orthogonal projections on subspaces converge at each point in the ambient space to the unique minimum-distance projection on the intersection of the two subspaces. More precisely, if R1 and R2 are closed subspaces of a Euclidean space and P1 and P2 respectively denote orthogonal projection on R1 and R2 , then for each vector b in that space,

§

§

lim (P1 P2 )i b = PR1 ∩ R2 b

i→∞

§

(1972)

Deutsch [99, thm.9.8, thm.9.35] shows rate of convergence for subspaces to be geometric [366, 1.4.4]; bounded above by κ2i+1 kbk , i = 0, 1, 2 . . . , where

722

APPENDIX E. PROJECTION

0≤κ<1 :

k (P1 P2 )i b − PR1 ∩ R2 b k ≤ κ2i+1 kbk

(1973)

This means convergence can be slow when κ is close to 1. Rate of convergence on intersecting halfspaces is also geometric. [100] [275] This von Neumann sense of alternating projection may be applied to convex sets that are not subspaces, although convergence is not necessarily to the unique minimum-distance projection on the intersection. Figure 158 illustrates one application where convergence is reasonably geometric and the result is the unique minimum-distance projection. Figure 159, in contrast, demonstrates convergence in one iteration to a fixed point (of the projection product)E.18 in the intersection of two halfspaces; a.k.a, feasibility problem. It was Dykstra who in 1983 [117] ( E.10.3) first solved this projection problem.

§

E.10.0.3

the bullets

Alternating projection has, therefore, various meaning dependent on the application or field of study; it may be interpreted to be: a distance problem, a feasibility problem (von Neumann), or a projection problem (Dykstra):

ˆ Distance.

Figure 160(a)(b). Find a unique point of projection P1 b ∈ C1 that attains the distance between any two closed convex sets C1 and C2 ; kP1 b − bk = dist(C1 , C2 ) , inf kP1 z − zk z∈C2

ˆ Feasibility. Figure 160(c), T C 6= ∅ .

(1974)

Given a number L of indexed closed convex sets Ck ⊂ R , find any fixed point in their intersection by iterating (i) a projection product starting from b ; Ã∞ L ! L YY \ Pk b ∈ Ck (1975) n

k

i=1 k=1

k=1

ˆ Optimization. Figure 160(c), T C 6= ∅ . Given a number of indexed T k

closed convex sets Ck ⊂ Rn , uniquely project a given point b on kP b − bk = inf T kx − bk x∈

Ck

Ck ;

(1976)

Fixed point of a mapping T : Rn → Rn is a point x whose image is identical under the map; id est, T x = x .

E.18

E.10. ALTERNATING PROJECTION

723

C2

a

(a)

C1

{y | (b − P1 b)T (y − P1 b) = 0} C2

b

(b)

C1

P1 b

y b Pb P1P2 b

(c)

C1

C2

Figure 160: (a) (distance) Intersection of two convex sets in R2 is empty. Method of alternating projection would be applied to find that point in C1 nearest C2 . (b) (distance) Given b ∈ C2 , then P1 b ∈ C1 is nearest b iff (y −P1 b)T (b −P1 b) ≤ 0 ∀ y ∈ C1 by the unique minimum-distance projection theorem ( E.9.1.0.2). When P1 b attains the distance between the two sets, hyperplane {y | (b − P1 b)T (y − P1 b) = 0} separates C1 from C2 . [57, 2.5.1] (c) (0 distance) Intersection is nonempty. T (optimization) We may want the point P b in Ck nearest point b . (feasibility) We may T instead be satisfied with a fixed point of the projection product P1 P2 b in Ck .

§

§

724

APPENDIX E. PROJECTION

E.10.1

Distance and existence

Existence of a fixed point is established: E.10.1.0.1 Theorem. Distance. [74] Given any two closed convex sets C1 and C2 in Rn , then P1 b ∈ C1 is a fixed point of the projection product P1 P2 if and only if P1 b is a point of C1 nearest C2 . ⋄ Proof. (⇒) Given fixed point a = P1 P2 a ∈ C1 with b , P2 a ∈ C2 in tandem so that a = P1 b , then by the unique minimum-distance projection theorem ( E.9.1.0.2) (b − a)T (u − a) ≤ 0 ∀ u ∈ C1 (a − b)T (v − b) ≤ 0 ∀ v ∈ C2 (1977) ⇔ ka − bk ≤ ku − vk ∀ u ∈ C1 and ∀ v ∈ C2

§

by Cauchy-Schwarz inequality |hx, yi| ≤ kxk kyk [285] (with equality iff x = κy where κ ∈ R [211, p.137]). (⇐) Suppose a ∈ C1 and ka − P2 ak ≤ ku − P2 uk ∀ u ∈ C1 and we choose u = P1 P2 a . Then ku − P2 uk = kP1 P2 a − P2 P1 P2 ak ≤ ka − P2 ak ⇔ a = P1 P2 a

(1978)

Thus a = P1 b (with b = P2 a ∈ C2 ) is a fixed point in C1 of the projection ¨ product P1 P2 .E.19

E.10.2

Feasibility and convergence

§

The set of all fixed points of any nonexpansive mapping is a closed convex set. [143, lem.3.4] [28, 1] The projection product P1 P2 is nonexpansive by Theorem E.9.3.0.1 because, for any vectors x, a ∈ Rn kP1 P2 x − P1 P2 ak ≤ kP2 x − P2 ak ≤ kx − ak

(1979)

If the intersection of two closed convex sets C1 ∩ C2 is empty, then the iterates converge to a point of minimum distance, a fixed point of the projection product. Otherwise, convergence is to some fixed point in their intersection E.19

Point b = P2 a can be shown, similarly, to be a fixed point of product P2 P1 .

E.10. ALTERNATING PROJECTION

725

(a feasible point) whose existence is guaranteed by virtue of the fact that each and every point in the convex intersection is in one-to-one correspondence with fixed points of the nonexpansive projection product. Bauschke & Borwein [28, 2] argue that any sequence monotonic in the sense of Fej´er is convergent.E.20

§

E.10.2.0.1 Definition. Fej´er monotonicity. [251] Given closed convex set C = 6 ∅ , then a sequence xi ∈ Rn , i = 0, 1, 2 . . . , is monotonic in the sense of Fej´er with respect to C iff kxi+1 − ck ≤ kxi − ck for all i ≥ 0 and each and every c ∈ C

(1980) △

Given x0 , b , if we express each iteration of alternating projection by xi+1 = P1 P2 xi ,

i = 0, 1, 2 . . .

(1981)

and define any fixed point a = P1 P2 a , then sequence xi is Fej´er monotone with respect to fixed point a because kP1 P2 xi − ak ≤ kxi − ak ∀ i ≥ 0

(1982)

by nonexpansivity. The nonincreasing sequence kP1 P2 xi − ak is bounded below hence convergent because any bounded monotonic sequence in R is convergent; [240, 1.2] [39, 1.1] P1 P2 xi+1 = P1 P2 xi = xi+1 . Sequence xi therefore converges to some fixed point. If the intersection C1 ∩ C2 is nonempty, convergence is to some point there by the distance theorem. Otherwise, xi converges to a point in C1 of minimum distance to C2 .

§

§

E.10.2.0.2 Example. Hyperplane/orthant intersection. Find a feasible point (1975) belonging to the nonempty intersection of two convex sets: given A ∈ Rm×n , β ∈ R(A) C1 ∩ C2 = Rn+ ∩ A = {y | y º 0} ∩ {y | Ay = β} ⊂ Rn

(1983)

the nonnegative orthant with affine subset A an intersection of hyperplanes. Projection of an iterate xi ∈ Rn on A is calculated P2 xi = xi − AT(AAT )−1 (Axi − β) E.20

(1876)

Other authors prove convergence by different means; e.g., [161] [59].

726

APPENDIX E. PROJECTION y2 C1 = R2+

θ

Pb = (

∞ Q 2 Q

Pk )b

j=1 k=1

b

y1

C2 = A = {y | [ 1 1 ] y = 1} Figure 161: From Example E.10.2.0.2 in R2 , showing von Neumann-style alternating projection to find feasible point belonging to intersection of nonnegative orthant with hyperplane A . Point P b lies at intersection of hyperplane with ordinate axis. In this particular example, the feasible point found is coincidentally optimal. Rate of convergence depends upon angle θ ; as it becomes more acute, convergence slows. [161, 3]

§

° ° ° ° ∞ Q 2 Q ° ° Pk )b° °xi − ( ° ° j=1 k=1

20

18

16

14

12

10

8

6

4

2

0

0

5

10

15

20

25

Figure 162: Geometric convergence Example E.10.2.0.2 in R1000 .

30

35

of

40

45

iterates

i in

norm,

for

E.10. ALTERNATING PROJECTION

727

while, thereafter, projection of the result on the orthant is simply xi+1 = P1 P2 xi = max{0, P2 xi }

(1984)

§

where the maximum is entrywise ( E.9.2.2.3). One realization of this problem in R2 is illustrated in Figure 161: For A = [ 1 1 ] , β = 1, and x0 = b = [ −3 1/2 ]T , the iterates converge to the feasible point P b = [ 0 1 ]T . To give a more palpable sense of convergence in higher dimension, we do this example again but now we compute an alternating projection for the case A ∈ R400×1000 , β ∈ R400 , and b ∈ R1000 , all of whose entries are independently and randomly set to a uniformly distributed real number in the interval [−1, 1] . Convergence is illustrated in Figure 162. 2 This application of alternating projection to feasibility is extensible to any finite number of closed convex sets.

§

E.10.2.0.3 Example. Under- and over-projection. [54, 3] Consider the following variation of alternating projection: We begin with some point x0 ∈ Rn then project that point on convex set C and then project that same point x0 on convex set D . To the first iterate we assign x1 = (PC (x0 ) + PD (x0 )) 21 . More generally, xi+1 = (PC (xi ) + PD (xi ))

1 , 2

i = 0, 1, 2 . . .

§

(1985)

Because the Cartesian product of convex sets remains convex, ( 2.1.8) we can reformulate this problem. Consider the convex set · ¸ C S, (1986) D

representing Cartesian product C × D . Now, those two projections PC and PD are equivalent to one projection on the Cartesian product; id est, ¸ ¸¶ · µ· PC (xi ) xi (1987) = PS PD (xi ) xi Define the subspace R,

½

v∈

·

Rn Rn

¾ ¸¯ ¯ ¯ [ I −I ] v = 0 ¯

(1988)

728

APPENDIX E. PROJECTION

By the results in Example E.5.0.0.6 µ· ¸¶ · ¸ ¸¶ µ· PC (xi ) PC (xi ) + PD (xi ) 1 xi = = PR PRS PD (xi ) PC (xi ) + PD (xi ) 2 xi

(1989)

This means the proposed variation of alternating projection is equivalent to an alternation of projection on convex sets S and R . If S and R intersect, these iterations will converge to a point in their intersection; hence, to a point in the intersection of C and D . We need not apply equal weighting to the projections, as supposed in 2 (1985). In that case, definition of R would change accordingly. E.10.2.1

Relative measure of convergence

Inspired by Fej´er monotonicity, the alternating projection algorithm from the example of convergence illustrated by Figure 162 employs a redundant ∞ Q L Q sequence: The first sequence (indexed by j) estimates point ( Pk )b in j=1 k=1

the presumably nonempty intersection of L convex sets, then the quantity ° Ã ∞ L ! ° ° ° YY ° ° Pk b° (1990) °xi − ° ° j=1 k=1

in second sequence xi is observed per iteration i for convergence. A priori knowledge of a feasible point (1975) is both impractical and antithetical. We need another measure: Nonexpansivity implies °Ã L ! ÃL ! ° ° ° Y Y ° ° Pℓ xk,i−1 − Pℓ xki ° = kxki − xk,i+1 k ≤ kxk,i−1 − xki k (1991) ° ° ° ℓ=1

ℓ=1

where

xki , Pk xk+1,i ∈ Rn ,

xL+1,i , x1,i−1

(1992)

xki represents unique minimum-distance projection of xk+1,i on convex set k at iteration i . So a good convergence measure is total monotonic sequence εi ,

L X k=1

kxki − xk,i+1 k ,

i = 0, 1, 2 . . .

where lim εi = 0 whether or not the intersection is nonempty. i→∞

(1993)

E.10. ALTERNATING PROJECTION

729

E.10.2.1.1 Example. Affine subset ∩ positive semidefinite cone. Consider the problem of finding X ∈ S n that satisfies X º 0,

hAj , X i = bj ,

j =1 . . . m

(1994)

given nonzero Aj ∈ S n and real bj . Here we take C1 to be the positive n semidefinite cone S+ while C2 is the affine subset of S n C2 = A , {X | tr(Aj X) = bj , j = 1 . . . m} ⊆ S n   svec(A1 )T .. svec X = b} = {X |  . T svec(Am )

(1995)

, {X | A svec X = b}

where b = [bj ] ∈ Rm , A ∈ Rm×n(n+1)/2 , and symmetric vectorization svec is defined by (51). Projection of iterate Xi ∈ S n on A is: ( E.5.0.0.6)

§

P2 svec Xi = svec Xi − A† (A svec Xi − b)

(1996)

Euclidean distance from Xi to A is therefore

dist(Xi , A) = kXi − P2 Xi kF = kA† (A svec Xi − b)k2 (1997) P Projection of P2 Xi , j λj qj qjT on the positive semidefinite cone ( 7.1.2) is found from its eigen decomposition ( A.5.1); P1 P2 Xi =

n X j=1

§

§

max{0 , λj } qj qjT

Distance from P2 Xi to the positive semidefinite cone is therefore v uX u n n dist(P2 Xi , S+ ) = kP2 Xi − P1 P2 Xi kF = t min{0 , λj }2

(1998)

(1999)

j=1

n When the intersection is empty A ∩ S+ = ∅ , the iterates converge to that positive semidefinite matrix closest to A in the Euclidean sense. Otherwise, convergence is to some point in the nonempty intersection. Barvinok ( 2.9.3.0.1) shows that if a point feasible with (1994) exists, n then there exists an X ∈ A ∩ S+ such that º ¹√ 8m + 1 − 1 (247) rank X ≤ 2 2

§

730

APPENDIX E. PROJECTION

E.10.2.1.2 Example. Semidefinite matrix completion. Continuing Example E.10.2.1.1: When m ≤ n(n + 1)/2 and the Aj matrices are distinct members of the standard orthonormal basis {Eℓq ∈ S n } (54) {Aj ∈ S n , j = 1 . . . m} ⊆ {Eℓq } =

(

) eℓ eT , ℓ = q = 1 . . . n ℓ √1 (e eT + eq eT ) , 1 ≤ ℓ < q ≤ n ℓ 2 ℓ q (2000)

and when the constants bj are set to constrained entries of variable X , [Xℓq ] ∈ S n {bj , j = 1 . . . m} ⊆

½

Xℓq √ , ℓ = q = 1...n Xℓq 2 , 1 ≤ ℓ < q ≤ n

¾

= {hX, Eℓq i}

(2001)

then the equality constraints in (1994) fix individual entries of X ∈ S n . Thus the feasibility problem becomes a positive semidefinite matrix completion problem. Projection of iterate Xi ∈ S n on A simplifies to (confer (1996)) P2 svec Xi = svec Xi − AT (A svec Xi − b)

(2002)

From this we can see that orthogonal projection is achieved simply by setting corresponding entries of P2 Xi to the known entries of X , while the entries of P2 Xi remaining are set to corresponding entries of the current iterate Xi . Using this technique, we find a positive semidefinite completion for 

4  3   ? 2

3 4 3 ?

? 3 4 3

 2 ?   3  4

(2003)

Initializing the unknown entries to 0, they all converge geometrically to 1.5858 (rounded) after about 42 iterations. Laurent gives a problem for which no positive semidefinite completion exists: [219]   1 1 ? 0  1 1 1 ?    (2004)  ? 1 1 1  0 ? 1 1

E.10. ALTERNATING PROJECTION

731

0

10

n dist(P2 Xi , S+ )

−1

10

0

2

4

6

8

10

12

14

16

18

i

Figure 163: Distance (confer (1999)) between positive semidefinite cone and iterate (2002) in affine subset A (1995) for Laurent’s completion problem; initially, decreasing geometrically.

Initializing unknowns to 0, by alternating projection we find the constrained matrix closest to the positive semidefinite cone,  1 1 0.5454 0  1 1 1 0.5454      0.5454 1 1 1 0 0.5454 1 1 

and we find the positive semidefinite (1995):  1.0521 0.9409  0.9409 1.0980   0.5454 0.9451 0.0292 0.5454

(2005)

matrix closest to the affine subset A 0.5454 0.9451 1.0980 0.9409

 0.0292 0.5454   0.9409  1.0521

(2006)

n These matrices (2005) and (2006) attain the Euclidean distance dist(A , S+ ). Convergence is illustrated in Figure 163. 2

732

APPENDIX E. PROJECTION

b H1 x22 x21 x12

H2

x11

H1 ∩ H2

Figure 164: H1 and H2 are the same halfspaces as in Figure 159. Dykstra’s alternating projection algorithm generates the alternations b , x21 , x11 , x22 , x12 , x12 . . . x12 . The path illustrated from b to x12 in R2 terminates at the desired result, P b in Figure 159. The {yki } correspond to the first two difference vectors drawn (in the first iteration i = 1), then oscillate between zero and a negative vector thereafter. These alternations are not so robust in presence of noise as for the example in Figure 158.

E.10.3

Optimization and projection

Unique projection on the nonempty intersection of arbitrary convex sets to find the closest point therein is a convex optimization problem. The first successful application of alternating projection to this problem is attributed to Dykstra [117] [58] who in 1983 provided an elegant algorithm that prevails today. In 1988, Han [164] rediscovered the algorithm and provided a primal−dual convergence proof. A synopsis of the history of alternating projectionE.21 can be found in [60] where it becomes apparent that Dykstra’s work is seminal.

E.21

For a synopsis of alternating projection applied to distance geometry, see [328, §3.1].

E.10. ALTERNATING PROJECTION E.10.3.1

733

Dykstra’s algorithm

Assume we are given some point b ∈ Rn and closed convex sets {Ck ⊂ Rn | k = 1 . . . L}. Let xki ∈ Rn and yki ∈ Rn respectively denote a primal and dual vector (whose meaning can be deduced from Figure 164 and Figure 165) associated with set k at iteration i . Initialize yk0 = 0 ∀ k = 1 . . . L

and

x1,0 = b

(2007)

Denoting by Pk t the unique minimum-distance projection of t on Ck , and for convenience xL+1,i = x1,i−1 (1992), iterate x1i calculation proceeds:E.22 for i = 1, 2, . . . until convergence { for k = L . . . 1 { t = xk+1,i − yk,i−1 xki = Pk t yki = Pk t − t } }

(2008)

§

Assuming a nonempty intersection, then the iterates converge to the unique minimum-distance projection of point b on that intersection; [99, 9.24] P b = lim x1i i→∞

(2009)

§

In the case all the Ck are affine, then calculation of yki is superfluous and the algorithm becomes identical to alternating projection. [99, 9.26] [135, 1] Dykstra’s algorithm is so simple, elegant, and represents such a tiny increment in computational intensity over alternating projection, it is nearly always arguably cost effective.

§

E.10.3.2

Normal cone

§

Glunt [142, 4] observes that the overall effect of Dykstra’s iterative T procedure is to drive t toward the translated normal cone to Ck at the solution P b (translated to P b). The normal cone gets its name from its graphical construction; which is, loosely speaking, to draw the outward-normals at P b (Definition E.9.1.0.1) to all the convex sets Ck touching P b . The relative interior of the normal cone subtends these normal vectors. E.22

We reverse order of projection (k = L . . . 1) in the algorithm for continuity of exposition.

734

APPENDIX E. PROJECTION ⊥ KH (0) 1 ∩ H2

⊥ KH (P b) + P b 1 ∩ H2

b

H1 0 Pb

H2

K , H1 ∩ H2

Figure 165: Two examples (truncated): Normal cone to H1 ∩ H2 at the origin, and at point P b on the boundary. H1 and H2 are the same halfspaces ∗ ⊥ (0) is simply −K . from Figure 164. The normal cone at the origin KH 1 ∩ H2

§

§

§

E.10.3.2.1 Definition. Normal cone. [249] [39, p.261] [185, A.5.2] [51, 2.1] [284, 3] [285, p.15] The normal cone to any set S ⊆ Rn at any particular point a ∈ Rn is defined as the closed cone KS⊥ (a) , {z ∈ Rn | z T (y − a) ≤ 0 ∀ y ∈ S} = −(S − a)∗

(2010)

an intersection of halfspaces about the origin in Rn , hence convex regardless of convexity of S ; it is the negative dual cone to the translate S − a ; the set of all vectors normal to S at a . △ Examples of normal cone construction are illustrated in Figure 64 and Figure 165: The normal cone at the origin in Figure 165 is the vector sum ( 2.1.8) of two normal cones; [51, 3.3, exer.10] for H1 ∩ int H2 6= ∅

§

§

⊥ ⊥ ⊥ KH (0) = KH (0) + KH (0) 1 ∩ H2 1 2

(2011)

This formula applies more generally to other points in the intersection. The normal cone to any affine set A at α ∈ A , for example, is the orthogonal complement of A − α . Projection of any point in the translated

E.10. ALTERNATING PROJECTION

735

E3

KE⊥3 (11T ) + 11T

Figure 166: A few renderings (including next page) of normal cone KE⊥3 to elliptope E 3 (Figure 122) at point 11T , projected on R3 . In [220, fig.2], normal cone is claimed circular in this dimension (severe numerical artifacts corrupt boundary and make interior corporeal, drawn truncated).

736

APPENDIX E. PROJECTION

normal cone KC⊥ (a ∈ C) + a on convex set C is identical to a ; in other words, point a is that point in C closest to any point belonging to the translated normal cone KC⊥ (a) + a ; e.g., Theorem E.4.0.0.1. When set S is a convex cone K , then the normal cone to K at the origin ⊥ (0) = −K KK

∗

§

(2012) ∗

is the negative dual cone. Any point belonging to −K , projected on K , projects on the origin. More generally, [99, 4.5] ⊥ KK (a) = −(K − a)∗ ∗

⊥ KK (a ∈ K) = −K ∩ a⊥

(2013) (2014)

T The normal cone to Ck at P b in Figure 159 is ray {ξ(b − P b) | ξ ≥ 0} illustrated in Figure 165. Applying Dykstra’s algorithm to that example, convergence to the desired result is achieved in two iterations as illustrated in Figure 164. Yet applying Dykstra’s algorithm to the example in Figure 158 does not improve rate of convergence, unfortunately, because the given point b and all the alternating projections already belong to the translated normal cone at the vertex of intersection.

E.10. ALTERNATING PROJECTION E.10.3.3

737

speculation

Dykstra’s algorithm always converges at least as quickly as classical alternating projection, never slower [99], and it succeeds where alternating projection fails. Rate of convergence is wholly dependent on particular geometry of a given problem. From these few examples we surmise, unique minimum-distance projection on blunt (not sharp or acute, informally) polyhedral cones having nonempty interior may be found by Dykstra’s algorithm in few iterations. But total number of alternating projections, constituting those iterations, can never be less than number of convex sets.

738

APPENDIX E. PROJECTION

Appendix F Notation and a few definitions b

(italic abcdef ghijklmnopqrstuvwxyz) column vector, scalar, logical condition

bi

i th entry of vector b = [b i , i = 1 . . . n] or i th b vector from a set or list {bj , j = 1 . . . n} or i th iterate of vector b

b i:j

or b(i : j) , truncated vector comprising i th through j th entry of vector b

bk (i : j)

or b i:j, k , truncated vector comprising i th through j th entry of vector bk

bT

vector transpose

bH

Hermitian (complex conjugate) transpose b ∗T

AT

Matrix transpose. Regarding A as a linear operator, AT is its adjoint.

A−2T AT1 A

skinny

matrix transpose of squared inverse first of various transpositions of a cubix or quartix A (p.650, p.655) matrix, scalar, or logical condition (italic ABCDEF GHIJKLM N OP QRST U V W XY Z) 

a skinny matrix; meaning, more rows than columns: 

§



. When

there are more equations than unknowns, we say that the system Ax = b is overdetermined. [146, 5.3] © 2001 Jon Dattorro. co&edg version 2009.09.22. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry, Mεβoo Publishing USA, 2005, v2009.09.22.

739

740 fat A F(C ∋ A) G(K) A−1

A1/2 and

APPENDIX F. NOTATION AND A FEW DEFINITIONS a fat matrix; meaning, more columns than rows: underdetermined

smallest face (154) that contains element A of set C

§

generators ( 2.13.4.2.1) of set K ; any collection of points and directions whose hull constructs K inverse of matrix A Moore-Penrose pseudoinverse of matrix A

√

positive square root

√ ℓ

positive ℓ th root

√ ◦

D A

¤

some set (calligraphic ABCDEFGHIJ KLMN OPQRST UVWX YZ)

A†

√ A

£

§

§

A1/2 is any matrix that A1/2A1/2 √ = A√. √ such n n For A ∈ S+ , A ∈ S+ is unique and A A = A . [51, 1.2] ( A.5.1.2) √ √ p = [ dij ] . (1293) Hadamard positive square root: D = ◦ D ◦ ◦ D Euler Fraktur A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

E

member of elliptope Et (1043) parametrized by scalar t

E

elliptope (1022)

E

elementary matrix

Eij

member of standard orthonormal basis for symmetric (54) or symmetric hollow (70) matrices 1 2 3   · · · 1 or A(i , j) , ij th entry of matrix A =  · · ·  2 · · · 3

Aij

§

or rank-one matrix ai aT j ( 4.7)

A(i , j) Ai

A is a function of i and j i th matrix from a set or i th principal submatrix or i th iterate of A

741 A(i , :)

i th row of matrix A

A(: , j)

j th column of matrix A [146, 1.1.8]

Ai:j, k:ℓ

or A(i : j , k : ℓ) , submatrix taken from i th through j th row and k th through ℓ th column

e.g.

exempli gratia, from the Latin meaning for sake of example

no.

number, from the Latin numero

a.i.

affinely independent ( 2.4.2.3)

c.i.

conically independent ( 2.10)

l.i.

linearly independent

§

w.r.t

with respect to

a.k.a

also known as

Re

real part

Im

imaginary part √ −1

ı or  ⊂ ⊃

§

∩ ∪

§

standard set theory, subset, superset, intersection, union

∈

membership, element belongs to, or element is a member of

∋

membership, contains as in C ∋ y (C contains element y)

Ä

such that

∃

there exists

∴

therefore

∀

for all, or over all

∝

proportional to

∞

infinity

742

APPENDIX F. NOTATION AND A FEW DEFINITIONS

≡

equivalent to

,

defined equal to, equal by definition

≈

approximately equal to

≃

isomorphic to or with

∼ =

congruent to or with Hadamard quotient as in, for x, y ∈ Rn ,

x , [ xi /yi , i = 1 . . . n ] ∈ Rn y

◦

Hadamard product of matrices: x ◦ y , [ xi yi , i = 1 . . . n ] ∈ Rn

⊗

Kronecker product of matrices ( D.1.2.1)

⊕

vector sum of sets X = Y ⊕ Z where every element x ∈ X has unique expression x = y + z where y ∈ Y and z ∈ Z ; [285, p.19] then the summands are algebraic complements. X = Y ⊕ Z ⇒ X = Y + Z . Now assume Y and Z are nontrivial subspaces. X = Y + Z ⇒ X = Y ⊕ Z ⇔ Y ∩ Z = 0 [286, 1.2] [99, 5.8]. Each element from a vector sum (+) of subspaces has a unique representation (⊕) when a basis from each subspace is linearly independent of bases from all the other subspaces.

§

§

§

⊖

likewise, the vector difference of sets

⊞

orthogonal vector sum of sets X = Y ⊞ Z where every element x ∈ X has unique orthogonal expression x = y + z where y ∈ Y , z ∈ Z , and y ⊥ z . [308, p.51] X = Y ⊞ Z ⇒ X = Y + Z . If Z ⊆ Y ⊥ then X = Y ⊞ Z ⇔ X = Y ⊕ Z . [99, 5.8] If Z = Y ⊥ then the summands are orthogonal complements.

§

±

plus or minus or plus and minus

⊥

as in A ⊥ B meaning A is orthogonal to B (and vice versa), where A and B are sets, vectors, or matrices. When A and B are vectors (or matrices under Frobenius norm), A ⊥ B ⇔ hA , Bi = 0 ⇔ kA + Bk2 = kAk2 + kBk2

\

as in \A means logical not A , or relative complement of set A ; id est, \A = {x ∈ / A} ; e.g., B\A , {x ∈ B | x ∈ / A} ≡ B ∩\A

743 ⇒ or ⇐ ⇔

if and only if (iff) or corresponds with or necessary and sufficient or logical equivalence

is

as in A is B means A ⇒ B ; conventional usage of English language imposed by logicians

; or :

does not imply

←

is replaced with; substitution, assignment

→

goes to, or approaches, or maps to

t → 0+ .. . . . .

sufficiency or necessity, implies; e.g., A ⇒ B ⇔ \A ⇐ \B

t goes to 0 from above; meaning, from the positive [185, p.2]

···

as in 1 · · · 1 meaning continuation, e.g., a sequence of ones in a row

...

as in i = 1 . . . N meaning, i is a sequence of successive integers beginning with 1 and ending with N

:

as in f : Rn → Rm meaning f is a mapping, or sequence of successive integers specified by bounds as in i : j (if j < i then sequence is descending)

f : M→R

meaning f is a mapping from ambient space M to ambient R , not necessarily denoting either domain or range

|

as in f (x) | x ∈ C means with the condition(s) or such that or evaluated for, or as in {f (x) | x ∈ C} means evaluated for each and every x belonging to set C

g|xp A,B

expression g evaluated at xp as in, for example, A , B ∈ SN means A ∈ SN and B ∈ SN

(A , B)

open interval between A and B in R , or variable pair perhaps of disparate dimension

[A , B ]

closed interval or line segment between A and B in R

( )

hierarchal, parenthetical, optional

744

APPENDIX F. NOTATION AND A FEW DEFINITIONS

{}

curly braces denote a set or list, e.g., {Xa | a º 0} the set comprising Xa evaluated for each and every a º 0 where membership of a to some space is implicit, a union

h i

angle brackets denote vector inner-product (28) (33)

[ ]

matrix or vector, or quote insertion, or citation

[dij ]

matrix whose ij th entry is dij

[xi ]

vector whose i th entry is xi

xp

particular value of x

x0

particular instance of x , or initial value of a sequence xi

x1

first entry of vector x , or first element of a set or list {xi }

xε

extreme point

x+

vector x whose negative entries are replaced with 0 ; x+ = 12 (x + |x|) (501) or clipped vector x or nonnegative part of x

x−

x− , 12 (x − |x|) or nonpositive part of x = x+ + x−

xˇ

known data

x⋆

optimal value of variable x

x∗

complex conjugate or dual variable or extreme direction of dual cone

f∗

convex conjugate function f ∗ (s) = sup{hs , xi − f (x) | x ∈ dom f }

PC x or P x Pk x

projection of point x on set C , P is operator or idempotent matrix projection of point x on set Ck or on range of implicit vector

§

δ(A)

( A.1) vector made from the main diagonal of A if A is a matrix; otherwise, diagonal matrix made from vector A

δ 2(A)

≡ δ(δ(A)). For vector or diagonal matrix Λ , δ 2(Λ) = Λ

δ(A)2

= δ(A)δ(A) where A is a vector

745 λ i (X)

i th entry of vector λ is function of X

λ(X)i

i th entry of vector-valued function of X

λ(A)

vector of eigenvalues of matrix A , (1405) typically arranged in nonincreasing order

σ(A)

vector of singular values of matrix A (always arranged in nonincreasing order), or support function in direction A

Σ P

π(γ)

diagonal matrix of singular values, not necessarily square sum

§

nonlinear permutation operator (or presorting function) arranges vector γ into nonincreasing order ( 7.1.3)

Ξ

permutation matrix

Π Q

doublet or permutation operator or matrix

ψ(Z)

product signum-like step function that returns a scalar for matrix argument (671), it returns a vector for vector argument (1511)

D

symmetric hollow matrix of distance-square, or Euclidean distance matrix

D

Euclidean distance matrix operator

DT(X)

adjoint operator

D(X)T

transpose of D(X)

D−1 (X)

inverse operator

D(X)−1

inverse of D(X)

D⋆

optimal value of variable D

D∗

dual to variable D

D◦

polar variable D

746

APPENDIX F. NOTATION AND A FEW DEFINITIONS

∂

partial derivative or partial differential or matrix of distance-square squared (1331) or boundary of set K as in ∂K

dij

(absolute) distance scalar

dij

distance-square scalar, EDM entry

V

geometric centering operator, V(D) = −V D V

VN

VN (D) = −VNTDVN

q

1 2

(942)

(956)

§

V

N ×N symmetric elementary, auxiliary, projector, geometric centering matrix, R(V ) = N (1T ) , N (V ) = R(1) , V 2 = V ( B.4.1)

VN

N ×N − 1 Schoenberg T N (VN ) = R(1) ( B.4.2)

VX

VX VXT ≡ V TX TX V

§

auxiliary

matrix,

R(VN ) = N (1T ) ,

(1136)

X

point list ((71) having cardinality N ) arranged columnar in Rn×N , or set of generators, or extreme directions, or matrix variable

G

Gram matrix X TX

r

affine dimension

k

number of conically independent generators or raw-data domain of Magnetic Resonance Imaging machine as in k-space

n

Euclidean (ambient spatial) dimension of list X ∈ Rn×N , or integer

N

cardinality of list X ∈ Rn×N , or integer

on

function f (x) on A means A is dom f , or projection of x on A means A is Euclidean body on which projection of x is made

onto

function f (x) maps onto M means f over its domain is a surjection with respect to M

one-to-one epi

injective map or unique correspondence between sets function epigraph

747 dom Rf R(A) span basis R(A)

N (A) Rn Rm×n × · m ¸ R Rn

function domain function range the subspace: range of A , or span basis R(A) ; R(A) ⊥ N (AT ) as in span A = R(A) = {Ax | x ∈ Rn } when A is a matrix columnar basis for range of A , or a minimal set constituting generators for the vertex-description of R(A) , or a linearly independent set of vectors spanning R(A) the subspace: nullspace of A ; N (A) ⊥ R(AT ) Euclidean n-dimensional real vector space (nonnegative integer n). R0 = 0, R = R1 or vector space of unspecified dimension. Euclidean vector space of m by n dimensional real matrices Cartesian product. Rm×n−m , Rm×(n−m) Rm × Rn = Rm+n

C n or C n×n

Euclidean complex vector space of respective dimension n and n×n

Rn+ or Rn×n +

nonnegative orthant in Euclidean vector space of respective dimension n and n×n

Rn− or Rn×n −

nonpositive orthant in Euclidean vector space of respective dimension n and n×n

Sn

subspace of real symmetric n×n matrices; the symmetric matrix subspace. S = S1 or symmetric subspace of unspecified dimension.

S n⊥

orthogonal complement of S n in Rn×n , the antisymmetric matrices

n S+

convex cone comprising all (real) symmetric positive semidefinite n×n matrices, the positive semidefinite cone

n int S+

interior of convex cone comprising all (real) symmetric positive semidefinite n×n matrices; id est, positive definite matrices

748

p

APPENDIX F. NOTATION AND A FEW DEFINITIONS

n S+ (ρ)

convex set of all positive semidefinite n×n matrices whose rank equals or exceeds ρ

EDMN

cone of N × N Euclidean distance matrices in the symmetric hollow subspace

EDMN

nonconvex cone of N × N Euclidean absolute distance matrices in the symmetric hollow subspace ( 6.3)

§

PSD

positive semidefinite

SDP

semidefinite program

SVD

singular value decomposition

SNR

signal to noise ratio

dB EDM

decibel Euclidean distance matrix

S1n

subspace comprising all symmetric n×n matrices having all zeros in first row and column (1925)

Shn

subspace comprising all symmetric hollow n×n matrices (0 main diagonal), the symmetric hollow subspace (61)

Shn⊥

orthogonal complement of Shn in S n (62), the set of all diagonal matrices

Scn

subspace comprising all geometrically centered symmetric n×n N matrices; geometric center subspace SN c = {Y ∈ S | Y 1 = 0} (1921)

Scn⊥ Rm×n c X⊥ x⊥ R(P )⊥ R⊥

orthogonal complement of Scn in S n (1923) subspace comprising all geometrically centered m×n matrices basis N (X T ) N (xT ) , {y ∈ Rn | xTy = 0} N (P T ) = {y ∈ Rn | hx, yi = 0 ∀ x ∈ R} (346); for R a subspace, orthogonal complement of R in Rn

749 K⊥ K K K

∗

◦

normal cone (2010) cone dual cone ∗

polar cone; K = −K

◦

KM+

monotone nonnegative cone

KM

monotone cone

Kλ ∗

Kλδ

spectral cone cone of majorization

H

halfspace

H−

halfspace described using an outward-normal (100) to the hyperplane partially bounding it

H+

halfspace described using an inward-normal (101) to the hyperplane partially bounding it

∂H

hyperplane; id est, partial boundary of halfspace

∂H

supporting hyperplane

∂H−

a supporting hyperplane having outward-normal with respect to set it supports

∂H+

a supporting hyperplane having inward-normal with respect to set it supports

d

vector of distance-square

dij

lower bound on distance-square dij

dij

upper bound on distance-square dij

AB

closed line segment between points A and B

AB

matrix multiplication of A and B

750 C decomposition expansion vector

APPENDIX F. NOTATION AND A FEW DEFINITIONS closure of set C orthonormal (1836) page 682, biorthogonal (1813) page 676 orthogonal (1846) page 685, biorthogonal (375) page 182 column vector in Rn

entry

scalar element or real variable constituting a vector or matrix

cubix

member of RM ×N ×L

quartix

member of RM ×N ×L×K

feasible set

most simply, the set of all variable values satisfying all constraints of an optimization problem

solution set

most simply, the set of all optimal solutions to an optimization problem; a subset of the feasible set and not necessarily a single point

optimal solution

solution to an optimization problem, distinguished throughout from feasible solution which may or may not be optimal

natural order

with reference to stacking columns in a vectorization means a vector made from superposing column 1 on top of column 2 then superposing the result on column 3 and so on; as in a vector made from entries of the main diagonal δ(A) means taken from left to right and top to bottom

operator tight

mapping to a vector space (a multidimensional function) with reference to a bound means a bound that can be met, with reference to an inequality means equality is achievable

g′

first derivative of possibly multidimensional function with respect to real argument

g ′′

second derivative with respect to real argument

→Y

dg

→Y

dg 2

first directional derivative of possibly multidimensional function g in direction Y ∈ RK×L (maintains dimensions of g) second directional derivative of g in direction Y

751 ∇

gradient from calculus, ∇f is shorthand for ∇x f (x). ∇f (y) means ∇y f (y) or gradient ∇x f (y) of f (x) with respect to x evaluated at y

∇2

second-order gradient

∆

distance scalar (Figure 25), or first-order difference matrix (799), or infinitesimal difference operator ( D.1.4)

△ijk

§

triangle made by vertices i , j , and k

I

Roman numeral one

I

identity matrix

I

index set, a discrete set of indices

∅

empty set, an implicit member of every set

0

real zero

0

origin or vector or matrix of zeros

O

sort-index matrix

O

order of magnitude information required, or computational intensity: O(N ) is first order, O(N 2 ) is second, and so on

1

real one

1

vector of ones

ei

vector whose i th entry is 1 (otherwise 0), i th member of the standard basis for Rn (55)

max maximize x

arg sup X

§

maximum [185, 0.1.1] or largest element of a totally ordered set find the maximum of a function with respect to variable x argument of operator or function, or variable of optimization

§

supremum of totally ordered set X , least upper bound, may or may not belong to set [185, 0.1.1]

752 arg sup f (x) subject to min minimize x

APPENDIX F. NOTATION AND A FEW DEFINITIONS argument x at supremum of function f ; not necessarily unique or a member of function domain specifies constraints to an optimization problem

§

minimum [185, 0.1.1] or smallest element of a totally ordered set find the function minimum with respect to variable x

§

inf X

infimum of totally ordered set X , greatest lower bound, may or may not belong to set [185, 0.1.1]

arg inf f (x)

argument x at infimum of function f ; not necessarily unique or a member of function domain

iff

if and only if , necessary and sufficient; also the meaning indiscriminately attached to appearance of the word “if ” in the statement of a mathematical definition, [118, p.106] [240, p.4] an esoteric practice worthy of abolition because of ambiguity thus conferred

rel

relative

int

interior

lim

limit

sgn

signum function or sign

round mod tr

round to nearest integer modulus function matrix trace

rank

as in rank A , rank of matrix A ; dim R(A)

dim

dimension, dim Rn = n , dim R(A ∈ Rm×n ) = rank(A)

aff dim aff

affine hull affine dimension

dim(x ∈ Rn ) = n ,

dim R(x ∈ Rn ) = 1,

753 card

cardinality, number of nonzero entries card x , kxk0 or N is cardinality of list X ∈ Rn×N (p.308)

conv

convex hull ( 2.3.2)

cone

conic hull ( 2.3.3)

cenv

convex envelope ( 7.2.2.1)

content

§

§

§

of high-dimensional bounded polyhedron, volume in 3 dimensions, area in 2, and so on

cof

matrix of cofactors corresponding to matrix argument

dist

distance between point or set arguments; e.g., dist(x , B)

vec

columnar vectorization of m × n matrix, Euclidean dimension mn (32)

svec

columnar vectorization of symmetric n × n matrix, dimension n(n + 1)/2 (51)

dvec

columnar vectorization of symmetric hollow n × n matrix, Euclidean dimension n(n − 1)/2 (68)

Á(x, y) º

angle between vectors x and y , subsets

Euclidean

or dihedral angle between affine

generalized inequality; e.g., A º 0 means: vector or matrix A must be expressible in a biorthogonal expansion having nonnegative coordinates with respect to extreme directions of some implicit pointed closed convex cone K ( 2.13.2.0.1, 2.13.7.1.1), or comparison to the origin with respect to some implicit pointed closed convex cone (2.7.2.2), n or (when K = S+ ) matrix A belongs to the positive semidefinite cone of symmetric matrices ( 2.9.0.1), or (when K = Rn+ ) vector A belongs to the nonnegative orthant (each vector entry is nonnegative, 2.3.1.1)

§

§

§

§

≻

strict generalized inequality, membership to cone interior

⊁

not positive definite

≥

scalar inequality, greater than or equal to; comparison of scalars, or entrywise comparison of vectors or matrices with respect to R+

754 nonnegative > positive

APPENDIX F. NOTATION AND A FEW DEFINITIONS for α ∈ Rn , α º 0 greater than for α ∈ Rn , α ≻ 0 ; id est, no zero entries when with respect to nonnegative orthant, no vector on boundary with respect to pointed closed convex cone K

⌊ ⌋

floor function, ⌊x⌋ is greatest integer not exceeding x

| |

entrywise absolute value of scalars, vectors, and matrices

log

natural (or Napierian) logarithm

det

matrix determinant

kxk kxkℓ

vector 2-norm s n P |xj |ℓ = ℓ

,

j=1 n P

j=1

or

|xj |ℓ

Euclidean norm kxk2 vector ℓ-norm for ℓ ≥ 1

= max{|xj | ∀ j}

infinity-norm

kxk22

= xTx = hx , xi

Euclidean norm square

kxk1

= 1T |x|

1-norm, dual infinity-norm

kxk0

= 1T |x|0

kXk2

§

vector ℓ-norm for 0 ≤ ℓ < 1 (violates 3.3 no.3)

kxk∞

(0 0 , 0)

(convex)

§

0-norm ( 4.5.1), cardinality of vector x (card x) p = sup kXak2 = σ1 = λ(X TX)1 (603) matrix 2-norm (spectral norm), kak=1

largest singular value, kXak2 ≤ kXk2 kak2 [146, p.56], kδ(x)k2 = kxk∞

kXk

= kXkF

Frobenius’ matrix norm

Bibliography [1] Edwin A. Abbott. Flatland: A Romance of Many Dimensions. Seely & Co., London, sixth edition, 1884. [2] Suliman Al-Homidan and Henry Wolkowicz. Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming, April 2004. orion.math.uwaterloo.ca/~hwolkowi/henry/reports/edmapr04.ps [3] Faiz A. Al-Khayyal and James E. Falk. Jointly constrained biconvex programming. Mathematics of Operations Research, 8(2):273–286, May 1983. http://www.convexoptimization.com/TOOLS/Falk.pdf [4] Abdo Y. Alfakih. On the uniqueness of Euclidean distance matrix completions. Linear Algebra and its Applications, 370:1–14, 2003. [5] Abdo Y. Alfakih. On the uniqueness of Euclidean distance matrix completions: the case of points in general position. Linear Algebra and its Applications, 397:265–277, 2005. [6] Abdo Y. Alfakih, Amir Khandani, and Henry Wolkowicz. Solving Euclidean distance matrix completion problems via semidefinite programming. Computational Optimization and Applications, 12(1):13–30, January 1999. http://citeseer.ist.psu.edu/alfakih97solving.html [7] Abdo Y. Alfakih and Henry Wolkowicz. On the embeddability of weighted graphs in Euclidean spaces. Research Report CORR 98-12, Department of Combinatorics and Optimization, University of Waterloo, May 1998. http://citeseer.ist.psu.edu/alfakih98embeddability.html Erratum: p.420 herein. [8] Abdo Y. Alfakih and Henry Wolkowicz. Matrix completion problems. In Henry Wolkowicz, Romesh Saigal, and Lieven Vandenberghe, editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, chapter 18. Kluwer, 2000. http://www.convexoptimization.com/TOOLS/Handbook.pdf [9] Abdo Y. Alfakih and Henry Wolkowicz. Two theorems on Euclidean distance matrices and Gale transform. Linear Algebra and its Applications, 340:149–154, 2002.

755

756

BIBLIOGRAPHY

[10] Farid Alizadeh. Combinatorial Optimization with Interior Point Methods and Semi-Definite Matrices. PhD thesis, University of Minnesota, Computer Science Department, Minneapolis Minnesota USA, October 1991. [11] Farid Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization, 5(1):13–51, February 1995. [12] Kurt Anstreicher and Henry Wolkowicz. On Lagrangian relaxation of quadratic matrix constraints. SIAM Journal on Matrix Analysis and Applications, 22(1):41–55, 2000. [13] Howard Anton. Elementary Linear Algebra. Wiley, second edition, 1977. [14] James Aspnes, David Goldenberg, and Yang Richard Yang. On the computational complexity of sensor network localization. In Proceedings of the First International Workshop on Algorithmic Aspects of Wireless Sensor Networks (ALGOSENSORS), volume 3121 of Lecture Notes in Computer Science, pages 32–44, Turku Finland, July 2004. Springer-Verlag. cs-www.cs.yale.edu/homes/aspnes/localization-abstract.html [15] D. Avis and K. Fukuda. A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete and Computational Geometry, 8:295–313, 1992. [16] Christine Bachoc and Frank Vallentin. New upper bounds for kissing numbers from semidefinite programming. ArXiv.org, October 2006. http://arxiv.org/abs/math/0608426 [17] Mih´aly Bakonyi and Charles R. Johnson. The Euclidean distance matrix completion problem. SIAM Journal on Matrix Analysis and Applications, 16(2):646–654, April 1995. [18] Keith Ball. An elementary introduction to modern convex geometry. In Silvio Levy, editor, Flavors of Geometry, volume 31, chapter 1, pages 1–58. MSRI Publications, 1997. www.msri.org/publications/books/Book31/files/ball.pdf [19] Richard G. Baraniuk. Compressive sensing [lecture notes]. IEEE Signal Processing Magazine, 24(4):118–121, July 2007. http://www.convexoptimization.com/TOOLS/GeometryCardinality.pdf [20] George Phillip Barker. 39:263–291, 1981.

Theory of cones.

Linear Algebra and its Applications,

[21] George Phillip Barker and David Carlson. Cones of diagonally dominant matrices. Pacific Journal of Mathematics, 57(1):15–32, 1975. [22] George Phillip Barker and James Foran. Self-dual cones in Euclidean spaces. Linear Algebra and its Applications, 13:147–155, 1976.

BIBLIOGRAPHY

757

[23] Dror Baron, Michael B. Wakin, Marco F. Duarte, Shriram Sarvotham, and Richard G. Baraniuk. Distributed compressed sensing. Technical Report ECE-0612, Rice University, Electrical and Computer Engineering Department, December 2006. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.85.4789 [24] Alexander I. Barvinok. Problems of distance geometry and convex properties of quadratic maps. Discrete & Computational Geometry, 13(2):189–202, 1995. [25] Alexander I. Barvinok. A remark on the rank of positive semidefinite matrices subject to affine constraints. Discrete & Computational Geometry, 25(1):23–31, 2001. http://citeseer.ist.psu.edu/304448.html [26] Alexander I. Barvinok. A Course in Convexity. American Mathematical Society, 2002. [27] Alexander I. Barvinok. Approximating orthogonal matrices by permutation matrices. Pure and Applied Mathematics Quarterly, 2:943–961, 2006. [28] Heinz H. Bauschke and Jonathan M. Borwein. On projection algorithms for solving convex feasibility problems. SIAM Review, 38(3):367–426, September 1996. [29] Steven R. Bell. The Cauchy Transform, Potential Theory, and Conformal Mapping. CRC Press, 1992. [30] Jean Bellissard and Bruno Iochum. Homogeneous and facially homogeneous self-dual cones. Linear Algebra and its Applications, 19:1–16, 1978. [31] Adi Ben-Israel. Linear equations and inequalities on finite dimensional, real or complex, vector spaces: A unified theory. Journal of Mathematical Analysis and Applications, 27:367–389, 1969. [32] Aharon Ben-Tal and Arkadi Nemirovski. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM, 2001. [33] Aharon Ben-Tal and Arkadi Nemirovski. Non-Euclidean restricted memory level method for large-scale convex optimization. Mathematical Programming, 102(3):407–456, January 2005. http://www2.isye.gatech.edu/~nemirovs/Bundle-Mirror rev fin.pdf [34] John J. Benedetto and Paulo J.S.G. Ferreira editors. Modern Sampling Theory: Mathematics and Applications. Birkh¨auser, 2001. [35] Christian R. Berger, Javier Areta, Krishna Pattipati, and Peter Willett. Compressed sensing − A look beyond linear programming. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, pages 3857–3860, 2008. [36] Radu Berinde, Anna C. Gilbert, Piotr Indyk, Howard J. Karloff, and Martin J. Strauss. Combining geometry and combinatorics: A unified approach to sparse signal recovery, April 2008. http://arxiv.org/abs/0804.4666

758

BIBLIOGRAPHY

[37] Abraham Berman. Cones, Matrices, and Mathematical Programming, volume 79 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, 1973. [38] Dimitri P. Bertsekas. Nonlinear Programming. Athena Scientific, second edition, 1999. [39] Dimitri P. Bertsekas, Angelia Nedi´c, and Asuman E. Ozdaglar. Convex Analysis and Optimization. Athena Scientific, 2003. [40] Rajendra Bhatia. Matrix Analysis. Springer-Verlag, 1997. [41] Pratik Biswas, Tzu-Chen Liang, Kim-Chuan Toh, Yinyu Ye, and Ta-Chung Wang. Semidefinite programming approaches for sensor network localization with noisy distance measurements. IEEE Transactions on Automation Science and Engineering, 3(4):360–371, October 2006. [42] Pratik Biswas, Tzu-Chen Liang, Ta-Chung Wang, and Yinyu Ye. Semidefinite programming based algorithms for sensor network localization. ACM Transactions on Sensor Networks, 2(2):188–220, May 2006. http://www.stanford.edu/~yyye/combined rev3.pdf [43] Pratik Biswas, Kim-Chuan Toh, and Yinyu Ye. A distributed SDP approach for large-scale noisy anchor-free graph realization with applications to molecular conformation. SIAM Journal on Scientific Computing, 30(3):1251–1277, March 2008. http://www.stanford.edu/~yyye/SISC-molecule-revised-2.pdf [44] Pratik Biswas and Yinyu Ye. Semidefinite programming for ad hoc wireless sensor network localization. In Proceedings of the Third International Symposium on Information Processing in Sensor Networks (IPSN), pages 46–54. IEEE, April 2004. http://www.stanford.edu/~yyye/adhocn2.pdf [45] Leonard M. Blumenthal. On the four-point property. Bulletin of the American Mathematical Society, 39:423–426, 1933. [46] Leonard M. Blumenthal. Theory and Applications of Distance Geometry. Oxford University Press, 1953. [47] Radu Ioan Bot¸, Ern¨o Robert Csetnek, and Gert Wanka. Regularity conditions via quasi-relative interior in convex programming. SIAM Journal on Optimization, 19(1):217–233, 2008. http://www.convexoptimization.com/TOOLS/Wanka.pdf [48] A. W. Bojanczyk and A. Lutoborski. The Procrustes problem for orthogonal Stiefel matrices. SIAM Journal on Scientific Computing, 21(4):1291–1304, February 1998. Date is electronic publication: http://citeseer.ist.psu.edu/500185.html [49] Ingwer Borg and Patrick Groenen. Modern Multidimensional Scaling. Springer-Verlag, 1997.

BIBLIOGRAPHY

759

[50] Jonathan M. Borwein and Heinz Bauschke. Projection algorithms and monotone operators, 1998. http://oldweb.cecm.sfu.ca/personal/jborwein/projections4.pdf [51] Jonathan M. Borwein and Adrian S. Lewis. Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer-Verlag, 2000. http://www.convexoptimization.com/TOOLS/Borwein.pdf [52] Jonathan M. Borwein and Warren B. Moors. Stability of closedness of convex cones under linear mappings. Journal of Convex Analysis, 16(3), 2009. http://www.convexoptimization.com/TOOLS/BorweinMoors.pdf [53] Richard Bouldin. The pseudo-inverse of a product. SIAM Journal on Applied Mathematics, 24(4):489–495, June 1973. http://www.convexoptimization.com/TOOLS/Pseudoinverse.pdf [54] Stephen Boyd and Jon Dattorro. Alternating projections, 2003. http://www.stanford.edu/class/ee392o/alt proj.pdf [55] Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994. [56] Stephen Boyd, Seung-Jean Kim, Lieven Vandenberghe, and Arash Hassibi. A tutorial on geometric programming. Optimization and Engineering, 8(1):1389–4420, March 2007. http://www.stanford.edu/~boyd/papers/gp tutorial.html [57] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. http://www.stanford.edu/~boyd/cvxbook [58] James P. Boyle and Richard L. Dykstra. A method for finding projections onto the intersection of convex sets in Hilbert spaces. In R. Dykstra, T. Robertson, and F. T. Wright, editors, Advances in Order Restricted Statistical Inference, pages 28–47. Springer-Verlag, 1986. [59] Lev M. Br`egman. The method of successive projection for finding a common point of convex sets. Soviet Mathematics, 162(3):487–490, 1965. AMS translation of Doklady Akademii Nauk SSSR, 6:688-692. [60] Lev M. Br`egman, Yair Censor, Simeon Reich, and Yael Zepkowitz-Malachi. Finding the projection of a point onto the intersection of convex sets via projections onto halfspaces, 2003. http://www.optimization-online.org/DB FILE/2003/06/669.pdf [61] Mike Brookes. Matrix reference manual: Matrix calculus, 2002. http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html [62] Richard A. Brualdi. Combinatorial Matrix Classes. Cambridge University Press, 2006.

760

BIBLIOGRAPHY

[63] Jian-Feng Cai, Emmanuel J. Cand`es, and Zuowei Shen. thresholding algorithm for matrix completion, October 2008. http://arxiv.org/abs/0810.3286

A singular value

[64] Emmanuel J. Cand`es. Compressive sampling. In Proceedings of the International Congress of Mathematicians, Madrid, volume III, pages 1433–1452. European Mathematical Society, August 2006. http://www.acm.caltech.edu/~emmanuel/papers/CompressiveSampling.pdf [65] Emmanuel J. Cand`es and Justin K. Romberg. ℓ1 -magic : Recovery of sparse signals via convex programming, October 2005. http://www.acm.caltech.edu/l1magic/downloads/l1magic.pdf [66] Emmanuel J. Cand`es, Justin K. Romberg, and Terence Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2):489–509, February 2006. http://arxiv.org/abs/math.NA/0409186 (2004 DRAFT) http://www.acm.caltech.edu/~emmanuel/papers/ExactRecovery.pdf [67] Emmanuel J. Cand`es, Michael B. Wakin, and Stephen P. Boyd. Enhancing sparsity by reweighted ℓ1 minimization. Journal of Fourier Analysis and Applications, 14(5-6):877–905, October 2008. http://arxiv.org/abs/0711.1612v1 [68] Michael Carter, Holly Jin, Michael Saunders, and Yinyu Ye. spaseloc: An adaptive subproblem algorithm for scalable wireless sensor network localization. SIAM Journal on Optimization, 17(4):1102–1128, December 2006. http://www.convexoptimization.com/TOOLS/JinSIAM.pdf Erratum: p.300 herein. [69] Lawrence Cayton and Sanjoy Dasgupta. Robust Euclidean embedding. In Proceedings of the 23 rd International Conference on Machine Learning (ICML), Pittsburgh Pennsylvania USA, 2006. icml2006.org/icml documents/camera-ready/022 Robust Euclidean Emb.pdf [70] Yves Chabrillac and Jean-Pierre Crouzeix. Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra and its Applications, 63:283–292, 1984. [71] Manmohan K. Chandraker, Sameer Agarwal, Fredrik Kahl, David Nist´er, and David J. Kriegman. Autocalibration via rank-constrained estimation of the absolute quadric. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, June 2007. http://vision.ucsd.edu/~manu/cvpr07 quadric.pdf [72] Rick Chartrand. Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Processing Letters, 14(10):707–710, October 2007. math.lanl.gov/Research/Publications/Docs/chartrand-2007-exact.pdf [73] Chi-Tsong Chen. Linear System Theory and Design. Oxford University Press, 1999.

BIBLIOGRAPHY

761

[74] Ward Cheney and Allen A. Goldstein. Proximity maps for convex sets. Proceedings of the American Mathematical Society, 10:448–450, 1959. Erratum: p.714 herein. [75] Mung Chiang. Geometric Programming for Communication Systems. Now, 2005. [76] St´ephane Chr´etien. An alternating l1 approach to the compressed sensing problem, 2008. http://arxiv.org/PS cache/arxiv/pdf/0809/0809.0660v1.pdf [77] Steven Chu. Autobiography from Les Prix Nobel, 1997. nobelprize.org/physics/laureates/1997/chu-autobio.html [78] Ruel V. Churchill and James Ward Brown. Complex Variables and Applications. McGraw-Hill, fifth edition, 1990. [79] Jon F. Claerbout and Francis Muir. Robust modeling of erratic data. Geophysics, 38:826–844, 1973. [80] J. H. Conway and N. J. A. Sloane. Springer-Verlag, third edition, 1999.

Sphere Packings, Lattices and Groups.

[81] John B. Conway. A Course in Functional Analysis. Springer-Verlag, second edition, 1990. [82] Richard W. Cottle, Jong-Shi Pang, and Richard E. Stone. The Linear Complementarity Problem. Academic Press, 1992. [83] G. M. Crippen and T. F. Havel. Distance Geometry and Molecular Conformation. Wiley, 1988. [84] Frank Critchley. Multidimensional scaling: a critical examination and some new proposals. PhD thesis, University of Oxford, Nuffield College, 1980. [85] Frank Critchley. On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and its Applications, 105:91–107, 1988. [86] Ronald E. Crochiere and Lawrence R. Rabiner. Multirate Digital Signal Processing. Prentice-Hall, 1983. [87] Lawrence B. Crowell. Quantum Fluctuations of Spacetime. World Scientific, 2005. [88] Joachim Dahl, Bernard H. Fleury, and Lieven Vandenberghe. Approximate maximum-likelihood estimation using semidefinite programming. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, volume VI, pages 721–724, April 2003. [89] George B. Dantzig. Linear Programming and Extensions. Princeton University Press, 1998. [90] Alexandre d’Aspremont, Laurent El Ghaoui, Michael I. Jordan, and Gert R. G. Lanckriet. A direct formulation for sparse PCA using semidefinite programming. In Lawrence K. Saul, Yair Weiss, and L´eon Bottou, editors, Advances in Neural Information Processing Systems 17, pages 41–48. MIT Press, Cambridge Massachusetts USA, 2005. http://books.nips.cc/papers/files/nips17/NIPS2004 0645.pdf

762

BIBLIOGRAPHY

[91] Jon Dattorro. Constrained least squares fit of a filter bank to an arbitrary magnitude frequency response, 1991. http://www.stanford.edu/~dattorro/Hearing.htm [92] Joel Dawson, Stephen Boyd, Mar Hershenson, and Thomas Lee. Optimal allocation of local feedback in multistage amplifiers via geometric programming. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48(1):1–11, January 2001. http://www.stanford.edu/~boyd/papers/fdbk alloc.html [93] Etienne de Klerk. Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications. Kluwer Academic Publishers, 2002. [94] Jan de Leeuw. Fitting distances by least squares. UCLA Statistics Series Technical Report No. 130, Interdivisional Program in Statistics, UCLA, Los Angeles California USA, 1993. http://citeseer.ist.psu.edu/deleeuw93fitting.html [95] Jan de Leeuw. Multidimensional scaling. In International Encyclopedia of the Social & Behavioral Sciences. Elsevier, 2001. http://preprints.stat.ucla.edu/274/274.pdf [96] Jan de Leeuw. Unidimensional scaling. In Brian S. Everitt and David C. Howell, editors, Encyclopedia of Statistics in Behavioral Science, volume 4, pages 2095–2097. Wiley, 2005. http://www.convexoptimization.com/TOOLS/deLeeuw2005.pdf [97] Jan de Leeuw and Willem Heiser. Theory of multidimensional scaling. In P. R. Krishnaiah and L. N. Kanal, editors, Handbook of Statistics, volume 2, chapter 13, pages 285–316. North-Holland Publishing, Amsterdam, 1982. [98] Erik D. Demaine, Francisco Gomez-Martin, Henk Meijer, David Rappaport, Perouz Taslakian, Godfried T. Toussaint, Terry Winograd, and David R. Wood. The distance geometry of music, 2007. http://arxiv.org/abs/0705.4085 [99] Frank Deutsch. Best Approximation in Inner Product Spaces. Springer-Verlag, 2001. [100] Frank Deutsch and Hein Hundal. The rate of convergence of Dykstra’s cyclic projections algorithm: The polyhedral case. Numerical Functional Analysis and Optimization, 15:537–565, 1994. [101] Frank Deutsch and Peter H. Maserick. Applications of the Hahn-Banach theorem in approximation theory. SIAM Review, 9(3):516–530, July 1967. [102] Frank Deutsch, John H. McCabe, and George M. Phillips. Some algorithms for computing best approximations from convex cones. SIAM Journal on Numerical Analysis, 12(3):390–403, June 1975. [103] Michel Marie Deza and Monique Laurent. Springer-Verlag, 1997.

Geometry of Cuts and Metrics.

BIBLIOGRAPHY

763

[104] Carolyn Pillers Dobler. A matrix approach to finding a set of generators and finding the polar (dual) of a class of polyhedral cones. SIAM Journal on Matrix Analysis and Applications, 15(3):796–803, July 1994. Erratum: p.202 herein. [105] Elizabeth D. Dolan, Robert Fourer, Jorge J. Mor´e, and Todd S. Munson. Optimization on the NEOS server. SIAM News, 35(6):4,8,9, August 2002. [106] Bruce Randall Donald. 3-D structure in chemistry and molecular biology, 1998. http://www.cs.duke.edu/brd/Teaching/Previous/Bio [107] David L. Donoho. Neighborly polytopes and sparse solution of underdetermined linear equations. Technical Report 2005-04, Stanford University, Department of Statistics, January 2005. www-stat.stanford.edu/~donoho/Reports/2005/NPaSSULE-01-28-05.pdf [108] David L. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306, April 2006. www-stat.stanford.edu/~donoho/Reports/2004/CompressedSensing091604.pdf [109] David L. Donoho. High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete & Computational Geometry, 35(4):617–652, May 2006. [110] David L. Donoho, Michael Elad, and Vladimir Temlyakov. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Transactions on Information Theory, 52(1):6–18, January 2006. www-stat.stanford.edu/~donoho/Reports/2004/StableSparse-Donoho-etal.pdf [111] David L. Donoho and Philip B. Stark. Uncertainty principles and signal recovery. SIAM Journal on Applied Mathematics, 49(3):906–931, June 1989. [112] David L. Donoho and Jared Tanner. Neighborliness of randomly projected simplices in high dimensions. Proceedings of the National Academy of Sciences, 102(27):9452–9457, July 2005. [113] David L. Donoho and Jared Tanner. Sparse nonnegative solution of underdetermined linear equations by linear programming. Proceedings of the National Academy of Sciences, 102(27):9446–9451, July 2005. [114] David L. Donoho and Jared Tanner. Counting faces of randomly projected polytopes when the projection radically lowers dimension. Journal of the American Mathematical Society, 22(1):1–53, January 2009. [115] Miguel Nuno Ferreira Fialho dos Anjos. New Convex Relaxations for the Maximum Cut and VLSI Layout Problems. PhD thesis, University of Waterloo, Ontario Canada, Department of Combinatorics and Optimization, 2001. cheetah.vlsi.uwaterloo.ca/~anjos/MFAnjosPhDThesis.pdf [116] John Duchi, Shai Shalev-Shwartz, Yoram Singer, and Tushar Chandra. Efficient projections onto the ℓ1 -ball for learning in high dimensions. In Proceedings of the 25 th International Conference on Machine Learning (ICML), pages 272–279, Helsinki Finland, July 2008. Association for Computing Machinery (ACM). http://icml2008.cs.helsinki.fi/papers/361.pdf

764

BIBLIOGRAPHY

[117] Richard L. Dykstra. An algorithm for restricted least squares regression. Journal of the American Statistical Association, 78(384):837–842, 1983. [118] Peter J. Eccles. An Introduction to Mathematical Reasoning: numbers, sets and functions. Cambridge University Press, 1997. [119] Carl Eckart and Gale Young. The approximation of one matrix by another of lower rank. Psychometrika, 1(3):211–218, September 1936. www.convexoptimization.com/TOOLS/eckart&young.1936.pdf [120] Alan Edelman, Tom´as A. Arias, and Steven T. Smith. The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 20(2):303–353, 1998. [121] John J. Edgell. Graphics calculator applications on 4-D constructs, 1996. http://archives.math.utk.edu/ICTCM/EP-9/C47/pdf/paper.pdf [122] Ivar Ekeland and Roger T´emam. Convex Analysis and Variational Problems. SIAM, 1999. [123] Julius Farkas. Theorie der einfachen Ungleichungen. Journal f¨ ur die reine und angewandte Mathematik, 124:1–27, 1902. dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D261364 [124] Maryam Fazel. Matrix Rank Minimization with Applications. PhD thesis, Stanford University, Department of Electrical Engineering, March 2002. http://www.cds.caltech.edu/~maryam/thesis-final.pdf [125] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. A rank minimization heuristic with application to minimum order system approximation. In Proceedings of the American Control Conference, volume 6, pages 4734–4739. American Automatic Control Council (AACC), June 2001. http://www.cds.caltech.edu/~maryam/nucnorm.html [126] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices. In Proceedings of the American Control Conference. American Automatic Control Council (AACC), June 2003. http://www.cds.caltech.edu/~maryam/acc03 final.pdf [127] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. Rank minimization and applications in system theory. In Proceedings of the American Control Conference. American Automatic Control Council (AACC), June 2004. http://www.cds.caltech.edu/~maryam/acc04-tutorial.pdf [128] J. T. Feddema, R. H. Byrne, J. J. Harrington, D. M. Kilman, C. L. Lewis, R. D. Robinett, B. P. Van Leeuwen, and J. G. Young. Advanced mobile networking, sensing, and controls. Sandia Report SAND2005-1661, Sandia National Laboratories, Albuquerque New Mexico USA, March 2005. www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/2005/051661.pdf

BIBLIOGRAPHY

765

[129] C´esar Fern´andez, Thomas Szyperski, Thierry Bruy`ere, Paul Ramage, Egon M¨osinger, and Kurt W¨ uthrich. NMR solution structure of the pathogenesis-related protein P14a. Journal of Molecular Biology, 266:576–593, 1997. [130] Richard Phillips Feynman, Robert B. Leighton, and Matthew L. Sands. The Feynman Lectures on Physics: Commemorative Issue, volume I. Addison-Wesley, 1989. [131] P. A. Fillmore and J. P. Williams. Some convexity theorems for matrices. Glasgow Mathematical Journal, 12:110–117, 1971. ¨ [132] Paul Finsler. Uber das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen. Commentarii Mathematici Helvetici, 9:188–192, 1937. [133] Anders Forsgren, Philip E. Gill, and Margaret H. Wright. Interior methods for nonlinear optimization. SIAM Review, 44(4):525–597, 2002. [134] Shmuel Friedland and Anatoli Torokhti. Generalized rank-constrained matrix approximations. SIAM Journal on Matrix Analysis and Applications, 29(2):656–659, March 2007. http://www2.math.uic.edu/~friedlan/frtor8.11.06.pdf [135] Norbert Gaffke and Rudolf Mathar. A cyclic projection algorithm via duality. Metrika, 36:29–54, 1989. [136] J´erˆ ome Galtier. Semi-definite programming as a simple extension to linear programming: convex optimization with queueing, equity and other telecom functionals. In 3`eme Rencontres Francophones sur les Aspects Algorithmiques des T´el´ecommunications (AlgoTel). INRIA, France, 2001. www-sop.inria.fr/mascotte/Jerome.Galtier/misc/Galtier01b.pdf [137] Laurent El Ghaoui. EE 227A: Convex Optimization and Applications, Lecture 11 − October 3. University of California, Berkeley, Fall 2006. Scribe: Nikhil Shetty. http://www.convexoptimization.com/TOOLS/Ghaoui.pdf [138] Laurent El Ghaoui and Silviu-Iulian Niculescu, editors. Advances in Linear Matrix Inequality Methods in Control. SIAM, 2000. [139] Philip E. Gill, Walter Murray, and Margaret H. Wright. Numerical Linear Algebra and Optimization, volume 1. Addison-Wesley, 1991. [140] Philip E. Gill, Walter Murray, and Margaret H. Wright. Practical Optimization. Academic Press, 1999. [141] James Gleik. Isaac Newton. Pantheon Books, 2003. [142] W. Glunt, Tom L. Hayden, S. Hong, and J. Wells. An alternating projection algorithm for computing the nearest Euclidean distance matrix. SIAM Journal on Matrix Analysis and Applications, 11(4):589–600, 1990. [143] K. Goebel and W. A. Kirk. Topics in Metric Fixed Point Theory. Cambridge University Press, 1990.

766

BIBLIOGRAPHY

[144] Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the Association for Computing Machinery, 42(6):1115–1145, November 1995. http://www-math.mit.edu/~goemans/maxcut-jacm.pdf [145] D. Goldfarb and K. Scheinberg. Interior point trajectories in semidefinite programming. SIAM Journal on Optimization, 8(4):871–886, 1998. [146] Gene H. Golub and Charles F. Van Loan. Matrix Computations. Johns Hopkins, third edition, 1996. [147] P. Gordan. Ueber die aufl¨osung linearer gleichungen mit reellen coefficienten. Mathematische Annalen, 6:23–28, 1873. [148] John Clifford Gower. Euclidean distance geometry. The Mathematical Scientist, 7:1–14, 1982. http://www.convexoptimization.com/TOOLS/Gower2.pdf [149] John Clifford Gower. Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra and its Applications, 67:81–97, 1985. http://www.convexoptimization.com/TOOLS/Gower1.pdf [150] John Clifford Gower and Garmt B. Dijksterhuis. Procrustes Problems. Oxford University Press, 2004. [151] John Clifford Gower and David J. Hand. Biplots. Chapman & Hall, 1996. [152] Alexander Graham. Kronecker Products and Matrix Calculus with Applications. Ellis Horwood Limited, 1981. [153] Michael Grant, Stephen Boyd, and Yinyu Ye. cvx: Matlab software for disciplined convex programming, 2007. http://www.stanford.edu/~boyd/cvx [154] Robert M. Gray. Toeplitz and circulant matrices: A review. Foundations and Trends in Communications and Information Theory, 2(3):155–239, 2006. http://www-ee.stanford.edu/~gray/toeplitz.pdf [155] T. N. E. Greville. Note on the generalized inverse of a matrix product. SIAM Review, 8:518–521, 1966. [156] R´emi Gribonval and Morten Nielsen. Highly sparse representations from dictionaries are unique and independent of the sparseness measure, October 2003. http://www.math.aau.dk/research/reports/R-2003-16.pdf [157] R´emi Gribonval and Morten Nielsen. Sparse representations in unions of bases. IEEE Transactions on Information Theory, 49(12):1320–1325, December 2003. http://www.math.aau.dk/~mnielsen/papers.htm [158] Karolos M. Grigoriadis and Eric B. Beran. Alternating projection algorithms for linear matrix inequalities problems with rank constraints. In Laurent El Ghaoui and Silviu-Iulian Niculescu, editors, Advances in Linear Matrix Inequality Methods in Control, chapter 13, pages 251–267. SIAM, 2000.

BIBLIOGRAPHY

767

[159] Peter Gritzmann and Victor Klee. On the complexity of some basic problems in computational convexity: II. volume and mixed volumes. Technical Report TR:94-31, DIMACS, Rutgers University, 1994. ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1994/94-31.ps [160] Peter Gritzmann and Victor Klee. On the complexity of some basic problems in computational convexity: II. volume and mixed volumes. In T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivi´c Weiss, editors, Polytopes: Abstract, Convex and Computational, pages 373–466. Kluwer Academic Publishers, 1994. [161] L. G. Gubin, B. T. Polyak, and E. V. Raik. The method of projections for finding the common point of convex sets. U.S.S.R. Computational Mathematics and Mathematical Physics, 7(6):1–24, 1967. [162] Osman G¨ uler and Yinyu Ye. Convergence behavior of interior-point algorithms. Mathematical Programming, 60(2):215–228, 1993. [163] P. R. Halmos. Positive approximants of operators. Indiana University Mathematics Journal, 21:951–960, 1972. [164] Shih-Ping Han. 40:1–14, 1988.

A successive projection method.

Mathematical Programming,

[165] Godfrey H. Hardy, John E. Littlewood, and George P´olya. Inequalities. Cambridge University Press, second edition, 1952. [166] Arash Hassibi and Mar Hershenson. Automated optimal design of switched-capacitor filters. In Proceedings of the Conference on Design, Automation, and Test in Europe, page 1111, March 2002. [167] Johan H˚ astad. Some optimal inapproximability results. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing (STOC), pages 1–10, El Paso Texas USA, 1997. Association for Computing Machinery (ACM). http://citeseer.ist.psu.edu/280448.html , 1999. [168] Johan H˚ astad. Some optimal inapproximability results. Journal of the Association for Computing Machinery, 48(4):798–859, July 2001. [169] Timothy F. Havel and Kurt W¨ uthrich. An evaluation of the combined use of nuclear magnetic resonance and distance geometry for the determination of protein conformations in solution. Journal of Molecular Biology, 182:281–294, 1985. [170] Tom L. Hayden and Jim Wells. Approximation by matrices positive semidefinite on a subspace. Linear Algebra and its Applications, 109:115–130, 1988. [171] Tom L. Hayden, Jim Wells, Wei-Min Liu, and Pablo Tarazaga. The cone of distance matrices. Linear Algebra and its Applications, 144:153–169, 1991. [172] Michiel Hazewinkel, editor. Encyclopaedia of Mathematics. Springer-Verlag, 2002. [173] Uwe Helmke and John B. Moore. Springer-Verlag, 1994.

Optimization and Dynamical Systems.

768

BIBLIOGRAPHY

[174] Bruce Hendrickson. Conditions for unique graph realizations. SIAM Journal on Computing, 21(1):65–84, February 1992. [175] T. Herrmann, Peter G¨ untert, and Kurt W¨ uthrich. Protein NMR structure determination with automated NOE assignment using the new software CANDID and the torsion angle dynamics algorithm DYANA. Journal of Molecular Biology, 319(1):209–227, May 2002. [176] Mar Hershenson. Design of pipeline analog-to-digital converters via geometric programming. In Proceedings of the IEEE/ACM International Conference on Computer Aided Design (ICCAD), pages 317–324, November 2002. [177] Mar Hershenson. Efficient description of the design space of analog circuits. In Proceedings of the 40 th ACM/IEEE Design Automation Conference, pages 970–973, June 2003. [178] Mar Hershenson, Stephen Boyd, and Thomas Lee. Optimal design of a CMOS OpAmp via geometric programming. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 20(1):1–21, January 2001. http://www.stanford.edu/~boyd/papers/opamp.html [179] Mar Hershenson, Dave Colleran, Arash Hassibi, and Navraj Nandra. Synthesizable full custom mixed-signal IP. Electronics Design Automation Consortium (EDA), 2002. http://www.eda.org/edps/edp02/PAPERS/edp02-s6 2.pdf [180] Mar Hershenson, Sunderarajan S. Mohan, Stephen Boyd, and Thomas Lee. Optimization of inductor circuits via geometric programming. In Proceedings of the 36 th ACM/IEEE Design Automation Conference, pages 994–998, 1999. http://www.stanford.edu/~boyd/papers/inductor opt.html [181] Nick Higham. Matrix Procrustes problems, 1995. http://www.ma.man.ac.uk/~higham/talks Lecture notes. [182] Richard D. Hill and Steven R. Waters. On the cone of positive semidefinite matrices. Linear Algebra and its Applications, 90:81–88, 1987. [183] Jean-Baptiste Hiriart-Urruty. Ensembles de Tchebychev vs. ensembles convexes: l’´etat de la situation vu via l’analyse convexe non lisse. Annales des Sciences Math´ematiques du Qu´ebec, 22(1):47–62, 1998. [184] Jean-Baptiste Hiriart-Urruty and Claude Lemar´echal. Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods. Springer-Verlag, second edition, 1996. [185] Jean-Baptiste Hiriart-Urruty and Claude Lemar´echal. Fundamentals of Convex Analysis. Springer-Verlag, 2001. [186] Alan J. Hoffman and Helmut W. Wielandt. The variation of the spectrum of a normal matrix. Duke Mathematical Journal, 20:37–40, 1953.

BIBLIOGRAPHY

769

[187] Alfred Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathematics, 76(3):620–630, July 1954. http://www.convexoptimization.com/TOOLS/AHorn.pdf [188] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 1987. [189] Roger A. Horn and Charles R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1994. [190] Alston S. Householder. The Theory of Matrices in Numerical Analysis. Dover, 1975. [191] Hong-Xuan Huang, Zhi-An Liang, and Panos M. Pardalos. Some properties for the Euclidean distance matrix and positive semidefinit matrix completion problems. Journal of Global Optimization, 25(1):3–21, January 2003. [192] Xiaoming Huo. Sparse Image Representation via Combined Transforms. PhD thesis, Stanford University, Department of Statistics, August 1999. www.convexoptimization.com/TOOLS/ReweightingFrom1999XiaomingHuo.pdf [193] Robert J. Marks II, editor. Advanced Topics in Shannon Sampling and Interpolation Theory. Springer-Verlag, 1993. [194] 5W Infographic. Wireless 911. Technology Review, 107(5):78–79, June 2004. http://www.technologyreview.com [195] George Isac. Complementarity Problems. Springer-Verlag, 1992. [196] Nathan Jacobson. Lectures in Abstract Algebra, vol. II - Linear Algebra. Van Nostrand, 1953. [197] Viren Jain and Lawrence K. Saul. Exploratory analysis and visualization of speech and music by locally linear embedding. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 3, pages 984–987, 2004. http://www.cs.ucsd.edu/~saul/papers/lle icassp04.pdf [198] Joakim Jald´en. Bi-criterion ℓ1 /ℓ2 -norm optimization. Master’s thesis, Royal Institute of Technology (KTH), Department of Signals Sensors and Systems, Stockholm Sweden, September 2002. www.ee.kth.se/php/modules/publications/reports/2002/IR-SB-EX-0221.pdf [199] Joakim Jald´en, Cristoff Martin, and Bj¨orn Ottersten. Semidefinite programming for detection in linear systems − Optimality conditions and space-time decoding. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, volume VI, April 2003. http://www.s3.kth.se/signal/reports/03/IR-S3-SB-0309.pdf [200] Florian Jarre. Convex analysis on symmetric matrices. In Henry Wolkowicz, Romesh Saigal, and Lieven Vandenberghe, editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, chapter 2. Kluwer, 2000. http://www.convexoptimization.com/TOOLS/Handbook.pdf

770

BIBLIOGRAPHY

[201] Holly Hui Jin. Scalable Sensor Localization Algorithms for Wireless Sensor Networks. PhD thesis, University of Toronto, Graduate Department of Mechanical and Industrial Engineering, 2005. www.stanford.edu/group/SOL/dissertations/holly-thesis.pdf [202] Charles R. Johnson and Pablo Tarazaga. Connections between the real positive semidefinite and distance matrix completion problems. Linear Algebra and its Applications, 223/224:375–391, 1995. [203] Charles R. Johnson and Pablo Tarazaga. Binary representation of normalized symmetric and correlation matrices. Linear and Multilinear Algebra, 52(5):359–366, 2004. [204] George B. Thomas, Jr. Calculus and Analytic Geometry. Addison-Wesley, fourth edition, 1972. [205] Mark Kahrs and Karlheinz Brandenburg, editors. Applications of Digital Signal Processing to Audio and Acoustics. Kluwer Academic Publishers, 1998. [206] Thomas Kailath. Linear Systems. Prentice-Hall, 1980. [207] Tosio Kato. Perturbation Theory for Linear Operators. Springer-Verlag, 1966. [208] Paul J. Kelly and Norman E. Ladd. Geometry. Scott, Foresman and Company, 1965. [209] Yoonsoo Kim and Mehran Mesbahi. On the rank minimization problem. In Proceedings of the American Control Conference, volume 3, pages 2015–2020. American Automatic Control Council (AACC), June 2004. [210] Ron Kimmel. Numerical Geometry of Images: Applications. Springer-Verlag, 2003.

Theory, Algorithms, and

[211] Erwin Kreyszig. Introductory Functional Analysis with Applications. Wiley, 1989. [212] Harold W. Kuhn. Nonlinear programming: a historical view. In Richard W. Cottle and Carlton E. Lemke, editors, Nonlinear Programming, pages 1–26. American Mathematical Society, 1976. [213] Takahito Kuno, Yasutoshi Yajima, and Hiroshi Konno. An outer approximation method for minimizing the product of several convex functions on a convex set. Journal of Global Optimization, 3(3):325–335, September 1993. [214] Amy N. Langville and Carl D. Meyer. Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press, 2006. [215] Jean B. Lasserre. A new Farkas lemma for positive semidefinite matrices. IEEE Transactions on Automatic Control, 40(6):1131–1133, June 1995. [216] Jean B. Lasserre and Eduardo S. Zeron. A Laplace transform algorithm for the volume of a convex polytope. Journal of the Association for Computing Machinery, 48(6):1126–1140, November 2001. http://arxiv.org/abs/math/0106168 (title revision).

BIBLIOGRAPHY

771

[217] Monique Laurent. A connection between positive semidefinite and Euclidean distance matrix completion problems. Linear Algebra and its Applications, 273:9–22, 1998. [218] Monique Laurent. A tour d’horizon on positive semidefinite and Euclidean distance matrix completion problems. In Panos M. Pardalos and Henry Wolkowicz, editors, Topics in Semidefinite and Interior-Point Methods, pages 51–76. American Mathematical Society, 1998. [219] Monique Laurent. Matrix completion problems. In Christodoulos A. Floudas and Panos M. Pardalos, editors, Encyclopedia of Optimization, volume III (Interior-M), pages 221–229. Kluwer, 2001. http://homepages.cwi.nl/~monique/files/opt.ps [220] Monique Laurent and Svatopluk Poljak. On a positive semidefinite relaxation of the cut polytope. Linear Algebra and its Applications, 223/224:439–461, 1995. [221] Monique Laurent and Svatopluk Poljak. On the facial structure of the set of correlation matrices. SIAM Journal on Matrix Analysis and Applications, 17(3):530–547, July 1996. [222] Monique Laurent and Franz Rendl. Semidefinite programming and integer programming. Optimization Online, 2002. http://www.optimization-online.org/DB HTML/2002/12/585.html [223] Monique Laurent and Franz Rendl. Semidefinite programming and integer programming. In K. Aardal, George L. Nemhauser, and R. Weismantel, editors, Discrete Optimization, volume 12 of Handbooks in Operations Research and Management Science, chapter 8, pages 393–514. Elsevier, 2005. [224] Charles L. Lawson and Richard J. Hanson. Solving Least Squares Problems. SIAM, 1995. [225] Jung Rye Lee. The law of cosines in a tetrahedron. Journal of the Korea Society of Mathematical Education Series B: The Pure and Applied Mathematics, 4(1):1–6, 1997. [226] Claude Lemar´echal. Note on an extension of “Davidon” methods to nondifferentiable functions. Mathematical Programming, 7(1):384–387, December 1974. http://www.convexoptimization.com/TOOLS/Lemarechal.pdf [227] Vladimir L. Levin. Quasi-convex functions and quasi-monotone operators. Journal of Convex Analysis, 2(1/2):167–172, 1995. [228] Scott Nathan Levine. Audio Representations for Data Compression and Compressed Domain Processing. PhD thesis, Stanford University, Department of Electrical Engineering, 1999. http://www-ccrma.stanford.edu/~scottl/thesis/thesis.pdf [229] Adrian S. Lewis. Eigenvalue-constrained faces. Linear Algebra and its Applications, 269:159–181, 1998.

772

BIBLIOGRAPHY

[230] Anhua Lin. Projection algorithms in nonlinear programming. PhD thesis, Johns Hopkins University, 2003. [231] Miguel Sousa Lobo, Lieven Vandenberghe, Stephen Boyd, and Herv´e Lebret. Applications of second-order cone programming. Linear Algebra and its Applications, 284:193–228, November 1998. Special Issue on Linear Algebra in Control, Signals and Image Processing. http://www.stanford.edu/~boyd/socp.html [232] David G. Luenberger. Optimization by Vector Space Methods. Wiley, 1969. [233] David G. Luenberger. Introduction to Dynamic Systems: Theory, Models, & Applications. Wiley, 1979. [234] David G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley, second edition, 1989. [235] Zhi-Quan Luo, Jos F. Sturm, and Shuzhong Zhang. Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming. SIAM Journal on Optimization, 8(1):59–81, 1998. [236] Zhi-Quan Luo and Wei Yu. An introduction to convex optimization for communications and signal processing. IEEE Journal On Selected Areas In Communications, 24(8):1426–1438, August 2006. [237] Morris Marden. Geometry of Polynomials. American Mathematical Society, second edition, 1985. [238] K. V. Mardia. Some properties of classical multi-dimensional scaling. Communications in Statistics: Theory and Methods, A7(13):1233–1241, 1978. [239] K. V. Mardia, J. T. Kent, and J. M. Bibby. Multivariate Analysis. Academic Press, 1979. [240] Jerrold E. Marsden and Michael J. Hoffman. Freeman, second edition, 1995.

Elementary Classical Analysis.

[241] Rudolf Mathar. The best Euclidean fit to a given distance matrix in prescribed dimensions. Linear Algebra and its Applications, 67:1–6, 1985. [242] Rudolf Mathar. Multidimensionale Skalierung. B. G. Teubner Stuttgart, 1997. [243] Nathan S. Mendelsohn and A. Lloyd Dulmage. The convex hull of sub-permutation matrices. Proceedings of the American Mathematical Society, 9(2):253–254, April 1958. http://www.convexoptimization.com/TOOLS/permu.pdf [244] Mehran Mesbahi and G. P. Papavassilopoulos. On the rank minimization problem over a positive semi-definite linear matrix inequality. IEEE Transactions on Automatic Control, 42(2):239–243, February 1997.

BIBLIOGRAPHY

773

[245] Mehran Mesbahi and G. P. Papavassilopoulos. Solving a class of rank minimization problems via semi-definite programs, with applications to the fixed order output feedback synthesis. In Proceedings of the American Control Conference, volume 1, pages 77–80. American Automatic Control Council (AACC), June 1997. [246] Sunderarajan S. Mohan, Mar Hershenson, Stephen Boyd, and Thomas Lee. Simple accurate expressions for planar spiral inductances. IEEE Journal of Solid-State Circuits, 34(10):1419–1424, October 1999. http://www.stanford.edu/~boyd/papers/inductance expressions.html [247] Sunderarajan S. Mohan, Mar Hershenson, Stephen Boyd, and Thomas Lee. Bandwidth extension in CMOS with optimized on-chip inductors. IEEE Journal of Solid-State Circuits, 35(3):346–355, March 2000. http://www.stanford.edu/~boyd/papers/bandwidth ext.html [248] E. H. Moore. On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society, 26:394–395, 1920. Abstract. [249] B. S. Mordukhovich. Maximum principle in the problem of time optimal response with nonsmooth constraints. Journal of Applied Mathematics and Mechanics, 40:960–969, 1976. [250] Jean-Jacques Moreau. D´ecomposition orthogonale d’un espace Hilbertien selon deux cˆ ones mutuellement polaires. Comptes Rendus de l’Acad´emie des Sciences, Paris, 255:238–240, 1962. [251] T. S. Motzkin and I. J. Schoenberg. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:393–404, 1954. [252] Neil Muller, Louren¸co Magaia, and B. M. Herbst. Singular value decomposition, eigenfaces, and 3D reconstructions. SIAM Review, 46(3):518–545, September 2004. [253] Katta G. Murty and Feng-Tien Yu. Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Internet edition, 1988. ioe.engin.umich.edu/people/fac/books/murty/linear complementarity webbook [254] Oleg R. Musin. An extension of Delsarte’s method. The kissing problem in three and four dimensions. In Proceedings of the 2004 COE Workshop on Sphere Packings, Kyushu University Japan, 2005. http://arxiv.org/abs/math.MG/0512649 [255] Stephen G. Nash and Ariela Sofer. Linear and Nonlinear Programming. McGraw-Hill, 1996. [256] Arkadi Nemirovski. Lectures on Modern Convex Optimization. http://www.convexoptimization.com/TOOLS/LectModConvOpt.pdf , 2005. [257] Yurii Nesterov and Arkadii Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. SIAM, 1994. [258] L. Nirenberg. Functional Analysis. New York University, New York, 1961. Lectures given in 1960-1961, notes by Lesley Sibner.

774

BIBLIOGRAPHY

[259] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer-Verlag, 1999. [260] Royal Swedish Academy of Sciences. Nobel prize in chemistry, 2002. http://nobelprize.org/chemistry/laureates/2002/public.html [261] C. S. Ogilvy. Excursions in Geometry. Dover, 1990. Citation: Proceedings of the CUPM Geometry Conference, Mathematical Association of America, No.16 (1967), p.21. [262] Ricardo Oliveira, Jo˜ ao Costeira, and Jo˜ ao Xavier. Optimal point correspondence through the use of rank constraints. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 2, pages 1016–1021, June 2005. ¨ [263] Onder Filiz and Aylin Yener. Rank constrained temporal-spatial filters for CDMA systems with base station antenna arrays. In Proceedings of the Johns Hopkins University Conference on Information Sciences and Systems, March 2003. labs.ee.psu.edu/labs/wcan/Publications/filiz-yener ciss03.pdf [264] Alan V. Oppenheim and Ronald W. Schafer. Discrete-Time Signal Processing. Prentice-Hall, 1989. [265] Robert Orsi, Uwe Helmke, and John B. Moore. A Newton-like method for solving rank constrained linear matrix inequalities. Automatica, 42(11):1875–1882, 2006. users.rsise.anu.edu.au/~robert/publications/OHM06 extended version.pdf [266] Brad Osgood. Notes on the Ahlfors mapping of a multiply connected domain, 2000. www-ee.stanford.edu/~osgood/Ahlfors-Bergman-Szego.pdf [267] M. L. Overton and R. S. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 13:41–45, 1992. [268] Pythagoras Papadimitriou. Parallel Solution of SVD-Related Problems, With Applications. PhD thesis, Department of Mathematics, University of Manchester, October 1993. [269] Panos M. Pardalos and Henry Wolkowicz, editors. Topics in Semidefinite and Interior-Point Methods. American Mathematical Society, 1998. [270] G´abor Pataki. Cone-LP’s and semidefinite programs: Geometry and a simplex-type method. In William H. Cunningham, S. Thomas McCormick, and Maurice Queyranne, editors, Integer Programming and Combinatorial Optimization, Proceedings of the 5th International IPCO Conference, Vancouver, British Columbia, Canada, June 3-5, 1996, volume 1084 of Lecture Notes in Computer Science, pages 162–174. Springer-Verlag, 1996. [271] G´abor Pataki. On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Mathematics of Operations Research, 23(2):339–358, 1998. http://citeseer.ist.psu.edu/pataki97rank.html Erratum: p.633 herein.

BIBLIOGRAPHY

775

[272] G´abor Pataki. The geometry of semidefinite programming. In Henry Wolkowicz, Romesh Saigal, and Lieven Vandenberghe, editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, chapter 3. Kluwer, 2000. http://www.convexoptimization.com/TOOLS/Handbook.pdf [273] Teemu Pennanen and Jonathan Eckstein. Generalized Jacobians of vector-valued convex functions. Technical Report RRR 6-97, RUTCOR, Rutgers University, May 1997. http://rutcor.rutgers.edu/pub/rrr/reports97/06.ps [274] Roger Penrose. A generalized inverse for matrices. In Proceedings of the Cambridge Philosophical Society, volume 51, pages 406–413, 1955. [275] Chris Perkins. A convergence analysis of Dykstra’s algorithm for polyhedral sets. SIAM Journal on Numerical Analysis, 40(2):792–804, 2002. [276] Florian Pfender and G¨ unter M. Ziegler. Kissing numbers, sphere packings, and some unexpected proofs. Notices of the American Mathematical Society, 51(8):873–883, September 2004. http://www.convexoptimization.com/TOOLS/Pfender.pdf [277] Benjamin Recht. Convex Modeling with Priors. PhD thesis, Massachusetts Institute of Technology, Media Arts and Sciences Department, 2006. www.media.mit.edu/physics/publications/theses/06.06.Recht.pdf [278] Benjamin Recht, Maryam Fazel, and Pablo A. Parrilo. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. Optimization Online, June 2007. http://arxiv.org/abs/0706.4138 [279] Alex Reznik. Problem 3.41, from Sergio Verd´ u. Multiuser Detection. Cambridge University Press, 1998. www.ee.princeton.edu/~verdu/mud/solutions/3/3.41.areznik.pdf , 2001. [280] Arthur Wayne Roberts and Dale E. Varberg. Convex Functions. Academic Press, 1973. [281] Sara Robinson. Hooked on meshing, researcher creates award-winning triangulation program. SIAM News, 36(9), November 2003. http://www.siam.org/news/news.php?id=370 [282] R. Tyrrell Rockafellar. Conjugate Duality and Optimization. SIAM, 1974. [283] R. Tyrrell Rockafellar. Lagrange multipliers in optimization. Mathematical Society Proceedings, 9:145–168, 1976. [284] R. Tyrrell Rockafellar. Lagrange multipliers and optimality. 35(2):183–238, June 1993.

SIAM-American SIAM Review,

[285] R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1997. First published in 1970.

776

BIBLIOGRAPHY

[286] C. K. Rushforth. Signal restoration, functional analysis, and Fredholm integral equations of the first kind. In Henry Stark, editor, Image Recovery: Theory and Application, chapter 1, pages 1–27. Academic Press, 1987. [287] Peter Santos. Application platforms and synthesizable analog IP. In Proceedings of the Embedded Systems Conference, San Francisco California USA, 2003. http://www.convexoptimization.com/TOOLS/Santos.pdf [288] Shankar Sastry. Nonlinear Systems: Springer-Verlag, 1999.

Analysis,

Stability,

and Control.

[289] Uwe Sch¨afer. A linear complementarity problem with a P-matrix. SIAM Review, 46(2):189–201, June 2004. [290] Isaac J. Schoenberg. Remarks to Maurice Fr´echet’s article “Sur la d´efinition axiomatique d’une classe d’espace distanci´es vectoriellement applicable sur l’espace de Hilbert”. Annals of Mathematics, 36(3):724–732, July 1935. http://www.convexoptimization.com/TOOLS/Schoenberg2.pdf [291] Isaac J. Schoenberg. Metric spaces and positive definite functions. Transactions of the American Mathematical Society, 44:522–536, 1938. http://www.convexoptimization.com/TOOLS/Schoenberg3.pdf [292] Peter H. Sch¨onemann. A generalized solution of the orthogonal Procrustes problem. Psychometrika, 31(1):1–10, 1966. [293] Peter H. Sch¨onemann, Tim Dorcey, and K. Kienapple. Subadditive concatenation in dissimilarity judgements. Perception and Psychophysics, 38:1–17, 1985. [294] Alexander Schrijver. On the history of combinatorial optimization (till 1960). In K. Aardal, G. L. Nemhauser, and R. Weismantel, editors, Handbook of Discrete Optimization, pages 1– 68. Elsevier, 2005. http://citeseer.ist.psu.edu/466877.html [295] Seymour Sherman. Doubly stochastic matrices and complex vector spaces. American Journal of Mathematics, 77(2):245–246, April 1955. http://www.convexoptimization.com/TOOLS/Sherman.pdf [296] Joshua A. Singer. Log-Penalized Linear Regression. PhD thesis, Stanford University, Department of Electrical Engineering, June 2004. http://www.convexoptimization.com/TOOLS/Josh.pdf [297] Anthony Man-Cho So and Yinyu Ye. A semidefinite programming approach to tensegrity theory and realizability of graphs. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 766–775, Miami Florida USA, January 2006. Association for Computing Machinery (ACM). http://www.convexoptimization.com/TOOLS/YeSo.pdf [298] Anthony Man-Cho So and Yinyu Ye. Theory of semidefinite programming for sensor network localization. Mathematical Programming: Series A and B, 109(2):367–384, January 2007. http://www.stanford.edu/~yyye/local-theory.pdf

BIBLIOGRAPHY

777

[299] J. M. Soriano. Global minimum point of a convex function. Applied Mathematics and Computation, 55(2-3):213–218, May 1993. http://www.convexoptimization.com/TOOLS/Soriano.pdf [300] Steve Spain. Polaris Wireless corp., 2005. http://www.polariswireless.com Personal communication. [301] Nathan Srebro, Jason D. M. Rennie, and Tommi S. Jaakkola. Maximum-margin matrix factorization. In Lawrence K. Saul, Yair Weiss, and L´eon Bottou, editors, Advances in Neural Information Processing Systems 17 (NIPS 2004), pages 1329–1336. MIT Press, 2005. [302] Wolfram Stadler, editor. Multicriteria Optimization in Engineering and in the Sciences. Springer-Verlag, 1988. [303] Henry Stark, editor. Image Recovery: Theory and Application. Academic Press, 1987. [304] Henry Stark. Polar, spiral, and generalized sampling and interpolation. In Robert J. Marks II, editor, Advanced Topics in Shannon Sampling and Interpolation Theory, chapter 6, pages 185–218. Springer-Verlag, 1993. [305] Willi-Hans Steeb. Matrix Calculus and Kronecker Product with Applications and C++ Programs. World Scientific Publishing Co., 1997. [306] Gilbert W. Stewart and Ji-guang Sun. Matrix Perturbation Theory. Academic Press, 1990. [307] Josef Stoer. On the characterization of least upper bound norms in matrix space. Numerische Mathematik, 6(1):302–314, December 1964. http://www.convexoptimization.com/TOOLS/Stoer.pdf [308] Josef Stoer and Christoph Witzgall. Dimensions I. Springer-Verlag, 1970.

Convexity and Optimization in Finite

[309] Gilbert Strang. Linear Algebra and its Applications. Harcourt Brace, third edition, 1988. [310] Gilbert Strang. Calculus. Wellesley-Cambridge Press, 1992. [311] Gilbert Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press, second edition, 1998. [312] Gilbert Strang. Course 18.06: Linear algebra, 2004. http://web.mit.edu/18.06/www ¨ Uber exponierte Punkte abgeschlossener Punktmengen. [313] Stefan Straszewicz. Fundamenta Mathematicae, 24:139–143, 1935. http://www.convexoptimization.com/TOOLS/Straszewicz.pdf [314] Jos F. Sturm and Shuzhong Zhang. On cones of nonnegative quadratic functions. Optimization Online, April 2001. www.optimization-online.org/DB HTML/2001/05/324.html

778

BIBLIOGRAPHY

[315] George P. H. Styan. A review and some extensions of Takemura’s generalizations of Cochran’s theorem. Technical Report 56, Stanford University, Department of Statistics, September 1982. [316] George P. H. Styan and Akimichi Takemura. Rank additivity and matrix polynomials. Technical Report 57, Stanford University, Department of Statistics, September 1982. [317] Jun Sun, Stephen Boyd, Lin Xiao, and Persi Diaconis. The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem. SIAM Review, 48(4):681–699, December 2006. http://stanford.edu/~boyd/papers/fmmp.html [318] Chen Han Sung and Bit-Shun Tam. A study of projectionally exposed cones. Linear Algebra and its Applications, 139:225–252, 1990. [319] Yoshio Takane. On the relations among four methods of multidimensional scaling. Behaviormetrika, 4:29–43, 1977. http://takane.brinkster.net/Yoshio/p008.pdf [320] Akimichi Takemura. On generalizations of Cochran’s theorem and projection matrices. Technical Report 44, Stanford University, Department of Statistics, August 1980. [321] Dharmpal Takhar, Jason N. Laska, Michael B. Wakin, Marco F. Duarte, Dror Baron, Shriram Sarvotham, Kevin F. Kelly, and Richard G. Baraniuk. A new compressive imaging camera architecture using optical-domain compression. In Proceedings of the SPIE Conference on Computational Imaging IV, volume 6065, February 2006. http://www.convexoptimization.com/TOOLS/csCamera-SPIE-dec05.pdf [322] Peng Hui Tan and Lars K. Rasmussen. The application of semidefinite programming for detection in CDMA. IEEE Journal on Selected Areas in Communications, 19(8), August 2001. [323] Pablo Tarazaga. Faces of the cone of Euclidean distance matrices: Characterizations, structure and induced geometry. Linear Algebra and its Applications, 408:1–13, 2005. [324] Warren S. Torgerson. Theory and Methods of Scaling. Wiley, 1958. [325] Lloyd N. Trefethen and David Bau, III. Numerical Linear Algebra. SIAM, 1997. [326] Michael W. Trosset. Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29:148–152, 1998. [327] Michael W. Trosset. Distance matrix completion by numerical optimization. Computational Optimization and Applications, 17(1):11–22, October 2000. [328] Michael W. Trosset. Extensions of classical multidimensional scaling: Computational theory. www.math.wm.edu/~trosset/r.mds.html , 2001. Revision of technical report entitled “Computing distances between convex sets and subsets of the positive semidefinite matrices” first published in 1997.

BIBLIOGRAPHY

779

[329] Michael W. Trosset and Rudolf Mathar. On the existence of nonglobal minimizers of the STRESS criterion for metric multidimensional scaling. In Proceedings of the Statistical Computing Section, pages 158–162. American Statistical Association, 1997. [330] Joshua Trzasko and Armando Manduca. Highly undersampled magnetic resonance image reconstruction via homotopic ℓ0 -minimization. IEEE Transactions on Medical Imaging, 28(1):106–121, January 2009. www.convexoptimization.com/TOOLS/SparseRecon.pdf [331] Joshua Trzasko, Armando Manduca, and Eric Borisch. Sparse MRI reconstruction via multiscale L0 -continuation. In Proceedings of the IEEE 14 th Workshop on Statistical Signal Processing, pages 176–180, August 2007. http://www.convexoptimization.com/TOOLS/L0MRI.pdf [332] P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice-Hall, 1993. [333] Jan van Tiel. Convex Analysis, an Introductory Text. Wiley, 1984. [334] Lieven Vandenberghe and Stephen Boyd. Semidefinite programming. SIAM Review, 38(1):49–95, March 1996. [335] Lieven Vandenberghe and Stephen Boyd. Connections between semi-infinite and semidefinite programming. In R. Reemtsen and J.-J. R¨ uckmann, editors, Semi-Infinite Programming, chapter 8, pages 277–294. Kluwer Academic Publishers, 1998. [336] Lieven Vandenberghe and Stephen Boyd. Applications of semidefinite programming. Applied Numerical Mathematics, 29(3):283–299, March 1999. [337] Lieven Vandenberghe, Stephen Boyd, and Shao-Po Wu. Determinant maximization with linear matrix inequality constraints. SIAM Journal on Matrix Analysis and Applications, 19(2):499–533, April 1998. [338] Robert J. Vanderbei. Convex optimization: Interior-point methods and applications, 1999. http://orfe.princeton.edu/~rvdb/pdf/talks/pumath/talk.pdf [339] Richard S. Varga. Gerˇsgorin and His Circles. Springer-Verlag, 2004. [340] Martin Vetterli and Jelena Kovaˇcevi´c. Wavelets and Subband Coding. Prentice-Hall, 1995. [341] Martin Vetterli, Pina Marziliano, and Thierry Blu. Sampling signals with finite rate of innovation. IEEE Transactions on Signal Processing, 50(6):1417–1428, June 2002. http://bigwww.epfl.ch/publications/vetterli0201.pdf http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.5630 ` B. Vinberg. The theory of convex homogeneous cones. Transactions of the [342] E. Moscow Mathematical Society, 12:340–403, 1963. American Mathematical Society and London Mathematical Society joint translation, 1965.

780

BIBLIOGRAPHY

[343] Marie A. Vitulli. A brief history of linear algebra and matrix theory, 2004. darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html [344] John von Neumann. Functional Operators, Volume II: The Geometry of Orthogonal Spaces. Princeton University Press, 1950. Reprinted from mimeographed lecture notes first distributed in 1933. [345] Michael B. Wakin, Jason N. Laska, Marco F. Duarte, Dror Baron, Shriram Sarvotham, Dharmpal Takhar, Kevin F. Kelly, and Richard G. Baraniuk. An architecture for compressive imaging. In Proceedings of the IEEE International Conference on Image Processing (ICIP), pages 1273–1276, October 2006. http://www.convexoptimization.com/TOOLS/CSCam-ICIP06.pdf [346] Roger Webster. Convexity. Oxford University Press, 1994. [347] Kilian Q. Weinberger and Lawrence K. Saul. Unsupervised learning of image manifolds by semidefinite programming. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 2, pages 988–995, 2004. http://www.cs.ucsd.edu/~saul/papers/sde cvpr04.pdf [348] Eric W. Weisstein. Mathworld – A Wolfram Web Resource, 2007. http://mathworld.wolfram.com/search [349] D. J. White. An analogue derivation of the dual of the general Fermat problem. Management Science, 23(1):92–94, September 1976. http://www.convexoptimization.com/TOOLS/White.pdf [350] Bernard Widrow and Samuel D. Stearns. Adaptive Signal Processing. Prentice-Hall, 1985. [351] Norbert Wiener. On factorization of matrices. Commentarii Mathematici Helvetici, 29:97–111, 1955. [352] Ami Wiesel, Yonina C. Eldar, and Shlomo Shamai (Shitz). Semidefinite relaxation for detection of 16-QAM signaling in MIMO channels. IEEE Signal Processing Letters, 12(9):653–656, September 2005. [353] Michael P. Williamson, Timothy F. Havel, and Kurt W¨ uthrich. Solution conformation of proteinase inhibitor IIA from bull seminal plasma by 1 H nuclear magnetic resonance and distance geometry. Journal of Molecular Biology, 182:295–315, 1985. [354] Willie W. Wong. Cayley-Menger determinant and generalized N-dimensional Pythagorean theorem, November 2003. Application of Linear Algebra: Notes on Talk given to Princeton University Math Club. http://www.princeton.edu/~wwong/papers/gp-r.pdf [355] William Wooton, Edwin F. Beckenbach, and Frank J. Fleming. Modern Analytic Geometry. Houghton Mifflin, 1975.

BIBLIOGRAPHY

781

[356] Margaret H. Wright. The interior-point revolution in optimization: History, recent developments, and lasting consequences. Bulletin of the American Mathematical Society, 42(1):39–56, January 2005. [357] Stephen J. Wright. Primal-Dual Interior-Point Methods. SIAM, 1997. [358] Shao-Po Wu. max-det Programming with Applications in Magnitude Filter Design. A dissertation submitted to the department of Electrical Engineering, Stanford University, December 1997. [359] Shao-Po Wu and Stephen Boyd. sdpsol: A parser/solver for semidefinite programming and determinant maximization problems with matrix structure, 1995. http://www.stanford.edu/~boyd/old software/SDPSOL.html [360] Shao-Po Wu and Stephen Boyd. sdpsol: A parser/solver for semidefinite programs with matrix structure. In Laurent El Ghaoui and Silviu-Iulian Niculescu, editors, Advances in Linear Matrix Inequality Methods in Control, chapter 4, pages 79–91. SIAM, 2000. http://www.stanford.edu/~boyd/sdpsol.html [361] Shao-Po Wu, Stephen Boyd, and Lieven Vandenberghe. FIR filter design via spectral factorization and convex optimization. In Biswa Nath Datta, editor, Applied and Computational Control, Signals, and Circuits, volume 1, chapter 5, pages 215–246. Birkh¨auser, 1998. http://www.stanford.edu/~boyd/papers/magdes.html [362] Naoki Yamamoto and Maryam Fazel. Computational approach to quantum encoder design for purity optimization. Physical Review A, 76(1), July 2007. http://arxiv.org/abs/quant-ph/0606106 [363] David D. Yao, Shuzhong Zhang, and Xun Yu Zhou. Stochastic linear-quadratic control via primal-dual semidefinite programming. SIAM Review, 46(1):87–111, March 2004. Erratum: p.226 herein. [364] Yinyu Ye. Semidefinite programming for Euclidean distance geometric optimization. Lecture notes, 2003. http://www.stanford.edu/class/ee392o/EE392o-yinyu-ye.pdf [365] Yinyu Ye. Convergence behavior of central paths for convex homogeneous self-dual cones, 1996. http://www.stanford.edu/~yyye/yyye/ye.ps [366] Yinyu Ye. Interior Point Algorithms: Theory and Analysis. Wiley, 1997. [367] D. C. Youla. Mathematical theory of image restoration by the method of convex projection. In Henry Stark, editor, Image Recovery: Theory and Application, chapter 2, pages 29–77. Academic Press, 1987. [368] Fuzhen Zhang. Matrix Theory: Basic Results and Techniques. Springer-Verlag, 1999.

[369] G¨ unter M. Ziegler. Kissing numbers: Surprises in dimension four. Emissary, pages 4–5, Spring 2004. http://www.msri.org/communications/emissary

782

Index 0-norm, 219, 279, 315, 317, 318, 325, 326, adjacency, 51 328, 348, 359, 754 adjoint Boolean, 279 operator, 49, 50, 156, 514, 739, 745 1-norm, 215–219, 221, 222, 251, 330, 446, self, 387, 569 550, 754 affine, 38, 226 ball, see ball algebra, 40, 60, 78, 92, 686 Boolean, 279 boundary, 93 nonnegative, 219, 220, 318, 325, 326, combination, 66, 71, 77, 141 328 dimension, 23, 38, 59, 423, 425, 426, 2-norm, 49, 211, 212, 215, 216, 332, 663, 754 435, 515 ball, see ball complementary, 515 Boolean, 279 cone, 132 matrix, 255, 530, 543, 600, 610, 617, low, 399 618, 628, 716, 754 minimization, 548, 560 inequality, 69, 754 Pr´ecis, 426 Schur-form, 255, 628 reduction, 472 Ax = b , 48, 85, 217, 218, 283, 291, 315, 317, spectral projection, 561 330, 332, 365, 672, 673 face, 92 Boolean, 279 hull, see hull nonnegative, 86, 87, 219, 318, 325, 326, independence, 70, 76, 77, 133, 137 328 preservation, 77 V , 618 map, see transformation δ , 569 membership relation, 214 ∞-norm, 215, 216, 222, 754 normal cone, 734 ball, see ball set, 38, 59, 61 dual, 754 parallel, 38 λ , 570, 745 subset, 61, 71 π , 221, 316, 537, 541, 563, 745 hyperplane intersection, 85 ψ , 287, 602 nullspace basis span, 85 σ √, 570, 745 projector, 675, 686 EDM , 481 transformation, 47, 48, 78, 173, 244, º , 753 259, 518 k-largest norm, 222 positive semidefinite cone, 112, 113 algebra active, 294 affine, 40, 60, 78, 92, 686 adaptive, 354, 359, 362 additivity, 101, 112, 584 cone, 156

783

784

INDEX

face, 91, 92 function linear, 210 support, 229 fundamental theorem, 595 linear, 84 interior, 40, 173 linear, 84, 569 projection, 686 algebraic complement, 93, 147, 194, 680, 684 multiplicity, 595 Alizadeh, 285 alternation, 463 alternative, see theorem ambient, 507, 520 vector space, 36 anchor, 300, 301, 395, 399 angle, 83 acute, 71 alternating projection, 726 brackets, 49 constraint, 386, 387, 390, 406 dihedral, 29, 83, 407, 469, 625 EDM cone, 119 halfspace, 72 inequality, 378, 467 matrix, 411 interpoint, 383 obtuse, 71 positive semidefinite cone, 119 relative, 408, 410 right, 121, 150, 172, 173 torsion, 29, 406 antidiagonal, 415 antisymmetry, 99, 584 artificial intelligence, 25 asymmetry, 35, 71 axioms of the metric, 375 axis of revolution, 120, 121, 625 ball 1-norm, 217, 218 nonnegative, 326 ∞-norm, 217 Euclidean, 39, 217, 218, 391, 675, 715

smallest, 63 nuclear norm, 66 packing, 391 spectral norm, 69, 716 Barvinok, 35, 109, 130, 133, 136, 270, 371 base, 93 station, 403 basis, 56, 181, 747 discrete cosine transform, 320 nonorthogonal, 181, 687 projection on, 687 orthogonal projection, 687 orthonormal, 56, 88, 684 pursuit, 351 standard matrix, 57, 59, 740 vector, 57, 144, 384, 751 bees, 30, 391 bijection, see bijective bijective, 49, 52, 114, 638 Gram form, 417 inner-product form, 421 invertible, 52, 672 isometry, 52, 463 linear, 51, 52, 56, 58, 135, 175, 541, 623 biorthogonal decomposition, 676, 684 expansion, 147, 160, 177, 181, 677 EDM, 495 projection, 688 unique, 179, 181, 184, 496, 614, 689 biorthogonality, 179, 182 condition, 177, 596, 614 bound lower, 530 polyhedral, 479 boundary, 39, 40, 63, 92 -point, 90 cone, 102, 169, 174, 176 EDM, 477, 493, 499 membership, 161, 169, 274 PSD, 40, 110, 115, 127–130, 274, 504 ray, 103 extreme, 107

INDEX of halfspace, 71 of hypercube slice, 362 of open set, 41 of point, 92 of ray, 93 relative, 39, 40, 63, 92 bounded, 63, 83 bowl, 652, 653 box, 81, 715 bridge, 407, 436, 486, 531, 547

785

certificate, 207, 274, 275, 295, 369 null, 161, 170 chain rule, 648 two argument, 648 Chu, 7 clipping, 222, 526, 715, 716, 744 closed, see set coefficient binomial, 222, 329 projection, 177, 502, 693, 700 cofactor, 470 calculus compaction, 303, 312, 386, 390, 484, 551 matrix, 641 comparable, 101, 178 of inequalities, 7 complement of variations, 165 algebraic, 93, 147, 194, 680, 684 Cand`es, 350 projection on, 680, 711 Carath´eodory’s theorem, 160, 698 orthogonal, 55, 83, 147, 297, 680, 684, cardinality, 380 711, 734, 742, 748 -one, 217, 326 relative, 88, 742 convex envelope, see convex complementarity, 194, 215, 278, 295, 316 geometry, 217, 326 linear, 198 minimization, 217, 218, 317, 348, 359 maximal, 264, 270, 293 Boolean, 279 problem, 197 nonnegative, 219, 315, 318, 325, 326, linear, 198 328 strict, 278 signed, 330 complementary problem, see cardinality minimization affine dimension, 515 quasiconcavity, 315 dimension, 84 regularization, 321, 349 eigenvalues, 680 Cartesian halfspaces, 73 axes, 38, 62, 79, 96, 104, 145, 157 inertia, 450, 590 cone, 37, 147, 158 slackness, 277 coordinates, 458 subspace, 55 product, 46, 271, 727, 747 compressed sensing, 214, 217–219, 318, 350, cone, 98, 154 351, 353, 360, 363 function, 214 nonnegative, 219, 220, 318, 325, 326, Cayley-Menger 328 determinant, 469, 471 problem, 219, 220, 317, 318, 325, 326, form, 427, 448, 449, 537, 562 328 cellular, 404 Procrustes, 338 telephone network, 403 compressive sampling, see compressed center concave, 209, 210, 223 geometric, 385, 389, 413, 456 condition gravity, 413 biorthogonality, 177, 596, 614 mass, 413 central path, 270 convexity

786 first order, 245, 248, 250, 252 second order, 249, 254 optimality dual, 153 first order, 165, 166, 194–197 semidefinite program, 277 Slater’s, 153, 206, 272, 277 unconstrained, 165, 239, 240, 702 cone, 69, 93, 753 algebra, 156 blade, 159 boundary, see boundary Cartesian, 37, 147, 158 circular, 45, 97, 120–123, 521 Lorentz, see cone, Lorentz right, 120 closed, see set, closed convex, 69, 96–98, 105 difference, 46, 155 dimension, 132 dual, 147, 148, 199, 201, 202, 453 algorithm, 199 construction, 149, 150 dual, 154, 167 examples, 156 facet, 177, 203 formula, 166, 167, 185, 192, 193 halfspace-description, 166 in subspace, 156, 161, 182, 186, 187 Lorentz, 158, 173 pointed, 142, 155 properties, 154 unique, 148, 176 vertex-description, 185, 199 EDM, √ 381, 475, 477, 478, 480 EDM , 478, 481, 544 boundary, 477, 493, 499 circular, 122 construction, 492 convexity, 481 decomposition, 516 dual, 507, 508, 518, 520–522 extreme direction, 495 face, 494, 496 interior, 477, 493

INDEX one-dimensional, 107 positive semidefinite, 496, 504, 512 projection, 544, 555, 559 empty set, 96 face, 102, 177, 203 smallest, 98, 117, 131, 169, 170, 220, 268, 329, 494, 503, 509, 519 halfline, 98 halfspace, 142, 151 hull, 155 ice cream, see cone, circular interior, see interior intersection, 98, 155 invariance, see invariance Lorentz circular, 97, 121 dual, 158, 173 polyhedral, 140, 158 majorization, 573 membership, see membership relation monotone, 190–192, 450, 564, 639 nonnegative, 188–190, 452, 453, 459, 461, 462, 538–540, 564, 573 negative, 178, 689 nonconvex, 94–96 normal, 194, 195, 197, 214, 501, 707, 713, 733–736 affine, 734 elliptope, 735 membership relation, 194 orthant, 194 one-dimensional, 93, 97, 133, 142 EDM, 107 PSD, 105 origin, 69, 96, 98, 99, 102, 105, 133 orthant, see orthant orthogonal complement, 157, 711, 734 pointed, 98, 99, 103–105, 142, 160, 179, 192 closed convex, 99, 135 dual, 142, 155 extreme, 99, 105, 142, 151 polyhedral, 104, 142, 175, 176 polar, 93, 148, 501, 711, 749 polyhedral, 69, 133, 139, 140, 142, 151

INDEX dual, 134, 156, 161, 167, 172, 175, 177 equilateral, 172 facet, 177, 203 halfspace-description, 139 majorization, 564 pointed, 104, 142, 175, 176 proper, 134 vertex-description, 69, 142, 161, 176 positive semidefinite, 43, 109–112, 511 boundary, 110, 115, 127, 274, 504 circular, 120–123, 223 dimension, 117 dual, 171 EDM, 512 extreme direction, 105, 111, 119 face, 117, 138, 503, 509, 519 hyperplane, supporting, 117, 118 inscription, 125 interior, 110, 273 inverse image, 112, 113 one-dimensional, 105 optimization, 269 polyhedral, 124 rank, 117, 267 visualization, 266 product, 98, 154 proper, 101, 199 dual, 156 quadratic, 97 ray, 93, 97, 142 recession, 151 second order, see cone, Lorentz self-dual, 161, 171, 172 shrouded, 172, 522 simplicial, 70, 144–146, 178 dual, 156, 178, 186 spectral, 448, 450, 537 dual, 453 orthant, 540 subspace, 97, 140 sum, 98, 155 supporting hyperplane, 98, 99, 177 tangent, 501

787 transformation, see transformation, linear two-dimensional, 133 union, 155 unique, 110, 148, 477 vertex, 99 congruence, 416, 447 conic combination, 71, 141 constraint, 196, 197, 223 coordinates, 147, 181, 184, 204 hull, see hull independence, 27, 70, 132–137, 285 versus dimension, 133 dual cone, 187 extreme direction, 135 preservation, 135 problem, 153, 196, 197, 214, 262 dual, 153, 262 section, 122 circular cone, 123 conjugate complex, 355, 574, 605, 623, 700, 739, 744 convex, 154, 158, 160, 161, 167, 177, 186, 194 function, 548, 744 gradient, 360 conservation dimension, 426, 603, 604 constellation, 22 constraint angle, 386, 387, 390, 406 binary, 279, 341 cardinality, 317, 348, 359 nonnegative, 219, 315, 318, 325, 326, 328 signed, 330 conic, 196, 197, 223 equality, 166 affine, 262 inequality, 197, 293 matrix Gram, 386, 387 orthogonal, see Procrustes, 369

788 orthonormal, 334 nonnegative, 86, 87, 197 norm, 332, 342, 603 spectral, 69 polynomial, 223, 339, 340 rank, see rank Schur-form, 69 sort, 462 tractable, 262 content, 471, 753 contour plot, 195, 247 contraction, 69, 198, 360, 362 convergence, 299, 463, 719 geometric, 721 global, 295, 299, 316, 369 local, 299, 317, 319, 331, 369, 371 convex, 21, 210 combination, 36, 71, 141 envelope, 548, 549 bilinear, 258 cardinality, 283, 550 rank, 310, 402, 548–550 equivalent, 310 form, 7, 261 geometry, 35 fundamental, 73, 78, 83, 373, 385 hull, see hull iteration cardinality, 219, 315, 317–319, 330, 331 convergence, 295, 299, 316, 317, 331, 371 rank, 294, 296, 299, 553 rank one, 334, 367, 565 stall, 317, 319, 320, 333, 342, 367, 371 optimization, 7, 21, 212, 262, 395 art, 8, 284, 302, 567 fundamental, 261 polyhedron, see polyhedron problem, 153, 166, 196, 216, 228, 235, 237, 264, 271, 542, 705 definition, 262 geometry, 21, 27 nonlinear, 261

INDEX statement as solution, 7, 261 tractable, 262 program, see problem quadratic, see function convexity, 209, 223 condition first order, 245, 248, 250, 252 second order, 249, 254 K-, 210 log, 259 norm, 69, 211, 212, 215, 216, 244, 255, 628, 754 strict, see function, convex coordinates conic, 147, 181, 184, 204 Crippen & Havel, 406 criterion EDM, 502, 506 dual, 514 Finsler, 506 metric versus matrix, 428 Schoenberg, 30, 385, 449, 519, 701 cubix, 642, 645, 750 curvature, 249, 257 decomposition, 750 biorthogonal, 676, 684 dyad, 608 eigen, 594 orthonormal, 682, 684 singular value, 597 compact, 597 full, 599 subcompact, 598 deconvolution, 351 deLeeuw, 454, 545, 555 Delsarte method, 392 dense, 91 derivative, 239 directional, 248, 641, 649, 652, 653, 657–659 dimension, 652, 662 gradient, 653 second, 654 gradient, 659 partial, 641, 746

INDEX table, 662 description conversion, 147 halfspace, 70, 73 affine, 85 dual cone, 166 polyhedral cone, 139 vertex, 70, 71, 106 affine, 85 of halfspace, 136, 137 polyhedral cone, 69, 142, 161, 176 polyhedron, 63 determinant, 256, 466, 580, 593, 662, 668 Cayley-Menger, 469, 471 inequality, 585 Kronecker, 572 product, 582 diagonal, 569 δ , 569 binary, 676 dominance, 126 inequality, 585 matrix, 594 commutative, 583 diagonalization, 597 values eigen, 594 singular, 598 diagonalizable, 179, 594 simultaneously, 118, 278, 296, 581, 583, 606 diagonalization, 42, 594 diagonal matrix, 597 expansion by, 179 symmetric, 596 diamond, 415 differential Fr´echet, 650 partial, 746 diffusion, 484 dilation, 459 dimension, 38 affine, see affine complementary, 84 conservation, 426, 603, 604

789 domain, 53 embedding, 60 Euclidean, 746 invariance, 54 positive semidefinite cone face, 117 Pr´ecis, 426 range, 53 rank, 132, 425, 426, 752 direction, 93, 103, 649 matrix, 294, 296, 310, 311, 349, 393, 554 Newton, 239 steepest descent, 239, 652 vector, 317 disc, 64 discretization, 166, 211, 249, 519 distance comparative, see distance, ordinal Euclidean, 380 geometry, 22, 403 matrix, see EDM ordinal, 457–464 origin to affine, 75, 217, 218, 675, 691 nonnegative, 219, 220, 325 property, 375 taxicab, 216–218, 251, 252 doublet, see matrix dual affine dimension, 515 dual, 154, 161, 167, 276, 281, 346 feasible set, 271 norm, 158, 628 problem, 151, 152, 206, 221, 263, 275, 277, 278, 346, 388, 488 projection, 706, 708 on cone, 709 on convex set, 706, 708 on EDM cone, 559 optimization, 707 strong, 152, 153, 277, 638 duality, 153, 388 gap, 152, 154, 272, 277, 282, 346 strong, 153, 206, 226, 277 weak, 275

790 dvec, 59 dyad, see matrix Dykstra algorithm, 719, 732, 733 edge, 89 EDM, √ 28, 373, 374, 381, 521 EDM , 481 closest, 535 composition, 443–446, 478, 479, 481 construction, 490 criterion, 502, 506 dual, 514 definition, 380, 488 Gram form, 383, 499 inner-product form, 408 interpoint angle, 383 relative-angle form, 410 dual, 514, 515 exponential entry, 443 graph, 377, 392, 396, 399, 400, 484 membership relation, 519 projection, 505 range, 491 test, numerical, 378, 385 unique, 442, 455, 456, 536 eigen, 594 decomposition, 594 matrix, 110, 596 spectrum, 448, 537 ordered, 450 unordered, 452 value, 579, 628 λ , 570, 745 coefficients, 695, 698 EDM, 427, 447, 460, 489 interlaced, 433, 447, 587 intertwined, 433 inverse, 596, 597, 602 Kronecker, 572 largest, 435, 629, 631, 632 left, 594 of sum, 586 principal, 348 product, 581, 583, 584, 634 real, 574, 594, 597 Schur, 591

INDEX smallest, 435, 444, 488, 629 sum of, 52, 580, 631–633 symmetric matrix, 597, 602 transpose, 595 unique, 120, 595 zero, 603, 611 vector, 630, 631 distinct, 595 EDM, 489 Kronecker, 572 left, 594 normal, 596 principal, 348, 350 real, 594, 597, 604 symmetric, 596 unique, 595 elbow, 529, 530 ellipsoid, 36, 41, 43, 332 invariance, 48 elliptope, 281, 283, 386, 395, 437, 438, 446, 499–501 smooth, 437 vertex, 282, 438, 439 embedding, 423 dimension, 60 empty interior, 39 set, 38, 39, 92, 627 affine, 38 closed, 40 cone, 96 hull, 61, 64, 69 open, 40 entry, 37, 210, 739, 744–746, 750 smallest/largest, 221 epigraph, 230, 253, 259 convex function, 230 form, 236, 558, 559 intersection, 229, 255, 259 equality constraint, 166 affine, 262 EDM, 512 normal equation, 674 errata, 202, 226, 300, 420, 633, 714

INDEX Euclidean ball, 39, 217, 218, 391, 675, 715 smallest, 63 distance, 374, 380 geometry, 22, 396, 403, 701 matrix, see EDM metric, 375, 430 fifth property, 375, 378, 379, 464, 467, 473 norm, see 2-norm projection, 128 space, 36, 747 exclusive mutually, 164 expansion, 182, 750 biorthogonal, 147, 160, 177, 181, 677 EDM, 495 projection, 688 unique, 179, 181, 184, 496, 614, 689 implied by diagonalization, 179 orthogonal, 57, 147, 180, 683, 685 w.r.t orthant, 181 exponential, 669 exposed, 88 direction, 106 extreme, 90, 99, 138 face, 89, 138 point, 90, 91, 106 density, 91 extreme, 88 boundary, 107 direction, 103, 108, 135 distinct, 103 dual cone, 134, 177, 182, 184, 187, 200–202 EDM cone, 107, 495, 513 one-dimensional, 105, 107 origin, 105 pointed cone, 99, 105, 142, 151 PSD cone, 105, 111, 119 shared, 203 supporting hyperplane, 177 unique, 103 exposed, 90, 99, 138 point, 68, 88, 90, 91, 270

791 cone, 99, 131 Fantope, 296 hypercube slice, 317 ray, 103, 499 face, 42, 89 affine, 92 algebra, 91, 92 exposed, see exposed isomorphic, 118, 388, 494 polyhedron, 138, 139 recognition, 25 smallest, 92, 169, see cone transitivity, 91 facet, 91, 177, 203 Fan, 631 Fantope, 64, 65, 120, 234, 295 extreme point, 296 Farkas’ lemma, 158, 163 positive definite, 273 not, 274 positive semidefinite, 271 fat, 85, 740 full-rank, 85 Fenchel, 209 Fermat point, 388, 389 finitely generated, 139 floor, 754 Forsgren & Gill, 261 Fourier transform, discrete, 53, 354 Frobenius, see norm full, 39 -dimensional, 39, 101 rank, 85 function, see operator affine, 47, 226–228, 241 supremum, 229 bilinear, 258 composition, 223, 244, 648 affine, 244, 259 concave, 209, 223, 258, 551, 630 conjugate, 548, 744 continuity, 210, 250, 259 convex, 209, 382, 411 matrix, 250

792

INDEX nonlinear, 210, 228 simultaneously, 232, 233, 253 strictly, 211, 212, 243, 249, 254, 255, 545, 551, 556, 652, 653 trivial, 212 vector, 210 differentiable, 228, 254 non, 210, 211, 216, 240, 259, 360, 361 distance Euclidean, 380, 381 taxicab, 251, 252 fractional, 232, 244, 249, 253 inverted, 223, 591, 662, 669 power, 224, 669 projector, 233 root, 223, 585, 597, 740 inverted, see function, fractional invertible, 48, 52, 53 linear, see operator monotonic, 209, 221, 243, 258, 259 multidimensional, 209, 238, 254, 750 affine, 226 negative, 223 objective, see objective presorting, 221, 316, 537, 541, 563, 745 product, 232, 244, 258 projection, see projector pseudofractional, 231 quadratic, 212, 236, 246, 255, 339, 409, 590, 699 distance, 380 nonnegative, 590 strictly, 212, 249, 254, 255, 556, 652, 653 quasiconcave, 257, 258, 287 cardinality, 315 rank, 127 quasiconvex, 127, 230, 256–258 continuity, 257, 259 quasilinear, 249, 258, 259, 602 quotient, see function, fractional ratio, see function, fractional smooth, 210, 216 sorting, 221, 316, 537, 541, 563, 745 step, 602

matrix, 287 stress, 249, 532, 553 support, 81, 228 algebra, 229 fundamental convex geometry, 73, 78, 83, 373, 385 optimization, 261 metric property, 375, 430 semidefiniteness test, 171, 574 subspaces, 48, 83, 672, 682 theorem algebra, 595 linear, 84 Gˆateaux differential, 650 Gaffke, 508 generators, 60, 71, 105, 167 c.i., 136 hull, 141 affine, 60, 76, 692 conic, 71, 136, 137 convex, 63 l.i., 136 minimal set, 63, 76, 136, 747 affine, 692 extreme, 106, 176 halfspace, 136, 137 hyperplane, 136 orthant, 211 positive semidefinite cone, 111, 171, 251 geometric center, 385, 389, 413, 456 operator, 419 subspace, 417, 418, 476, 486, 510–512, 548, 606, 703 Hahn-Banach theorem, 73, 78, 83 mean, 669 multiplicity, 595 Gerˇsgorin, 124 gimbal, 624, 625 global positioning system, 24, 301 Gower, 373 gradient, 165, 195, 228, 238, 248, 641, 660 composition, 648 conjugate, 360

INDEX derivative, 659 directional, 653 dimension, 238 first order, 659 image, 357 monotonic, 243 norm, 663, 665, 681, 691, 692, 702, 706 product, 645 second order, 660, 661 sparsity, 352 table, 662 zero, 165, 239, 240, 702 Gram matrix, 308, 383, 395 constraint, 386, 387 Hadamard product, 50, 544, 570, 571, 648, 742 quotient, 662, 669, 742 square root, 740 halfline, 93 halfspace, 71–73, 151 boundary, 71 description, 70, 73 affine, 85 dual cone, 166 polyhedral cone, 139 vertex-, 136, 137 interior, 93 polarity, 71 Hayden & Wells, 475, 489, 555 Hessian, 239, 641 hexagon, 390 homogeneity, 101, 215, 259, 381 honeycomb, 30 Horn & Johnson, 575, 576 horn, flared, 97 hull, 59 affine, 38, 59–63, 78, 141 EDM cone, 419, 507 empty set, 61 point, 39, 61 positive semidefinite cone, 61 rank-1 correlation matrices, 61 unique, 60 conic, 62, 69, 141 empty set, 69

793 convex, 60, 62, 63, 141, 374 cone, 155 dyads, 66 empty set, 64 extreme directions, 105 orthogonal matrices, 69 orthonormal matrices, 69 outer product, 64 permutation matrices, 68 positive semidefinite cone, 119 projection matrices, 64 rank-one matrices, 66, 67 rank-one symmetric matrices, 119 unique, 63 hyperboloid, 608 hyperbox, 81, 715 hypercube, 51, 81, 217, 715 nonnegative, 283 slice nonnegative, 317, 362 hyperdimensional, 646 hyperdisc, 653 hyperplane, 71–76, 241 hypersphere radius, 74 independent, 85 movement, 74, 87, 220 normal, 73 separating, 83, 156, 161, 163 strictly, 83 supporting, 78, 80, 195, 245–248 cone, 117, 118, 148, 149, 169, 177, 507 polarity, 78 strictly, 81, 98, 99, 246 trivially, 78, 170 unique, 245, 246, 248 vertex-description, 76 hypersphere, 64, 74, 386, 391, 439 circumhypersphere, 427 hypograph, 231, 259, 664 idempotent, see matrix iff, 752 image, 47 inverse, 47, 48, 173 cone, 112, 113, 156, 175

794 magnetic resonance, 352 indefinite, 446 independence affine, 70, 76, 77, 133, 137 preservation, 77 conic, 27, 70, 132–137, 285 versus dimension, 133 dual cone, 187 extreme direction, 135 preservation, 135 linear, 37, 70, 133, 136, 327, 613 preservation, 37 inequality angle, 378, 467 matrix, 411 Bessel, 683 Cauchy-Schwarz, 724 constraint, 197, 293 determinant, 585 diagonal, 585 generalized, 101, 147, 158 dual, 27, 160, 171 partial order, 99 Hardy-Littlewood-P´olya, 538 Jensen, 244 linear, 35, 71, 161 matrix, 27, 173–175, 271, 272, 274 log, 664 norm, 280, 754 triangle, 215 rank, 585 singular value, 69 spectral, 69, 448 trace, 585 triangle, 375, 376, 409, 430–432, 434, 442, 465, 466, 468–470, 479, 481 norm, 215 strict, 432 unique, 465 variation, 165 inertia, 447, 590 complementary, 450, 590 Sylvester’s law, 582 infimum, 260, 627, 630, 681, 702, 752 inflection, 249

INDEX injective, 52, 53, 102, 205, 417, 746 invertible, 52, 671, 672 non, 54 nullspace, 52 innovation rate, 351 interior, 39, 40, 92 -point, 39 method, 264, 394, 517 algebra, 40, 173 cone, 174, 176 EDM, 477, 493 membership, 160 positive semidefinite, 110, 273 empty, 39 of halfspace, 93 of point, 39, 40 of ray, 93 relative, 39, 40, 92 transformation, 173 intersection, 46 cone, 98, 155 epigraph, 229, 255, 259 halfspaces, 73, 155 hyperplane, 85 convex set, 79 line with boundary, 41 nullspaces, 88 planes, 86 positive semidefinite cone affine, 130, 274 geometric center subspace, 607 line, 397 subspaces, 88 tangential, 43 invariance closure, 39, 139, 174 cone, 98, 135, 155, 477 convexity, 46 dimension, 54 ellipsoid, 48 Gram form, 413 inner-product form, 416 isometric, 23, 398, 412, 416, 454, 455 orthogonal, 53, 463, 530, 623, 626

INDEX reflection, 414 rotation, 120, 414 scaling, 259, 381 set, 445 translation, 412–414, 419, 440, 488, 703 inversion conic coordinates, 205 Gram form, 419 injective, 52 matrix, see matrix, inverse nullspace, 48, 52 operator, 48, 52, 53, 173 is, 743 isedm(), 378, 385 isometry, 52, 463, 532 isomorphic, 51, 55, 57, 742 face, 118, 388, 494 isometrically, 42, 52, 114, 171 non, 606 range, 52 isomorphism, 51, 174, 420, 422, 463, 476 isometric, 52, 56, 59, 463 symmetric hollow subspace, 59 symmetric matrix subspace, 56 isotonic, 459 iterate, 719, 729, 730, 733 iteration alternating projection, 558, 719, 725, 733 convex, see convex

795 vector, 648 vectorization, 571

Lagrange multiplier, 166 sign, 206 Lagrangian, 206, 214, 346, 358, 388 Lasso, 351 lattice, 304–307, 313, 314 regular, 30, 303 Laurent, 443 law of cosines, 408 least 1-norm, 217, 218, 280, 326 energy, 386, 551 norm, 217, 280, 674 squares, 301, 674 Legendre-Fenchel transform, 548 Lemar´echal, 21, 81 line, 241, 254 tangential, 44 linear algebra, 84, 569 bijection, see bijective complementarity, 198 function, see operator independence, 37, 70, 133, 136, 327, 613 preservation, 37 inequality, 35, 71, 161 matrix, 27, 173–175, 271, 272, 274 operator, 46, 57, 210, 226, 228, 383, 419, 421, 463, 497, 569, 704 injective, 52 Jacobian, 239 noninjective, 54 Karhunen−Lo´eve transform, 454 projector, 679, 684, 686, 707 kissing, 152, 217, 218, 325–327, 391, 675 program, 82, 86, 87, 154, 215, 228, 261, number, 391 263–265, 392 KKT conditions, 197, 535 dual, 154, 263 Krein-Rutman, 156 prototypical, 154, 263 Kronecker product, 114, 570–572, 638, 646, regression, 351 647, 664 transformation, see transformation eigen, 572 list, 28 inverse, 354 localization, 24, 397 positive semidefinite, 380, 584 sensor network, 300 projector, 720 standardized test, 303 pseudoinverse, 673 unique, 23, 301, 302, 396, 399 trace, 572 wireless, 403

796 log, 664, 668 det, 256, 403, 551, 659, 667 machine learning, 25 majorization, 573 cone, 573 symmetric hollow, 573 manifold, 25, 26, 68, 69, 117, 482, 483, 530, 565, 623, 634 map, see transformation isotonic, 457, 460 USA, 28, 457, 458 Markov process, 488 mater, 642 Mathar, 508 Matlab, 33, 132, 279, 312, 319, 324, 333, 341, 348, 393, 458, 461, 600 MRI, 350, 355, 360, 362 matrix, 642 angle interpoint, 383, 386 relative, 410 antisymmetric, 55, 574, 747 antihollow, 58, 527 auxiliary, 385, 618, 622 orthonormal, 621 projector, 618 Schoenberg, 382, 620 table, 622 binary, see matrix, Boolean Boolean, 282, 439 bordered, 433, 447 calculus, 641 circulant, 256, 596, 619 permutation, 355 symmetric, 355 completion, see problem correlation, 61, 438, 439 covariance, 385 determinant, see determinant diagonal, see diagonal direction, 294, 296, 310, 311, 349, 393, 554 distance Euclidean, see EDM taxicab, 251, 446

INDEX doublet, 413, 615, 703 nullspace, 615, 616 range, 615, 616 dyad, 608, 610 -decomposition, 608 independence, 612, 613 negative, 610 nullspace, 610, 611 projection on, 694, 696 projector, 611, 676, 693, 694 range, 610, 611 sum, 594, 597, 614 symmetric, 119, 612 EDM, see EDM elementary, 445, 616, 618 nullspace, 616, 617 range, 616, 617 exponential, 256, 669 fat, 54, 85, 740 Fourier, 53, 354 Gale, 399 geometric centering, 385, 413, 477, 486, 618 Gram, 308, 383, 395 constraint, 386, 387 Hermitian, 574 hollow, 55, 57 Householder, 617 auxiliary, 619 idempotent, 679 nonsymmetric, 675 range, 675, 676 symmetric, 681, 684 inverse, 254, 585, 596, 612, 658, 659 injective, 672 symmetric, 597 Jordan form, 579, 595 measurement, 526 nonsingular, 596, 597 normal, 52, 580, 596, 598 normalization, 283 nullspace, see nullspace orthogonal, 335, 336, 414, 597, 623 manifold, 623 product, 623

INDEX symmetric, 624 orthonormal, 53, 64, 69, 296, 334, 414, 599, 621, 631, 673, 683 manifold, 69 partitioned, 589–593 permutation, 68, 335, 336, 429, 572, 597, 623 circulant, 355 extreme point, 68 product, 623 symmetric, 355, 618, 637 positive definite inverse, 585 positive real numbers, 250 positive semidefinite, 574, 577, 579, 587 difference, 130 from extreme directions, 120 Kronecker, 380, 584 nonsymmetric, 579 pseudoinverse, 673 square root, 585, 597, 740 product, see product projection, 114, 619, 682 Kronecker, 720 nonorthogonal, 675 orthogonal, 681 product, 719 pseudoinverse, 114, 182, 240, 280, 589, 671 Kronecker, 673 nullspace, 671, 672 positive semidefinite, 673 product, 672 range, 671, 672 SVD, 600 symmetric, 602, 673 transpose, 179 unique, 671 quotient Hadamard, 662, 669, 742 range, see range rank, see rank reflection, 414, 415, 624, 637 rotation, 414, 623–625, 635 of range, 624

797 rotation of, 626 Schur-form, 589 simple, 609 skinny, 53, 182, 739 sort index, 459 square root, 740, see matrix, positive semidefinite squared, 255 standard basis, 57, 59, 740 Stiefel, see matrix, orthonormal stochastic, 68, 335 sum eigenvalue, 586 rank, 127, 581 symmetric, 55, 597 inverse, 597 pseudoinverse, 602, 673 real numbers, 250 subspace of, 55 symmetrized, 575 trace, see trace unitary, 623 symmetric, 354, 356 max cut problem, 344 maximum variance unfolding, 482, 488 membership relation, 63, 99, 110, 158, 186, 272, 753 affine, 214 boundary, 161, 169, 274 discretized, 166, 167, 519 EDM, 519 in subspace, 186 interior, 160, 273 normal cone, 194 metric, 52, 251, 374 postulate, 375 property, 375, 430 fifth, 375, 378, 379, 464, 467, 473 space, 375 minimal element, 100, 101, 212, 213 set, see set minimization, 165, 194, 303, 653 affine function, 227 cardinality, 217, 218, 317, 348, 359

798 Boolean, 279 nonnegative, 219, 315, 318, 325, 326, 328 signed, 330 fractional, inverted, see function norm 1-, 216, see problem 2-, 216, 388, 389 ∞-, 216 Frobenius, 216, 236, 237, 240, 531, 702 on hypercube, 81 rank, 294, 334, 348, 365, 367, 553, 561 trace, 297, 298, 397, 548, 550, 551, 591 minimum cardinality, see minimization element, 100, 101, 212, 213 global unique, 209, 653 value unique, 212 MIT, 363 molecular conformation, 24, 406 monotonic, 235, 243, 299, 725, 728 Fej´er, 725 function, 209, 221, 243, 258, 259 gradient, 243 Moore-Penrose conditions, 671 Moreau, 197, 711, 712 MRI, 350, 352 multidimensional function, 209, 238, 254 affine, 226 objective, 100, 101, 212, 213, 315, 359 scaling, 24, 454 ordinal, 459 multilateration, 301, 403 multiobjective optimization, 100, 101, 212–214, 315, 359 multipath, 403 multiplicity, 581, 595 nearest neighbors, 484 neighborhood graph, 483, 484 Nemirovski, 525 nesting, 432

INDEX Newton, 392, 641 direction, 239 nondegeneracy, 375 nonexpansive, 683, 714 nonnegative, 428, 754 constraint, 86, 87, 197 part, 222, 526, 715, 716, 744 nonnegativity, 215, 375 nonvertical, 241 norm, 212, 214, 216, 219, 315, 754 0, 602 0-, see 0-norm 1-, see 1-norm 2-, see 2-norm ∞-, see ∞-norm k-largest, 222 ball, see ball constraint, 332, 342, 603 spectral, 69 convexity, 69, 211, 212, 215, 216, 255, 628, 754 dual, 158, 628 Euclidean, see 2-norm Frobenius, 51, 212, 216, 530 minimization, 216, 236, 237, 531, 702 gradient, 663, 665, 681, 691, 692, 706 inequality, 280, 754 triangle, 215 Ky Fan, 628 least, 217, 280, 674 nuclear, 550, 628 ball, 66 orthogonally invariant, 53, 463, 623, 626 property, 214 regularization, 343, 674 Schur-form, 236, 255, 628 spectral, 255, 530, 543, 600, 610, 618, 628, 716, 754 ball, 69, 716 inequality, 69, 754 square, 212, 216, 244, 530, 665, 691, 692, 702, 754

244,

240, 702,

530,

617,

681,

INDEX normal, 73, 195, 706 cone, see cone equation, 674 facet, 177 inward, 71, 507 outward, 71 vector, 706 Notation, 739 NP-hard, 553 nuclear magnetic resonance, 406 nullspace, 84, 619, 704 basis, 85 doublet, 615, 616 dyad, 610, 611 elementary matrix, 616, 617 form, 83 hyperplane, 72 injective, 52 intersection, 88 orthogonal complement, 747 product, 582, 610, 614, 682 pseudoinverse, 671, 672

799

linear, 46, 57, 210, 226, 228, 383, 419, 421, 463, 497, 569, 704 injective, 52 noninjective, 54 projector, 679, 684, 686, 707 nonlinear, 210, 228 nullspace, 413, 417, 419, 421, 704 permutation, 221, 316, 537, 541, 563, 745 projection, see projector unitary, 53 optimality condition directional derivative, 653 dual, 153 first order, 165, 166, 194–197 KKT, 197, 535 semidefinite program, 277 Slater’s, 153, 206, 272, 277 unconstrained, 165, 239, 240, 702 constraint conic, 196, 197 equality, 166 inequality, 197 objective, 82 nonnegative, 197 convex, 261 global, 8, 212, 235, 261 strictly, 545, 551 perturbation, 290 function, 82, 153, 165, 194, 196 optimization, 21, 261 linear, 228, 310, 397 combinatorial, 68, 82, 220, 279, 335, multidimensional, 100, 101, 212, 213, 341, 344, 439, 567 315, 359 conic, see problem nonlinear, 228 convex, 7, 21, 212, 262, 395 polynomial, 339 art, 8, 284, 302, 567 quadratic, 236 real, 212 fundamental, 261 value, 275, 290 multiobjective, 100, 101, 212–214, 315, 359 unique, 263 tractable, 262 offset, 412 vector, 100, 101, 212–214, 315, 359 on, 746 order one-to-one, 746, see injective onto, 746, see surjective natural, 49, 569, 750 open, see set nonincreasing, 539, 597, 598, 745 operator, 750 of projection, 526 adjoint, 49, 50, 156, 514, 739, 745 partial, 63, 97, 99–101, 110, 112, 168, 177, 178, 584, 753 self, 387, 569 invertible, 48, 52, 53, 173 generalized inequality, 99

800

INDEX

norm ball, 218 property, 99, 101 total, 99, 100, 751, 752 permutation, 68, 335, 338 origin, 36 stochastic, 68, 335, 338 cone, 69, 96, 98, 99, 102, 105, 133 transformation projection of, 217, 218, 675, 691 linear, 135, 139, 161 projection on, 711, 716 unbounded, 38, 42, 140, 141 translation to, 424 vertex, 68, 140, 218 Orion nebula, 22 vertex-description, 63, 141 orthant, 37, 144, 181, 194 polynomial, 590 dual, 173 constraint, 223, 339, 340 nonnegative, 161, 513 objective, 339 orthogonal, 49 quadratic, see function complement, 55, 83, 147, 297, 680, 684, polytope, 139 711, 734, 742, 748 positive, 754 equivalence, 626 strictly, 429 expansion, 57, 147, 180, 683, 685 power, see function, fractional invariance, 53, 463, 530, 623, 626 Pr´ecis, 426 orthonormal, 49 precision, numerical, 139, 264, 282, 312, decomposition, 682 358, 392, 525, 555 matrix, see matrix presolver, 327 orthonormality condition, 180 primal overdetermined, 739 feasible set, 271 problem, 153, 263 PageRank, 488 via dual, 278 parallel, 38, 685 principal pattern recognition, 24 component analysis, 348, 454 Penrose conditions, 671 eigenvalue, 348 pentahedron, 470 eigenvector, 348, 350 pentatope, 144, 470 submatrix, 395, 465, 470, 494, 587 perpendicular, 49, 681, 702 leading, 430, 432 perturbation, 285, 287 theorem, 587 optimality, 290 principle phantom, 352 halfspace-description, 73 plane, 71 separating hyperplane, 83 segment, 103 supporting hyperplane, 78 point problem boundary, 90 1-norm, 216–220, 280, 330 of hypercube slice, 362 nonnegative, 219, 220, 318, 325, 326, exposed, see exposed 328 fixed, 299, 360, 704, 722–725 Boolean, 68, 82, 279, 341, 439 inflection, 249 cardinality, 317, 348, 359 polychoron, 93, 139, 144, 469, 470 Boolean, 279 polyhedron, 42, 60, 82, 138, 436 nonnegative, 219, 315, 318, 325, 326, bounded, 63, 68, 220 328 face, 138, 139 halfspace-description, 138 signed, 330

INDEX

801

combinatorial, 335 combinatorial, 68, 82, 220, 279, 335, 341, 344, 439, 567 compressed sensing, 338 complementarity, 197 diagonal, 640 linear, 198 linear program, 637 completion, 303–307, 311, 355, 376, maximization, 637 377, 396, 399, 400, 435, 442, 443, orthogonal, 634 472, 473, 480, 482, 483, 567 two sided, 636, 638 geometry, 482 orthonormal, 334 semidefinite, 730, 731 symmetric, 639 compressed sensing, see compressed translation, 635 concave, 262 unique solution, 635 conic, 153, 196, 197, 214, 262 product, 258, 647 dual, 153, 262 Cartesian, 46, 271, 727, 747 convex, 153, 166, 196, 216, 228, 235, cone, 98, 154 237, 264, 271, 542, 705 function, 214 definition, 262 commutative, 256, 278, 581–583, 606, geometry, 21, 27 719 nonlinear, 261 determinant, 582 statement as solution, 7, 261 direct, 646 tractable, 262 eigenvalue, 581, 583, 584, 634 dual, see dual matrix, 583, 584 epigraph form, 236, 558, 559 fractional, see function equivalent, 296 function, see function feasibility, 142, 169, 170 gradient, 645 semidefinite, 267, 275, 368 Hadamard, 50, 544, 570, 571, 648, 742 fractional, inverted, see function inner max cut, 344 matrix, 383 minimax, 63, 153, 154 vector, 49, 258, 264, 351, 409, 694, nonlinear, 395 744 convex, 261 vectorized matrix, 49, 579 open, 30, 126, 299, 453, 496, 567 Kronecker, 114, 570–572, 638, 646, 647, primal, see primal 664 Procrustes, see Procrustes eigen, 572 proximity, 525, 527, 530, 533, 544, 555 inverse, 354 EDM, nonconvex, 540 positive semidefinite, 380, 584 in spectral norm, 543 projector, 720 rank heuristic, 550, 552 pseudoinverse, 673 SDP, Gram form, 547 vector, 648 semidefinite program, 536, 546, 557 vectorization, 571 same, 296 nullspace, 582, 610, 614, 682 stress, 249, 532, 553 orthogonal, 623 tractable, 262 outer, 64 procedure permutation, 623 rank reduction, 286 positive definite, 576 Procrustes nonsymmetric, 576

802 positive semidefinite, 583, 584, 587 nonsymmetric, 583 projector, 512, 694, 719 pseudofractional, see function pseudoinverse, 672 Kronecker, 673 range, 582, 610, 614, 682 rank, 581, 582 tensor, 646 trace, see trace zero, 606 program, 82, 261 convex, see problem geometric, 8, 265 integer, 68, 82, 439 linear, 82, 86, 87, 154, 215, 228, 261, 263–265, 392 dual, 154, 263 prototypical, 154, 263 nonlinear convex, 261 quadratic, 264, 265, 462 second-order cone, 264, 265 semidefinite, 154, 215, 228, 261–265, 337 dual, 154, 263 equality constraint, 112 Gram form, 559 prototypical, 154, 263 Schur-form, 223, 236, 237, 255, 463, 546, 558, 559, 628, 714 projection, 51, 671 I − P , 680, 711 algebra, 686 alternating, 391, 719, 720 convergence, 724, 726, 728, 731 distance, 722, 723 Dykstra algorithm, 719, 732, 733 feasibility, 722–724 iteration, 719, 733 on affine/psd cone, 729 on EDM cone, 557 on halfspaces, 721 on orthant/hyperplane, 725, 726 optimization, 722, 723, 732

INDEX over/under, 727 biorthogonal expansion, 688 coefficient, 177, 502, 693, 700 cyclic, 721 direction, 681, 684, 696 nonorthogonal, 678, 679, 693 dual, see dual, projection easy, 715 Euclidean, 128, 527, 530, 534, 536, 544, 556, 706 matrix, see matrix minimum-distance, 165, 679, 682, 702, 705 nonorthogonal, 384, 543, 675, 678, 679, 693 eigenvalue coefficients, 695 oblique, 217, 218, 532, see projection, nonorthogonal of convex set, 48 of hypercube, 51 of matrix, 701, 703 of origin, 217, 218, 675, 691 of PSD cone, 116 on 1-norm ball, 717 on affine, 48, 686, 687, 691 hyperplane, 691, 693 of origin, 217, 218, 675, 691 vertex-description, 692 on basis nonorthogonal, 687 orthogonal, 687 on box, 715 on cardinality k , 716 on complement, 680, 711 on cone, 710, 712, 716 dual, 93, 147, 711, 712 EDM, 544, 545, 555, 556, 560, 562 Lorentz (second-order), 717 monotone nonnegative, 452, 462, 538–540, 716 polyhedral, 717 truncated, 713 on cone, PSD, 128, 533, 535, 536, 716 geometric center subspace, 512 rank constrained, 239, 536, 542, 630

INDEX on convex set, 165, 705–707 in affine subset, 717 in subspace, 717 on convex sets, 721 on dyad, 694, 696 on ellipsoid, 332 on Euclidean ball, 715 on function domain, 231 on geometric center subspace, 512, 703 on halfspace, 693 on hyperbox, 715 on hypercube, 715 on hyperplane, 691, 693, see projection, on affine on hypersphere, 715 on intersection, 717 on matrices rank constrained, 716 on matrix, 693, 694, 696 on origin, 711, 716 on orthant, 539, 713, 715 in subspace, 539 on range, 674 on rowspace, 54 on simplex, 717 on slab, 693 on spectral norm ball, 716 on subspace, 48, 93, 147, 682, 711 elementary, 690 hollow symmetric, 534, 715 matrix, 701 symmetric matrices, 715 on vector cardinality k , 716 nonorthogonal, 693 orthogonal, 696 one-dimensional, 696 order of, 526, 528, 717 orthogonal, 674, 681, 696 eigenvalue coefficients, 698 semidefiniteness test as, 699 range/rowspace, 705 spectral, 486, 537, 540 norm, 543, 716 unique, 540

803 successive, 721 two sided, 701, 703, 705 umbral, 49 unique, 679, 682, 702, 705–707 vectorization, 502, 703 projector biorthogonal, 685 commutative, 719 non, 512, 717, 720 complement, 681 convexity, 679, 684, 705, 707 direction, 681, 684, 696 nonorthogonal, 678, 679, 693 dyad, 611, 676, 693, 694 Kronecker, 720 linear operator, 54, 679, 684, 686, 707 nonorthogonal, 384, 679, 686, 693 orthogonal, 682 orthonormal, 685 product, 512, 694, 719 range, 682 prototypical complementarity problem, 198 compressed sensing, 326 coordinate system, 147 linear program, 263 SDP, 154, 263, 276, 293 signal, 351 pyramid, 471, 472 quadrant, 37 quadratic, see function quadrature, 414 quartix, 644, 750 quasi-, see function quotient Hadamard, 662, 669, 742 Rayleigh, 700 range, 62, 84, 702, 747 dimension, 53 doublet, 615, 616 dyad, 610, 611 elementary matrix, 616, 617 form, 83 idempotent, 675, 676

804 orthogonal complement, 747 product, 582, 610, 614, 682 pseudoinverse, 671, 672 rotation, 624 rank, 581, 582, 598, 599, 752 -ρ subset, 115 constraint, 294, 334, 339, 340, 348, 365, 367, 553, 561 projection on matrices, 716 projection on PSD cone, 239, 536, 542, 630 convex envelope, see convex convex iteration, see convex diagonalizable, 581 dimension, 132, 425, 426, 752 affine, 427, 553 full, 85 heuristic, 550, 552, 553 inequality, 585 log det, 551, 553 lowest, 267 minimization, 294, 334, 348, 365, 367, 553, 561 one, 439, 588, 608, 610, 612 convex iteration, 334, 367, 565 hull, see hull, convex modification, 612, 616 subset, 239 symmetric, 119 partitioned matrix, 592 positive semidefinite cone, 267 face, 117 Pr´ecis, 426 product, 581, 582 projection matrix, 680, 683 quasiconcavity, 127 reduction, 265, 285, 286, 291 procedure, 286 regularization, 315, 334, 344, 366, 367, 553, 565 Schur-form, 592 sum, 127 positive semidefinite, 127, 581 trace heuristic, 297, 298, 548–551 ray, 93

INDEX boundary of, 93 cone, 93, 97, 142 boundary, 103 EDM, 107 positive semidefinite, 105 extreme, 103, 499 interior, 93 Rayleigh quotient, 700 realizable, 22, 390, 532, 533 reconstruction, 454 isometric, 398, 458 isotonic, 457–460, 462, 464 list, 454 unique, 377, 399, 400, 416, 417 recursion, 360, 517, 569, 587 reflection, 414, 415, 637 invariant, 414 vector, 624 reflexivity, 99, 584 regular lattice, 30, 303 simplex, 470 tetrahedron, 467, 481 regularization cardinality, 321, 349 norm, 343, 674 rank, 315, 334, 344, 366, 367, 553, 565 relaxation, 68, 82, 220, 281, 283, 397, 439, 461 Riemann, 264 robotics, 25, 31 root, see function, fractional rotation, 414, 624, 625, 635 invariant, 414 of matrix, 626 of range, 624 vector, 124, 414, 623 round, 752 saddle value, 152, 153 scaling, 454 multidimensional, 24, 454 ordinal, 459 unidimensional, 553 Schoenberg

INDEX auxiliary matrix, 382, 620 criterion, 30, 385, 449, 519, 701 Schur -form, 589 anomaly, 237 constraint, 69 convex set, 112, 113 norm, 236, 255, 628 rank, 592 semidefinite program, 223, 236, 237, 255, 463, 546, 558, 559, 628, 714 sparse, 591 complement, 309, 397, 589, 590, 592 self-distance, 375 semidefinite domain of test, 574 program, see program symmetry versus, 574 sensor, 300 network localization, 300, 395, 399 sequence, 725 set Cartesian product, 46 closed, 36, 39, 40, 71–73, 173 cone, 109, 115, 135, 139, 140, 155, 161, 173, 174, 271–273, 477 polyhedron, 135, 138, 139 connected, 36 convex, 36 invariance, 46 projection on, 165, 705–707 difference, 46, 155 empty, 38, 39, 92, 627 affine, 38 closed, 40 cone, 96 hull, 61, 64, 69 open, 40 feasible, 45, 68, 165, 194, 212, 261, 750 dual, 271 primal, 271 intersection, see intersection invariant, 445 level, 195, 226, 229, 238, 247, 258, 262 affine, 228

805 minimal, 63, 76, 136, 747 affine, 692 extreme, 106, 176 halfspace, 136, 137 hyperplane, 136 orthant, 211 positive semidefinite cone, 111, 171, 251 nonconvex, 40, 94–97 open, 36, 38, 40, 41, 160 projection of, 48 Schur-form, 112, 113 smooth, 437 solution, 212, 750 sublevel, 195, 230, 239, 248, 253, 258, 590 convex, 231, 247 quasiconvex, 258 sum, 46 superlevel, 231, 258 union, 46 shape, 23 Sherman-Morrison, 612, 659 shift, 412 shroud, 415, 416, 500 cone, 172, 522 SIAM, 262 signal dropout, 320 processing digital, 320, 351, 363, 454 simplex, 143, 144 area, 469 content, 471 nonnegative, 325, 326 regular, 470 unit, 143, 144 volume, 471 singular value, 598, 628 σ , 570, 745 decomposition compact, 597 full, 599 geometrical, 601 pseudoinverse, 600

806 subcompact, 598 symmetric, 602 eigenvalue, 598, 602 inequality, 69 inverse, 600 largest, 69, 255, 628, 716, 754 normal matrix, 52, 598 sum of, 52, 66, 628, 629 symmetric matrix, 602 triangle inequality, 66 skinny, 182, 739 slab, 38, 309, 693 slack variable, 263, 271, 293 slice, 121, 173, 403, 652, 653 slope, 239 solid, 139, 143, 467 solution analytical, 236, 261, 627 feasible, 212, 750 numerical, 311 optimal, 82, 86, 87, 212, 750 problem statement as, 7, 261 set, 212, 750 trivial, 37 unique, 45, 211, 212, 267, 397, 543, 545, 551, 560 vertex, 82, 220, 338 sort function, 221, 316, 537, 541, 563, 745 index matrix, 459 largest entries, 221 monotone nonnegative, 539 smallest entries, 221 span, 62, 747 sparsity, 217–220, 279, 315–319, 325–328, 330, 331, 348, 351 gradient, 352 nonnegative, 219, 318, 325, 326, 328 theorem, 219, 318, 351 spectral cone, 448, 450, 537 dual, 453 orthant, 540 inequality, 69, 448

INDEX norm, 255, 530, 543, 600, 610, 617, 618, 628, 716, 754 ball, 69, 716 inequality, 69, 754 Schur-form, 255, 628 projection, see projection sphere packing, 391 steepest descent, 239, 652 strict complementarity, 278 feasibility, 272 positivity, 429 triangle inequality, 432 subject to, 752 submatrix principal, 395, 465, 470, 494, 587 leading, 430, 432 theorem, 587 subspace, 36 antihollow, 58 antisymmetric, 58, 527 complementary, 55 fundamental, 48, 672, 682 geometric center, 417, 418, 476, 486, 510–512, 548, 606, 703 orthogonal complement, 117, 413, 488, 703 hollow, 57, 418, 511 intersection, 88 hyperplanes, 85 nullspace basis span, 85 proper, 36 representation, 83 symmetric hollow, 57 tangent, 117 translation invariant, 413, 419, 488, 703 successive approximation, 721 projection, 721 sum, 46 eigenvalues of, 586 matrix rank, 127, 581 of eigenvalues, 52, 580

INDEX of singular values, 52, 66, 628, 629 vector, 46 unique, 55, 147, 613 supremum, 63, 229, 255, 259, 260, 627–629, 751 surjective, 53, 417, 419, 421, 538, 746 linear, 421, 422, 439, 441 svec, 56 Swiss roll, 26 symmetry, 375 tangent line, 43, 397 subspace, 117 tangential, 43, 81, 398 Taylor series, 641, 651, 657–659 tensor, 644, 645 tesseract, 51 tetrahedron, 143, 144, 325, 469 angle inequality, 467 regular, 467, 481 theorem 0 eigenvalues, 603 alternating projection distance, 724 alternative, 161, 162, 164, 170 EDM, 521 semidefinite, 274 weak, 164 Barvinok, 130 Bunt-Motzkin, 706 Carath´eodory, 160, 698 cone faces, 102 intersection, 98 conic coordinates, 204 convexity condition, 245 decomposition dyad, 608 directional derivative, 653 discretized membership, 167 dual cone intersection, 199 duality, weak, 275 EDM, 439 eigenvalue of difference, 586

807 order, 585 sum, 586 zero, 603 elliptope vertices, 439 exposed, 107 extreme existence, 89 extremes, 105, 106 Farkas’ lemma, 158, 163 not positive definite, 274 positive definite, 273 positive semidefinite, 271 fundamental algebra, 595 algebra, linear, 84 convex optimization, 261 generalized inequality and membership, 158 Gerˇsgorin discs, 124 gradient monotonicity, 243 Hahn-Banach, 73, 78, 83 halfspaces, 73 Hardy-Littlewood-P´olya, 538 hypersphere, 439 inequalities, 538 intersection, 46 inverse image, 47, 50 inverse image closedness, 173 Klee, 105 line, 254, 259 linearly independent dyads, 613 majorization, 573 mean value, 658 Minkowski, 140 monotone nonnegative sort, 539 Moreau decomposition, 197, 712 Motzkin, 706 transposition, 164 nonexpansivity, 714 pointed cones, 99 positive semidefinite, 578 cone subsets, 128 matrix sum, 127 principal submatrix, 587 symmetric, 588 projection

808 algebraic complement, 712 minimum-distance, 706, 707 on affine, 687 on cone, 710 on convex set, 706, 707 on PSD geometric intersection, 512 on subspace, 48 via dual cone, 713 via normal cone, 707 projector rank/trace, 683 semidefinite, 588 proper-cone boundary, 102 Pythagorean, 339, 409, 718 range of dyad sum, 614 rank affine dimension, 427 partitioned matrix, 592 Schur-form, 592 trace, 680 real eigenvector, 594 sparse sampling, 318, 351 sparsity, 219 Tour, 443 Weyl, 140 eigenvalue, 586 tight, 377, 750 trace, 569, 570, 628, 629, 665 /rank gap, 549 eigenvalues, 580 heuristic, 297, 298, 548, 550, 551 inequality, 584, 585 Kronecker, 572 maximization, 486 minimization, 297, 298, 397, 548, 550, 551, 591 of product, 50, 581, 584 product, 572, 584 projection matrix, 680 zero, 606 trajectory, 445, 506 sensor, 401 transformation affine, 47, 48, 78, 173, 244, 259, 518 positive semidefinite cone, 112, 113

INDEX bijective, see bijective Fourier, discrete, 53, 354 injective, see injective invertible, 48, 52, 53 linear, 37, 48, 77, 135, 672 cone, 98, 102, 135, 139, 161, 174, 176 cone, dual, 156, 161 inverse, 48, 477, 672 polyhedron, 135, 139, 161 rigid, 23, 398 similarity, 698 surjective, see surjective transitivity, 91, 99, 584 trefoil, 484, 485, 487 triangle, 376 inequality, 375, 376, 409, 430–432, 434, 442, 465, 466, 468–470, 479, 481 norm, 215 strict, 432 unique, 465 triangulation, 23 trilateration, 23, 45, 396 tandem, 399 unbounded below, 82, 86, 87, 163 underdetermined, 365, 740 unfolding, 484, 488 unfurling, 482 unimodal, 256, 257 unique, 8 eigenvalues, 595 extreme direction, 103 localization, 301, 302 minimum element, 101, 212, 213 solution, 211, 398 unraveling, 484, 485, 487 untieing, 485, 487 value absolute, 214, 249 minimum unique, 212 objective, 275, 290 unique, 263 variable matrix, 76

INDEX slack, 263, 271, 293 variation calculus, 165 total, 357 vec, 49, 569, 571 vector, 36, 750 binary, 61, 283, 341, 437 direction, 317 dual, 733 normal, 706 optimization, 100, 101, 212–214, 315, 359 parallel, 685 Perron, 447 primal, 733 product inner, 49, 258, 264, 351, 409, 579, 694, 744 Kronecker, 648 quadrature, 414 reflection, 624 rotation, 124, 414, 623 space, 36, 747 standard basis, 57, 144, 384, 751 vectorization, 49 projection, 502 symmetric, 56 hollow, 59 Venn, 140, 265, 528, 529 vertex, 41, 45, 68, 90, 91, 218 cone, 99 description, 70, 71, 106 affine, 85 dual cone, 185, 199 halfspace, 136, 137 hyperplane, 76 polyhedral cone, 69, 142, 161, 176 polyhedron, 63, 141 projection, 692 elliptope, 282, 438, 439 polyhedron, 68, 140, 218 solution, 82, 220, 338 Vitulli, 609 vortex, 492

809 Wıκımization, 33, 132, 165, 198, 199, 286, 324, 341, 342, 348, 362, 371, 378, 385, 388, 458, 460, 461, 608, 716 W¨ uthrich, 24 wavelet, 351, 353 wedge, 103, 104, 158 wireless location, 24, 300, 403 Wolkowicz, 637 womb, 642 zero, 602 definite, 607 entry, 603 gradient, 165, 239, 240, 702 matrix product, 606 norm-, 602 trace, 606

810

Convex Optimization & Euclidean Distance Geometry, Dattorro

Mεβoo Publishing USA