Optimization: Structure and Applications (Springer Optimization and Its Applications)

OPTIMIZATION

Springer Optimization and Its Applications VOLUME 32 Managing Editor Panos M. Pardalos (University of Florida) Editor–Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University)

Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences. The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multiobjective programming, description of software packages, approximation techniques and heuristic approaches.

OPTIMIZATION Structure and Applications Edited By CHARLES PEARCE School of Mathematical Sciences, The University of Adelaide, Adelaide, Australia EMMA HUNT School of Economics & School of Mathematical Sciences, The University of Adelaide, Adelaide, Australia

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Editors Charles Pearce Department of Applied Mathematics University of Adelaide 70 North Terrace Adelaide SA 5005 Australia [email protected]

Emma Hunt Department of Applied Mathematics University of Adelaide 70 North Terrace Adelaide SA 5005 Australia [email protected]

ISSN 1931-6828 ISBN 978-0-387-98095-9 e-ISBN 978-0-387-98096-6 DOI 10.1007/978-0-387-98096-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009927130 Mathematics Subject Classification (2000): 49-06, 65Kxx, 65K10, 76D55, 78M50 c Springer Science+Business Media, LLC 2009  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover illustration: Picture provided by Elias Tyligadas Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

This volume is dedicated with great affection to the late Alex Rubinov, who was invited plenary speaker at the mini-conference. He is sorely missed. ♥

Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Part I Optimization: Structure 1

On the nondifferentiability of cone-monotone functions in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Jonathan Borwein and Rafal Goebel 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2

Duality and a Farkas lemma for integer programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean B. Lasserre 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Summary of content . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Duality for the continuous problems P and I . . . . . . . . . . . . . . 2.2.1 Duality for P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Duality for integration . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Comparing P, P∗ and I, I∗ . . . . . . . . . . . . . . . . . . . . . . 2.2.4 The continuous Brion and Vergne formula . . . . . . . . . 2.2.5 The logarithmic barrier function . . . . . . . . . . . . . . . . . 2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Duality for the discrete problems Id and Pd . . . . . . . . . . . . . . 2.3.1 The Z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The dual problem I∗d . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Comparing I∗ and I∗d . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 17 18 18 19 19 20 21 22 22 23 24 24

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2.3.4 The “discrete” Brion and Vergne formula . . . . . . . . . 2.3.5 The discrete optimization problem Pd . . . . . . . . . . . . 2.3.6 A dual comparison of P and Pd . . . . . . . . . . . . . . . . . 2.4 A discrete Farkas lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The case when A ∈ Nm×n . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

Some nonlinear Lagrange and penalty functions for problems with a single constraint . . . . . . . . . . . . . . . . . . . . . . . . . J. S. Giri and A. M. Rubinov 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The relationship between extended penalty functions and extended Lagrange functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Generalized Lagrange functions . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 The Lagrange function approach . . . . . . . . . . . . . . . . . 3.5.2 Penalty function approach . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence of truncates in l1 optimal feedback control . . Robert Wenczel, Andrew Eberhard and Robin Hill 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 System-theoretic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Basic system concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Feedback stabilization of linear systems . . . . . . . . . . . 4.4 Formulation of the optimization problem in l1 . . . . . . . . . . . . 4.5 Convergence tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Verification of the constraint qualification . . . . . . . . . . . . . . . . 4.6.1 Limitations on the truncation scheme . . . . . . . . . . . . . 4.7 Convergence of approximates . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Some extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 26 27 29 30 31 33 34 34 35 36 37 38 41 41 43 44 47 51 51 52 53 55 55 58 60 60 62 64 67 71 76 78 85 88 92

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Asymptotical stability of optimal paths in nonconvex problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Musa A. Mamedov 5.1 Introduction and background . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The main conditions of the turnpike theorem . . . . . . . . . . . . . 5.3 Definition of the set D and some of its properties . . . . . . . . . . 5.4 Transformation of Condition H3 . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Sets of 1st and 2nd type: Some integral inequalities . . . . . . . . 5.5.1 ............................................. 5.5.2 ............................................. 5.5.3 ............................................. 5.5.4 ............................................. 5.5.5 ............................................. 5.6 Transformation of the functional (5.2) . . . . . . . . . . . . . . . . . . . 5.6.1 ............................................. 5.6.2 ............................................. 5.7 The proof of Theorem 13.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 ............................................. 5.7.2 ............................................. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pontryagin principle with a PDE: a unified approach . . . . . B. D. Craven 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Pontryagin for an ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Pontryagin for an elliptic PDE . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Pontryagin for a parabolic PDE . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A turnpike property for discrete-time control systems in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander J. Zaslavski 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Stability of the turnpike phenomenon . . . . . . . . . . . . . . . . . . . . 7.3 A turnpike is a solution of the problem (P) . . . . . . . . . . . . . . . 7.4 A turnpike result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mond–Weir Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Mond 8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Convexity and Wolfe duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Fractional programming and some extensions of convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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95 95 97 100 101 105 105 106 107 108 113 117 117 119 123 123 129 133 135 135 136 138 139 140 141

143 143 146 149 151 155 157 157 158 159

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8.4 Mond–Weir dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Second order duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Symmetric duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Computing the fundamental matrix of an M /G/1–type Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emma Hunt 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Algorithm H: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Probabilistic construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Algorithm H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Algorithm H: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H, G and convergence rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 9.7 A special case: The QBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Algorithms CR and H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 A comparison of probabilistic and invariant subspace methods for the block M /G/1 Markov chain . . . . . . . . . . . . . . Emma Hunt 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Error measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Experiment G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Experiment G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 The Daigle and Lucantoni teletraffic problem . . . . . . 10.3.4 Experiment G6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Experiment G7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Interpolating maps, the modulus map and Hadamard’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. S. Dragomir, Emma Hunt and C. E. M. Pearce 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 A refinement of the basic inequality . . . . . . . . . . . . . . . . . . . . . 11.3 Inequalities for Gf and Hf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 More on the identric mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 The mapping Lf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160 160 162 163 165

167 167 170 172 174 175 178 181 185 187

189 189 190 192 193 194 196 201 202 204

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Part II Optimization: Applications 12 Estimating the size of correcting codes using extremal graph problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergiy Butenko, Panos Pardalos, Ivan Sergienko, Vladimir Shylo and Petro Stetsyuk 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Finding lower bounds and exact solutions for the largest code sizes using a maximum independent set problem . . . . . . 12.2.1 Finding the largest correcting codes . . . . . . . . . . . . . . 12.3 Lower Bounds for Codes Correcting One Error on the Z-Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 The partitioning method . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 The partitioning algorithm . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Improved lower bounds for code sizes . . . . . . . . . . . . . 12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 New perspectives on optimal transforms of random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. G. Howlett, C. E. M. Pearce and A. P. Torokhti 13.1 Introduction and statement of the problem . . . . . . . . . . . . . . . 13.2 Motivation of the statement of the problem . . . . . . . . . . . . . . . 13.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Comparison of the transform T 0 and the GKLT . . . . . . . . . . . 13.6 Solution of the unconstrained minimization problem (13.3) . 13.7 Applications and further modifications and extensions . . . . . 13.8 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Optimal capacity assignment in general queueing networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. K. Pollett 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The residual-life approximation . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Optimal allocation of effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Data networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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227 229 232 236 237 239 239 241 242 245 245 247 248 249 251 252 253 254 258 258 261 261 262 263 264 267 268 271 271

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15 Analysis of a simple control policy for stormwater management in two connected dams . . . . . . . . . . . . . . . . . . . . . . Julia Piantadosi and Phil Howlett 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 A discrete-state model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 The transition matrix for a specific control policy . . 15.2.3 Calculating the steady state when 1 < m < k . . . . . . 15.2.4 Calculating the steady state for m = 1 . . . . . . . . . . . . 15.2.5 Calculating the steady state for m = k . . . . . . . . . . . . 15.3 Solution of the matrix eigenvalue problem using Gaussian elimination for 1 < m < k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Stage 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 The general rules for stages 2 to m − 2 . . . . . . . . . . . 15.3.3 Stage m − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.4 The general rules for stages m to k − 2m . . . . . . . . . . 15.3.5 Stage k − 2m + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.6 The general rule for stages k − 2m + 2 to k − m − 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.7 The final stage k − m − 1 . . . . . . . . . . . . . . . . . . . . . . . 15.4 The solution process using back substitution for 1 < m < k . 15.5 The solution process for m = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 The solution process for m = k . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Justification of inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.1 Existence of the matrix W0 . . . . . . . . . . . . . . . . . . . . . . 15.8.2 Existence of the matrix Wp for 1 ≤ p ≤ m − 1 . . . . . 15.8.3 Existence of the matrix Wq for m ≤ q ≤ k − m − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Optimal design of linear consecutive–k–out–of–n systems . Malgorzata O’Reilly 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Applications and generalizations of linear consecutive–k–out–of–n systems . . . . . . . . . . . . . . . . . 16.1.3 Studies of consecutive–k–out–of–n systems . . . . . . . . 16.1.4 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Propositions for R and M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Preliminaries to the main proposition . . . . . . . . . . . . . . . . . . . . 16.4 The main proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273 273 274 274 275 276 279 280 280 281 281 283 284 285 286 287 287 290 292 292 295 296 296 298 305 306 307 307 307 308 309 311 312 315 318 321

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Procedures to improve designs not satisfying necessary conditions for the optimal design . . . . . . . . . . . . . . . . . . . . . . . . 324 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 17 The (k+1)-th component of linear consecutive–k–out–of–n systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Malgorzata O’Reilly 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 General result for n > 2k, k ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Results for n = 2k + 1, k > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Results for n = 2k + 2, k > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Procedures to improve designs not satisfying the necessary conditions for the optimal design . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Optimizing properties of polypropylene and elastomer compounds containing wood flour . . . . . . . . . . . . . . . . . . . . . . . . . Pavel Spiridonov, Jan Budin, Stephen Clarke and Jani Matisons 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 Sample preparation and tests . . . . . . . . . . . . . . . . . . . . 18.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Density of compounds . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Comparison of compounds obtained in a Brabender mixer and an injection-molding machine . . . . . . . . . . 18.3.3 Compatibilization of the polymer matrix and wood flour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.4 Optimization of the compositions . . . . . . . . . . . . . . . . 18.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Constrained spanning, Steiner trees and the triangle inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prabhu Manyem 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Upper bounds for approximation . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 The most expensive edge is at most a minimum spanning tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.2 MaxST is at most (n − 1)MinST . . . . . . . . . . . . . . . . . 19.3 Lower bound for a CSP approximation . . . . . . . . . . . . . . . . . . . 19.3.1 E-Reductions: Definition . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2 SET COVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.3 Reduction from SET COVER . . . . . . . . . . . . . . . . . . . 19.3.4 Feasible Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327 327 329 330 334 337 340 341 343 344 344 344 345 345 345 346 349 350 352 353 355 355 359 359 359 360 360 361 361 362

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19.3.5 Proof of E-Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 19.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 20 Parallel line search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. C. Peachey, D. Abramson and A. Lewis 20.1 Line searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Nimrod/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Execution time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 A model for execution time . . . . . . . . . . . . . . . . . . . . . 20.3.2 Evaluation time a Bernoulli variate . . . . . . . . . . . . . . . 20.3.3 Simulations of evaluation time . . . . . . . . . . . . . . . . . . . 20.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Accelerating convergence by incomplete iterations . . . . . . . . . 20.4.1 Strategies for aborting jobs . . . . . . . . . . . . . . . . . . . . . . 20.4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Alternative Mathematical Programming Models: A Case for a Coal Blending Decision Process . . . . . . . . . . . . . . . . . . . . . Ruhul A. Sarker 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Mathematical programming models . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Single period model (SPM) . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Multiperiod nonlinear model (MNM) . . . . . . . . . . . . . 21.2.3 Upper bound linear model (MLM) . . . . . . . . . . . . . . . 21.2.4 Multiperiod linear model (MLM) . . . . . . . . . . . . . . . . 21.3 Model flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Case-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Case-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Case-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Problem size and computation time . . . . . . . . . . . . . . . . . . . . . . 21.5 Objective function values and fluctuating situation . . . . . . . . 21.6 Selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

369 369 370 373 373 373 375 375 377 377 378 381 381 383 383 385 386 389 390 391 392 392 394 395 395 396 397 398 398

About the Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

List of Figures

3.1 3.2 3.3

P (f0 , f1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 L(x; 52 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 L+ s 1 (x; 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1 12.1 12.2 13.1 13.2

A closed-loop control system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A scheme of the Z-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithm for finding independent set partitions . . . . . . . . . . . . Illustration of the performance of our method. . . . . . . . . . . . . . Typical examples of a column reconstruction in the matrix X (image “Lena”) after filtering and compression of the observed noisy image (Figure 13.1b) by transforms H (line with circles) and T 0 (solid line) of the same rank. In both subfigures, the plot of the column (solid line) virtually coincides with the plot of the estimate by the transform T 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density of (a) polypropylene and (b) SBS in elastomer compounds for different blending methods. . . . . . . . . . . . . . . . . Comparison of tensile strength of the compounds obtained in an injection-molding machine and in a Brabender mixer. . . Influence of wood flour fractions and the modifier on the tensile strength of injection-molded specimens of the (a) PP and (b) SBS compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative cost of the (a) PP and (b) SBS compounds depending on the content of wood flour and maleated polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photographs of the PP compounds containing 40% wood flour of different fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Constrained Steiner Tree and some of its special cases. . . . . A CSPI instance reduced from SET COVER (not all edges shown). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18.1 18.2 18.3

18.4

18.5 19.1 19.2

3

62 236 240 256

257 347 348

349

351 352 357 361

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19.3

Feasible solution for our instance of CSPI (not all edges shown). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 A sample configuration file. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Architecture of Nimrod/O. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Performance with Bernoulli job times. . . . . . . . . . . . . . . . . . . . . 20.4 Test function g(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Results of simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Incomplete evaluation points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.7 Strategy 1 with exponential distribution of job times. . . . . . . . 20.8 Strategy 1 with rectangular distribution of job times. . . . . . . . 20.9 Results for Strategy 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.10 Results for Strategy 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Simple case problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

364 371 372 374 375 376 377 379 379 380 380 393

List of Tables

9.1 10.1 10.2 10.3 10.4 10.5

The interlacing property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment G2 continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iterations required with various traffic levels: Experiment G3 Iterations required with various traffic levels: Experiment G3 continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Experiment G4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Experiment G4 continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Experiment G5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Experiment G6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Experiment G7: a transient process . . . . . . . . . . . . . . . . . . . . . . . 12.1 Lower bounds obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Exact algorithm: Computational results . . . . . . . . . . . . . . . . . . . 12.3 Exact solutions found . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Lower bounds obtained in: a [27]; b [6]; c [7]; d [12]; e (this chapter) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Partitions of asymmetric codes found . . . . . . . . . . . . . . . . . . . . . 12.6 Partitions of constant weight codes obtained in: a (this chapter); b [4]; c [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 New lower bounds. Previous lower bounds were found in: a [11]; b [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Ratios ρij of the error associated with the GKLT H to that of the transform T 0 with the same compression ratios . . . . . . . 15.1 Overflow lost from the system for m = 1, 2, 3, 4 . . . . . . . . . . . . . 16.1 Invariant optimal designs of linear consecutive–k–out–of–n systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Physical characteristics of the wood flour fractions . . . . . . . . . . 19.1 Constrained Steiner Tree and special cases: References . . . . . . 19.2 E-Reduction of a SET COVER to a CSPI : Costs and delays of edges in G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

184 194 195 196 198 199 200 200 202 202 204 231 235 236 237 240 241 241 255 295 310 345 358 362

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21.1 21.2

List of Tables

Relative problem size of ULM, MLM and MNM . . . . . . . . . . . . 395 Objective function values of ULM, MLM and MNM . . . . . . . . 397

Preface

This volume comprises a selection of material based on presentations at the Eighth Australian Optimization Day, held in McLaren Vale, South Australia, in September 2001, and some additional invited contributions by distinguished colleagues, here and overseas. Optimization Day is an annual miniconference in Australia which dates from 1994. It has been successful in bringing together Australian researchers in optimization and related areas for the sharing of ideas and the facilitation of collaborative work. These meetings have also attracted some collaborative researchers from overseas. This particular meeting was remarkable in the efforts made by some of the participants to ensure being present. It took place within days of the September 11 tragedy in New York and the financial collapse of a major Australian airline. These events left a number of us without air tickets on the eve of the conference. Some participants arrived in South Australia by car, having driven up to several thousand kilometers to join the meeting. This volume has two parts, one concerning mathematical structure and the other applications. The first part begins with a treatment of nondifferentiability of cone-monotone functions in Banach spaces, showing that whereas several regularity properties of cone-monotone functions in finite-dimensional spaces carry over to a separable Banach space provided the cone has an interior, further generalizations are not readily possible. The following chapter concerns a comparison between linear and integer programming, particularly from a duality perspective. A discrete Farkas lemma is provided and it is shown that the existence of a nonnegative integer solution to a linear equation can be tested via a linear program. Next, there is a study of connections between generalized Lagrangians and generalized penalty functions for problems with a single constraint. This is followed by a detailed theoretical analysis of convergence of truncates in 1 optimal feedback control. The treatment permits consideration of the frequently occurring case of an objective function lacking interiority of domain. The optimal control theme continues with a study of asymptotic stability of optimal paths in nonconvex problems. The purpose of the chapter is to avoid the convexity conditions usually

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assumed in turnpike theory. The succeeding chapter proposes a unified approach to Pontryagin’s principle for optimal control problems with dynamics described by a partial differential equation. This is followed by a study of a turnpike property for discrete-time control systems in metric spaces. A treatment of duality theory for nonlinear programming includes comparisons of alternative approaches and discussion of how Mond–Weir duality and Wolfe duality may be combined. There are two linked chapters centered on the use of probabilistic structure for designing an improved algorithm for the determination of the fundamental matrix of a block-structured M/G/1 Markov chain. The approach via probabilistic structure makes clear in particular the nature of the relationship between the cyclic reduction algorithms and the Latouche–Ramaswami algorithm in the QBD case. Part I concludes with a chapter developing systematic classes of refinements of Hadamard’s inequality, a cornerstone of convex analysis. Although Part II of this volume is concerned with applications, a number of the chapters also possess appreciable theoretical content. Part II opens with the estimation of the sizes of correcting codes via formulation in terms of extremal graph problems. Previously developed algorithms are used to generate new exact solutions and estimates. The second chapter addresses the issue of optimal transforms of random vectors. A new transform is presented which has advantages over the Karhunen–Lo`eve transform. Theory is developed and applied to an image reconstruction problem. The following chapter considers how to assign service capacity in a queueing network to minimize expected delay under a cost constraint. Next there is analysis of a control policy for stormwater management in a pair of connected tandem dams, where a developed mathematical technology is proposed and exhibited. Questions relating to the optimal design of linear consecutive-k-out-of-n systems are treated in two related chapters. There is a study of optimizing properties of plastics containing wood flour; an analysis of the approximation characteristics of constrained spanning and Steiner tree problems in weighted undirected graphs where edge costs and delays satisfy the triangle inequality; heuristics for speeding convergence in line search; and the use of alternative mathematical programming formulations for a real-world coal-blending problem under different scenarios. All the contributions to this volume had the benefit of expert refereeing. We are grateful to the following reviewers for their help: Mirta In´es Aranguren (Universidad Nacional de Mar del Plata, Argentina), Eduardo Casas (Universidad de Cantabria, Spain), Aurelian Cernea (University of Bucharest), Pauline Coolen–Schrijner (University of Durham), Bruce Craven (University of Melbourne), Yu-Hong Dai (Chinese Academy of Sciences, Beijing), Sever Dragomir (Victoria University of Technology, Melbourne), Elfadl Khalifa Elsheikh (Cairo University), Christopher Frey (N. Carolina State University), Frank Kelly (University of Cambridge), Peter Kloeden (Johann Wolfgang Goethe-Universit¨ at, Frankfurt), Denis Lander (RMIT University), Roy Leipnik (UCSB), Musa Mamedov (University of

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Ballarat), Jie Mi (Florida International University), Marco Muselli (Institute of Electronics, Computer and Telecommunication Engineering, Genova), Malgorzata O’Reilly (University of Tasmania), Stavros Papastavridis (University of Athens), Serpil Pehlivan (S¨ uleyman Demirel University, Turkey), Danny Ralph (Cambridge University), Sekharipuram S. Ravi (University at Albany, New York), Dan Rosenkrantz (University at Albany, New York), Alexander Rubinov (University of Ballarat), Hanif D. Sherali (Virginia Polytechnic Institute), Moshe Sniedovich (University of Melbourne), Nicole Stark (USDA Forest Products Laboratory, Madison), Nasser Hassan Sweilam (Cairo University), Fredi Tr¨ oltzsch (Technische Universit¨at, Berlin), Erik van Doorn (University of Twente, Netherlands), Frank K. Wang (National Chiao University, Taiwan, R.O.C.), Jianxing Yin (SuDa University, China), Alexander Zaslavski (Technion-Israel Institute of Technology), and Ming Zuo (University of Alberta). We wish to thank John Martindale for his involvement in bringing about a firm arrangement for publication of this volume with Kluwer; Elizabeth Loew for shepherding the book through to publication with Springer following the Kluwer–Springer merger; and Panos Pardalos for his support throughout. A special thank you is due to Jason Whyte, through whose technical skill, effort and positive morale many difficulties were overcome. Charles Pearce & Emma Hunt

Chapter 1

On the nondifferentiability of cone-monotone functions in Banach spaces Jonathan Borwein and Rafal Goebel

Abstract In finite-dimensional spaces, cone-monotone functions – a special case of which are coordinate-wise nondecreasing functions – possess several regularity properties like almost everywhere continuity and differentiability. Such facts carry over to a separable Banach space, provided that the cone has interior. This chapter shows that further generalizations are not readily possible. We display several examples of cone–monotone functions on various Banach spaces, lacking the regularity expected from their finite-dimensional counterparts. Key words: Monotone functions, ordered Banach spaces, generating cones, differentiability

1.1 Introduction Functions for which f (y) ≥ f (x) whenever y − x is an element of a given convex cone K are called cone monotone with respect to K (or, simply, Kmonotone). The simplest examples are provided by nondecreasing functions on the real line. These have several immediate regularity properties, the most intuitive of which may be the at most countable number of discontinuities. Regularity properties of coordinate-wise nondecreasing functions on IRn , that is, functions f for which f (y) ≥ f (x) whenever yi ≥ xi for i = 1, 2, . . . , n, Jonathan Borwein Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC, CANADA V5A 1S6 e-mail: [email protected] Rafal Goebel Center for Control Engineering and Computation ECE, University of California, Santa Barbara, CA 93106-9650 U. S. A. e-mail: [email protected] C. Pearce, E. Hunt (eds.), Structure and Applications, Springer Optimization and Its Applications 32, DOI 10.1007/978-0-387-98096-6 1, c Springer Science+Business Media, LLC 2009 

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were first collected by Chabrillac and Crouzeix [5] and include measurability, almost everywhere continuity, and almost everywhere Fr´echet differentiability. Note that nondecreasing functions, whether on the real line or IRn , are cone monotone with respect to the nonnegative cone, either [0, +∞) or [0, +∞)n . Recently, Borwein, Burke and Lewis [2] showed that functions on a separable Banach space, monotone with respect to a convex cone with nonempty interior, are differentiable except at points of an appropriately understood null set. The main goal of the current chapter is to demonstrate how possible extensions of this result, or other generalizations of finite-dimensional results on regularity of cone-monotone functions, fail in a general Banach space. Motivation for studying coordinate-wise nondecreasing functions in Chabrillac and Crouzeix, and cone-monotone functions by Borwein, Burke and Lewis, comes in part from the connections of such functions with Lipschitz, and more generally, directionally Lipschitz functions. Interest in the latter stems from the work of Burke, Lewis and Overton [4] on approximation of the Clarke subdifferential using gradients, an important idea in practical optimization. It turns out that such approximations, like in the Lipschitz case, are possible in the more general directionally Lipschitz setting. Before summarizing the properties of nondecreasing functions in finite dimensions, we illustrate their connection with Lipschitz functions. Consider a Lipschitz function l : IRn → IR and a K > 0 satisfying |l(x) − l(y)| ≤ K x − y ∞ ,

for all x, y ∈ IRn .

Let z ∈ IRn be given by zi = K, i = 1, 2, . . . , n, and define a function f : IRn → IR by f (x) = l(x) + z, x. Then for x and y such that yi ≥ xi , i = 1, 2, . . . , n, we have f (y) − f (x) = z, y − x + l(y) − l(x)  n   ≥K (yi − xi ) − max (yi − xi ) ≥ 0. i=1

i=1,2,...,n

In effect, f is coordinate-wise nondecreasing. Thus a Lipschitz function decomposes into a sum of a linear and a nondecreasing function, and inherits the regularity properties of the latter. Borwein, Burke and Lewis show that directionally Lipschitz functions on more general spaces admit a similar (local) decomposition, into a sum of a linear function and a cone-monotone one (with respect to a convex cone with interior). Theorem 1 (Monotone functions in finite dimensions). Suppose that f : IRn → IR satisfies f (x) ≤ f (y) whenever xi ≤ yi , i = 1, 2, . . . , n. Then: (a) f is measurable. (b) If, for some d with di > 0 for i = 1, 2, . . . , n, the function t → f (x0 + td) is lower semicontinuous at t = 0, then f is lower semicontinuous at x0 . Similarly for upper semicontinuity.

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On the nondifferentiability of cone-monotone functions in Banach spaces

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(c) f is almost everywhere continuous. (d) If f is Gˆ ateaux differentiable at x0 , then it is Fr´echet differentiable at x0 . (e) Let f be the lower semicontinuous hull of f . Then f is continuous at x0 if and only if f is. Similarly, f is Gˆ ateaux differentiable at x0 if and only ateaux differentiable, their derivatives f is, and if these functions are Gˆ agree. (f ) f is almost everywhere Fr´echet differentiable. For details and proofs consult Chabrillac and Crouzeix [5]. Statements (c) and (f ) generalize the Lebesgue monotone differentiability theorem and in fact can be deduced from the two-dimensional version given by S. Saks [9]; for details consult Borwein, Burke and Lewis [2]. The Banach space version of (c) and (f ) of Theorem 1, proved by Borwein, Burke and Lewis, requires a notion of a null set in a Banach space. We recall that a Banach space does not admit a Haar measure unless it is finite-dimensional, and proceed to make the following definitions – for details on these, and other measure-related notions we use in this chapter, we refer the reader to Benyamini and Lindenstrauss [1]. Let X be a separable Banach space. A probability measure μ on X is called Gaussian if for every x∗ ∈ X ∗ , the measure μx∗ on the real line, defined by μx∗ (A) = μ{y | x∗ , y ∈ A}, has a Gaussian distribution. It is additionally called nondegenerate if for every x∗ = 0 the distribution μx∗ is nondegenerate. A Borel set C ⊂ X is called Gauss null if μ(C) = 0 for every nondegenerate Gaussian measure on X. It is known that the set of points where a given Lipschitz function f : X → IR is not Gˆ ateaux differentiable is Gauss null. This in fact holds for functions with values in a space with the Radon–Nikodym property (Benyamini and Lindenstrauss [1] Theorem 6.42), whereas it fails completely for the stronger notion of Fr´echet differentiability. Theorem 2 (Borwein, Burke and Lewis). Let X be a separable space and let K ⊂ X be a convex cone with non-empty interior. If f : X → IR is K-monotone, then it is continuous and Hadamard (so also Gˆ ateaux) differentiable except at the points of a Gauss null set. In what follows, we show that all the assumptions in the above theorem are necessary, and, more generally, demonstrate how the properties of conemonotone functions described in Theorem 1 fail to extend to a general Banach space. Note that the results of Theorems 1 and 2 hold if the functions are allowed to take on infinite values, as long as appropriate meaning is given to the continuity or differentiability of such functions. Indeed, composing a possibly infinite-valued function with, for example, an inverse tangent does not change its monotonicity properties, while leading to finite values. We do not address this further, and work with finite-valued functions. Moreover, we only work with monotone functions which are in fact nondecreasing (homotone) and note that nonincreasing (antitone) functions can be treated in a symmetric fashion.

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1.2 Examples We begin the section by showing that K-monotonicity of a function f : X → IR, with X finite- or infinite-dimensional, carries little information about the function if the cone K is not generating: K − K = X. (Recall that K − K is always a linear subspace of X.) An interesting example of a nongenerating cone K is provided by nonnegative and nondecreasing functions in X = C[0, 1]: K − K is the non-closed set of all functions of bounded variation, dense in (but not equal to) X. Note that the example below, as well as Example 2, could be considered in a general vector space. Example 1 (a nongenerating cone). Suppose that K − K = X. Let L ⊃ K − K be a hyperplane in X, not necessarily closed, and let l be the (onedimensional) algebraic complement of L in X, that is, L + l = X, L ∩ l = 0. Such L and l can be constructed using one of the versions of the Separation Principle: for any point l0 not in the intrinsic core (relative algebraic interior) of the convex set K − K (as K − K is linear, for any point not in K − K) there exists a linear functional (not necessarily continuous) φ on X such that

φ, l0  > φ, K − K; see Holmes [6], page 21. Now let L = ker φ, l = IR l0 . Define Pl (x) to be the projection of x onto l – the unique point of l such that x ∈ Pl (x) + L. Given any function g : l → IR, let f (x) = g(Pl (x)). The function f : X → IR is K-monotone: if y ≥K x, that is, y − x ∈ K, then Pl (x) = Pl (y), and in effect, f (y) = f (x). Now note that at any point x ∈ X, the function f has the properties of g “in the direction of l.” Consequently, in this direction, f may display any desired irregular behavior. In light of the example above, in what follows we only discuss generating cones. A cone K is certainly generating when int K = ∅, as then the linear subspace K − K has interior. More generally, K is generating when the core (algebraic interior) of K is not empty. These conditions are met by nonnegative cones in C[a, b] or l∞ but not by all generating cones — consider for example nonnegative cones in c0 or lp , 1 ≤ p < ∞. A condition equivalent to K − K = X is that for any x, y ∈ X there exists an upper bound of x and y: an element z ∈ X such that z ≥K x, z ≥K y. Consequently, nonnegative cones are generating in Banach lattices, as in such spaces, the maximum of two elements (with respect to the nonnegative cone) is defined. Banach lattices include Lp , lp , C[a, b] and c0 spaces. When the subspace K − K is dense in X, K is generating whenever K − K is additionally closed; equivalently, when the difference of polar cones to K is closed. Finally, in some cases, measure-theoretic arguments lead to conclusions that int(K − K) is nonempty, under some assumptions on K. We take advantage of such arguments in Example 4.

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On the nondifferentiability of cone-monotone functions in Banach spaces

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In what follows, we assume that the spaces in question are infinitedimensional. The next example involves a nonnegative cone defined through a Hamel basis. Example 2 (lack of continuity, general Banach space). Consider an ordered Hamel basis in X (every element of X is a finite linear combination of elements of such a basis, the latter is necessarily uncountable). Let K be a cone of elements, all of which have nonnegative coordinates. This cone is closed, convex, generating and has empty interior. Define a function f : X → IR by setting f (x) = 1 if the last nonzero coordinate of x is positive, and f (x) = 0 otherwise. Then f is K-monotone: if y − x ∈ K, then the last coordinates where x and y differ must satisfy yα > xα . If the α-th coordinate is the last nonzero one for either x or y, then f (y) ≥ f (x). In the opposite case, the last nonzero coordinates for x and y agree, and f (x) = f (y). In any neighborhood of any point in X, there are points y with f (y) = 0 as well as points z with f (z) = 1. Indeed, if xα is the last nonzero coordinate of x, we can perturb x by assigning a small negative or positive value to some xβ , with β > α. In effect, f is neither lower nor upper semicontinuous. However, for any nonzero x ∈ X, the function t → f (x + tx) is continuous at t = 0 (multiplying the last nonzero coordinate by 1 + t does not change its sign). Moreover, the lower semicontinuous hull of f is the constant function l(x) = 0, and the upper semicontinuous hull of f is the constant function u(x) = 1. Both of these are smooth. The preceding example demonstrated the failure of (c) and (e) of Theorem 1 in a general Banach space. We note that the cone K in this example is a variation on one given by Klee [7], of a pointed convex cone dense in X (pointed means that K ∩ −K = 0). Such a set is obtained by considering all elements of X for which the last nonzero coordinate in a given Hamel basis is positive. In the example below, we use a Schauder basis to construct a conemonotone function violating (c) and (e) of Theorem 1 similarly to Example 2, however continuous at every point in a dense set of directions. Example 3 (lack of continuity but continuity on a dense set of directions). basis of X, and let {x∗j }∞ Let {xi }∞ i=1 be a Schauder i=1 be the associated ∞ ∈ X. We assume that the projections, so that x = i=1 x, x∗i xi for any x  ∞ basis is unconditional, that is, for any x the sum i=1 i x, x∗i xi converges ∞ for any combinations of i = ±1, and consequently, j=1 x, x∗ij xij converges ∞ for any subsequence {ij }j=1 . The standard bases in c0 and lp with p < +∞ satisfy this condition. Define a cone K ⊂ X by K = co(cone{xi }∞ i=1 ), a closed convex hull of the cone generated by xi ’s – equivalently, K = {x | x, x∗i  ≥ 0 for all i = 1, 2, . . .}. As the basis is unconditional, any x can be decomposed into a sum of an element with positive coordinates and an element with negative coordinates, and thus the cone K is generating. Let f : X → IR be given by

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f (x) = lim sup sign x, x∗j + , j→∞

where a+ = max{0, a}. Then f is K-monotone. Indeed, if x ≤K y, that is, y − x ∈ K, then y − x, x∗j  ≥ 0 – equivalently y, x∗j  ≥ x, x∗j  – for all x∗j . This implies that f (x) ≤ f (y). Note that the sets {x | f (x) = 0} and {x | f (x) = 1} are dense in X: we have f (x) = 0for any x ∈ span({xi }∞ i=1 ) whereas f (x) = 1 for any x ∈ ∞ −i span({xi }∞ i=1 ) + i=1 2 xi / xi . As a result, f (x) is nowhere continuous, whereas for any x ∈ X there exists d ∈ X such that f (x + td) is continuous at t = 0. In fact, f (x + txi ) is continuous in t, with f (x + txi ) = f (x) for any t and any xi . In greater generality, f (x + y) = f (x) for any y ∈ span({xi }∞ i=1 ), as for large enough j, we have x + y, x∗j  = x, x∗j . Thus there exists a set D dense in X such that, for every x ∈ X and every d ∈ D, the function t → f (x + td) is continuous. As in Example 2 the lower and upper semicontinuous hulls of f are the constant functions l(x) = 0 and u(x) = 1, respectively, and in particular, they are smooth. We now need to introduce another notion of a null set in a separable Banach space, more general than that of a Gauss null set, described in the comments preceding Theorem 2. A Borel set C ⊂ X is called Haar null if there is a Borel probability measure μ on X such that μ(C + x) = 0 for all x ∈ X. Haar null sets include all Gauss null sets. The nonnegative cone in lp , 1 ≤ p < ∞, is Haar null but not Gauss null (in fact it is not σ-directionally null, a notion weaker than Gauss null), whereas the nonnegative cone in c0 is not Haar null. In the example below, we use a fact that follows from Theorem 6.4 in Benyamini and Lindenstrauss [1]: if a set S is not Haar null, then S − S contains a neighborhood of 0. Example 4 (continuity, but only on a dense subset of a separable and nonreflexive Banach space). Let X be a separable, nonreflexive space, and Y ⊂ X a hyperplane not containing 0. Let C ⊂ Y be a closed convex set, with empty interior and not Haar null with respect to Y . In X, consider K = IR+ C = {rC | r ∈ [0, +∞)}, and note that this set is a closed convex cone: any description of C as {x ∈ X | x, aγ  ≥ bγ , γ ∈ Γ } for aγ ∈ X ∗ , bγ ∈ IR leads to K = {x ∈ X | x, aγ  ≥ bγ , γ ∈ Γ, bγ = 0}. Moreover, K has empty interior and is not Haar null. Indeed, suppose that μ(K) = 0 for some Borel probability measure on X. Then μ (C) = 0 where μ is a Borel probability measure on Y defined by μ (A) = μ(IR+ A), and this contradicts C being non Haar null. Also note that K − K = X, as K − K is a cone, and, since K is not Haar null, K − K is a neighborhood of 0. Define a function f : X → IR by  0 if x ∈ −K, f (x) = 1 if x ∈ −K.

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On the nondifferentiability of cone-monotone functions in Banach spaces

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We check that f is K-monotone. The only way this could fail is if for some y ≥K x, f (y) = 0 and f (x) = 1. But f (y) = 0 states that y ∈ −K, y ≥K x implies x = y − k for some k ∈ K, and since −K is a convex cone, x = y + (−k) ∈ −K + (−K) = −K. Thus x ∈ −K and f (x) cannot equal 1. Thus f is K-monotone. Additionally, as K is closed and convex, the function f is, respectively, lower semicontinuous and quasiconvex (level sets {x | f (x) ≤ r} are closed and convex). Moreover, f is generically continuous by Fort’s theorem (see Borwein, Fitzpatrick and Kenderov [3]). However, for every x ∈ −K, there exists d ∈ X such that t → f (x + td) is not continuous at t = 0. Indeed, suppose that this failed for some x0 ∈ −K. Then for every d ∈ X there exists (d) > 0 so that |t| < (d) implies f (x0 + td) = 0, and so x0 + td ∈ −K. Thus x0 is an absorbing point of a closed convex set −K, and, as X is barelled, x0 ∈ int(−K). But the latter set is empty. To sum up, f is continuous on a dense set but it is not continuous (and not differentiable) at any point of the given non Haar null set. The closed convex set C ⊂ Y in the example above was chosen to be not Haar null and has no interior. Such a set exists in any nonreflexive space, and in fact can be chosen to contain a translate of every compact subset of the space – see Benyamini and Lindenstrauss [1]. In c0 , the nonnegative cone is such a set (this requires the cone to be not Haar null), whereas the Haar null nonnegative cone in l1 is not. Still, in l1 , and in fact in any nonreflexive space, a cone satisfying the mentioned conditions can be found. Indeed, suppose the set C of Example 4 contains translates of all compact subsets of Y . We show that the constructed cone K contains a translate of every compact subset of X. Pick any compact D ⊂ X. Let g ∈ X ∗ be such that Y = g −1 (1). Shift D by z1 so that mind∈D+z1 g(d) = 1, and moreover, so that (D + z1 ) ∩ C = ∅. Pick any v ∈ (D + z1 ) ∩ C, and let E ⊂ Y be the projection of D onto Y in the direction v. Then E is a compact subset of Y , and thus for some z2 ∈ ker g, E + z2 ⊂ C. Now note that E + z2 is exactly the projection in the direction v onto Y of the set D + z1 + z2 , which implies that the latter set is a subset of C + IR+ v. Now C + IR+ v ⊂ K, as C ⊂ K and v ∈ C. In effect, K contains D + z1 + z2 . We now address another question on regularity of cone-monotone functions. Weak and strong notions of lower semicontinuity for convex functions agree. One cannot expect the same to hold for monotone functions, as the following example demonstrates. Example 5 (Lipschitz continuity, but no weak continuity). Let X = c0 with the supremum norm. The nonnegative cone K is closed, convex, has empty interior but is not Haar null. Fix any a ∈ X with a > 0 (a has positive coordinates) and define f : X → IR by f (x) =

x+

x+ . + (a − x)+

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The denominator is never 0, as at least one of the summands is always positive, and thus f is continuous. In fact, x+ + (a − x)+ ≥ a , and since both the numerator and denominator are Lipschitz, so is f . Note also that f (X) = [0, 1], with f (x) = 0 if and only if x ≤ 0, and f (x) = 1 if and only if x ≥ a. We check that f is monotone. For any y ≥ x we have y + ≥ x+ , and (a − x)+ ≥ (a − y)+ , since a − x ≥ a − y. Then y +

y + x+ x+ ≥ + ≥ + , + + + (a − y) x + (a − y) x + (a − x)+

where the first inequality stems from the fact that for a fixed β ≥ 0, the function α → α/(α + β) is nondecreasing. Thus f is monotone. Let {en }∞ n=1 be the standard unit vectors in X. Notice that for any fixed α > 0 and large enough n, we have (x − αen )+ = x+ and (a − x + αen )+ = max{ (a − x)+ , α}. In effect, (x − αen )+ (x − αen )+ + (a − x + αen )+ x+ → + x + max{ (a − x)+ , α}

f (x − αen ) =

as n → ∞. Note that the last expression is less than f (x) whenever x+ > 0 and (a − x)+ < α. Similar analysis leads to f (x + αen ) →

max{ x+ , α} , max{ x+ , α} + (a − x)+

with the limit greater than f (x) when (a − x)+ > 0 and x+ < α. For a given α, the vectors αen converge weakly to 0. The constant α can be fixed arbitrarily large, and thus the function f is not weakly lower semicontinuous at any x with x+ > 0 (equivalently x ∈ −K), and not weakly upper semicontinuous at any x with (a − x)+ > 0 (equivalently x ∈ a + K). Consider any x with xn < 0 for all n. It is easy to verify that lim

t→0

f (x + th) − f (x) =0 t h

for all h ∈ c00 , that is, for sequences h with finitely many nonzero entries (in fact, the difference quotient is then 0 for all small enough t). As c00 is dense in X, and f is Lipschitz continuous, f is Gˆateaux differentiable at x, with ateaux derivative 0 ∈ X ∗ the derivative equal to 0 ∈ X ∗ . Similarly, f has Gˆ at every x such that xn > an for all n. Theorem 2 of Burke, Borwein and Lewis states that functions monotone with respect to a cone with interior are Gˆ ateaux differentiable outside a Gauss null set. In the example below, we show a failure of that conclusion for a cone with empty interior, even in a Hilbert space.

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On the nondifferentiability of cone-monotone functions in Banach spaces

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We first recall that the nonnegative cone in c0 is not Haar null, and so is not Gaussian null, whereas those in lp , 1 < p < ∞ are Haar null but not Gauss null. To see that the nonnegative cone in l2 is not Gauss null, observe for example that it contains the interval J = {x ∈ l2 | 0 ≤ xn ≤ 1/8n , n = 1, 2, . . .} and apply the fact that the closed convex hull of any norm-convergent to 0 sequence with dense span is not Gauss null (Theorem 6.24 in Benyamini and Lindenstrauss). The non Gauss null interval J will be used in the example below. Example 6 (Holder continuity, lack of Gˆ ateaux differentiability). We show that the Holder continuous function  f (x) = x+ , monotone with respect to the nonnegative cone K, fails to be Gˆ ateaux differentiable at any point of −K, both in c0 with the supremum norm and in the Hilbert space l2 with the standard l2 –norm. We discuss the c0 case first. Pick any x ∈ −K. If xn = 0 for some n, ateaux differentiable (the then considering xn + ten shows that f is not Gˆ directional derivative in the direction of en is infinite). Suppose that xn < 0 for all n. Let h be given by hn = (−xn )1/3 , and consider tk = 2(−xk )2/3 converging to 0 as k → ∞. We have √ xk + tk hk f (x + tk h) f (x + tk h) − f (x) = ≥ tk tk tk 1/2 (−xk ) 1 = = , 2(−xk )2/3 2(−xk )1/6 and the last expression diverges to +∞ as k → ∞. Thus f is not differentiable at x. We now turn to l2 , and first show that f fails to be Gˆ ateaux differentiable on the non Gauss null interval −J = {x ∈ l2 | − 1/8n ≤ xn ≤ 0, n = 1, 2, . . .}. Indeed, for any x ∈ −J, consider h with hn = 1/2n and tk = 1/2k . Then  √  −1/8k + 1/4k f (x + tk h) − f (x) xk + tk hk ≥ ≥ = 1 − 1/2k , tk tk 1/2k and, if the function was Gˆ ateaux differentiable at x, the directional derivative in the direction h would be at least 1. But this is impossible, as x provides a global minimum for f .

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To see that f fails to be Gˆ ateaux differentiable at any point x ∈ −K, note that for some sequence ni , we have −1/8i ≤ xni ≤ 0. An argument as above, with hni = 1/2i and 0 otherwise, leads to the desired conclusion. A slight variation on Example 6 leads to a continuous function on c0 , monotone with respect to the nonnegative cone K, but not Gˆ ateaux differentiable at any point of a dense set c00 − K (any point with finite number of positive coefficients). Let {ki }∞ i=1 be dense in K, and define ∞   (x − ki )+ f (x) = . 2i i=1 Monotonicity is easy to check, and f is continuous as (x − ki )+ ≤ x+ . If ateaux at x. Indeed, in such a case we for some i, x ≤K ki , then f is not Gˆ have, for h ≥K 0 and t > 0,   (x − ki + th)+ − (x − ki )+ f (x + th) − f (x) ≥ . t 2i t Picking t and h as in Example 6 (for x − Ki ) leads to the desired conclusion. The close relationship of cone-monotone and Lipschitz functions suggests that badly behaved cone-monotone functions will exist in spaces where irregular Lipschitz functions do. For example, p(x) = lim sup |xn | n→∞

is a nowhere Gˆateaux differentiable continuous seminorm in l∞ , see Phelps [8]. Arguments similar to those given by Phelps show that the function f : l∞ → IR, given by − f (x) = lim sup x+ n − lim sup xn , n→∞

n→∞

though monotone with respect to the nonnegative cone, is not Gˆ ateaux differentiable outside c0 , that is, at any x for which at least one of lim sup x+ n − − and lim sup x− n is positive. (Recall α = −α when α < 0 and α = 0 otherwise.) Indeed, suppose that lim sup x+ n = α > 0, and choose a subsequence nk so that lim xnk = α. Define h by hn2i = 1, hn2i+1 = −1, i = 1, 2, . . ., and hn = 0 for n = nk , k = 1, 2, . . .. Notice that for t close enough to 0, lim sup(x + th)+ n = α + |t| (if t > 0 then (x + th)n2i = xn2i + t, i = 1, 2, . . ., and these terms provide the lim sup(x + th)+ n ; if t < 0 one should consider − (x + th)n2i+1 = xn2i+1 − t). On the other hand, lim sup(x + th)− n = lim sup xn , and in effect, the limit of (f (x + th) − f (x))/t as t → 0 does not exist. The case of lim sup x− n = β > 0 can be treated in a symmetric fashion. Borwein, Burke and Lewis [2] show that any directionally Lipschitz function decomposes (locally) into a linear function, and a cone-monotone one (with respect to a cone with interior). Consequently, nondifferentiable

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On the nondifferentiability of cone-monotone functions in Banach spaces

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Lipschitz functions lead to local examples of nonregular cone-monotone functions. On spaces where there exist nowhere differentiable globally Lipschitz functions, like l∞ or l1 (Γ ) with Γ uncountable, one can in fact construct nowhere Gˆ ateaux cone-monotone functions; we carry this out explicitly in our final example. We note that the technique of Example 7 can be used to construct cone-monotone functions (with respect to cones with nonempty interiors) from any given Lipschitz function, on spaces like c0 and lp . Also note that spaces which admit a nowhere Fr´echet differentiable convex function (spaces which are not Asplund spaces) also admit a nowhere Fr´echet renorm (and so a nowhere Fr´echet globally Lipschitz function); the situation is not well understood for Gˆ ateaux differentiability. Example 7 (a nowhere Gˆ ateaux differentiable function on l∞ ). As discussed above, p(x) = lim supn→∞ |xn | is nowhere Gˆateaux differentiable on l∞ . We use this fact to construct a nowhere Gˆ ateaux differentiable function, monotone with respect to a cone with interior. the

Let e1 be the first of the standard unit vectors in l∞ , and consider function f (x) = p(x) + e1 , x = p(x) + x1 and the cone K = IR+ IB 1/2 (e1 ) (the cone generated by the closed ball of radius 1/2 centered at e1 ). Then K has interior and f is K-monotone. Indeed, as for any x ∈ IB 1/2 (e1 ), x1 ≥ 1/2 while xr ≤ 1/2, r = 2, 3, . . ., we have, for any k ∈ K, k = k1 (for the supremum norm). As p(x) is Lipschitz continuous, with constant 1, we obtain, for any x ∈ X, k ∈ K, p(x + k) − p(x) ≥ − k ≥ − e1 , k, which translates to p(x + k) + e1 , x + k ≥ p(x) + e1 , x, and this means that f is K-monotone. As p is nowhere Gˆateaux differentiable, so is f . Acknowledgments The first author’s research was partially supported by NSERC and by the Canada Research Chair Programme. The second author performed this research at the Centre for Experimental and Constructive Mathematics at Simon Fraser University and at the Department of Mathematics at University of British Columbia.

References 1. Y. Benyamini and J. Lindenstrauss,Geometric Nonlinear Functional Analysis, Vol. 1 (American Mathematical Society, Providence, RI, 2000). 2. J. M. Borwein, J. Burke and A. S. Lewis, Differentiability of cone–monotone functions on separable Banach space, Proc. Amer. Math. Soc., 132 (2004), 1067–1076. 3. J. M. Borwein, S. Fitzpatrick and P. Kenderov, Minimal convex uscos and monotone operators on small sets, Canad. J. Math. 43 (1991), 461–477.

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J. Borwein and R. Goebel

4. J. Burke, A. S. Lewis and M. L. Overton, Approximating subdifferentials by random sampling of gradients, Math. Oper. Res. 27 (2002), 567–584. 5. Y. Chabrillac and J.-P. Crouzeix, Continuity and differentiability properties of monotone real functions of several real variables, Nonlinear Analysis and Optimization (Louvain-la-Neuve, 1983); Math. Programming Stud. 30 (1987), 1–16. 6. R. B. Holmes, Geometric Functional Analysis and its Applications (Springer-Verlag, New York, 1975). 7. V. Klee, Convex sets in linear spaces, Duke Math. J. 8 (1951), 433–466. 8. R. Phelps, Convex Functions, Monotone Operators and Differentiability (SpringerVerlag, New York, 1993). 9. S. Saks, Theory of the Integral, English translation, second edition (Stechert, New York, 1937).

Chapter 2

Duality and a Farkas lemma for integer programs Jean B. Lasserre

Abstract We consider the integer program max{c x | Ax = b, x ∈ Nn }. A formal parallel between linear programming and continuous integration, and discrete summation, shows that a natural duality for integer programs can be derived from the Z-transform and Brion and Vergne’s counting formula. Along the same lines, we also provide a discrete Farkas lemma and show that the existence of a nonnegative integral solution x ∈ Nn to Ax = b can be tested via a linear program. Key words: Integer programming, counting problems, duality

2.1 Introduction In this paper we are interested in a comparison between linear and integer programming, and particularly in a duality perspective. So far, and to the best of our knowledge, the duality results available for integer programs are obtained via the use of subadditive functions as in Wolsey [21], for example, and the smaller class of Chv´ atal and Gomory functions as in Blair and Jeroslow [6], for example (see also Schrijver [19, pp. 346–353]). For more details the interested reader is referred to [1, 6, 19, 21] and the many references therein. However, as subadditive, Chv´ atal and Gomory functions are only defined implicitly from their properties, the resulting dual problems defined in [6] or [21] are conceptual in nature and Gomory functions are used to generate valid inequalities for the primal problem. We claim that another natural duality for integer programs can be derived from the Z-transform (or generating function) associated with the counting Jean B. Lasserre LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse C´edex 4, FRANCE e-mail: [email protected] C. Pearce, E. Hunt (eds.), Structure and Applications, Springer Optimization and Its Applications 32, DOI 10.1007/978-0-387-98096-6 2, c Springer Science+Business Media, LLC 2009 

15

16

J.B. Lasserre

version (defined below) of the integer program. Results for counting problems, notably by Barvinok [4], Barvinok and Pommersheim [5], Khovanskii and Pukhlikov [12], and in particular, Brion and Vergne’s counting formula [7], will prove especially useful. For this purpose, we will consider the four related problems P, Pd , I and Id displayed in the diagram below, in which the integer program Pd appears in the upper right corner. Continuous Optimization Discrete Optimization − | −  x : f (b, c) := max c x P P : f (b, c) := max c d d  Ax = b Ax = b ←→ s.t. s.t. x ∈ Rn+ x ∈ Nn  −  Integration Summation − | −    Id : fd (b, c) := εc x εc x ds I : f (b, c) := x∈Ω(b) Ω(b)   ←→ Ax = b Ax = b Ω(b) := Ω(b) := x ∈ Rn+ x ∈ Nn Problem I (in which ds denotes the Lebesgue measure on the affine variety {x ∈ Rn | Ax = b} that contains the convex polyhedron Ω(b)) is the integration version of the linear program P, whereas Problem Id is the counting version of the (discrete) integer program Pd . Why should these four problems help in analyzing Pd ? Because first, P and I, as well as Pd and Id , are simply related, and in the same manner. Next, as we will see, the nice and complete duality results available for P, I and Id extend in a natural way to Pd .

2.1.1 Preliminaries In fact, I and Id are the respective formal analogues in the algebra (+, ×) of P and Pd in the algebra (⊕, ×), where in the latter, the addition a ⊕ b stands for max(a, b); indeed, the “max” in P and Pd can be seen as an idempotent integral (or Maslov integral) in this algebra (see, for example, Litvinov et al. [17]). For a nice parallel between results in probability ((+, ×) algebra) and optimization ((max, +) algebra), the reader is referred to Bacelli et al. [3, Section 9]. Moreover, P and I, as well as Pd and Id , are simply related via εf (b,c) = lim f (b, rc)1/r ; r→∞

εfd (b,c) = lim f d (b, rc)1/r . r→∞

(2.1)

2

Duality and a Farkas lemma for integer programs

17

Equivalently, by continuity of the logarithm, f (b, c) = lim

r→∞

1 ln f (b, rc); r

fd (b, c) = lim

r→∞

1 ln fd (b, rc), r

(2.2)

a relationship that will be useful later. Next, concerning duality, the standard Legendre-Fenchel transform which yields the usual dual LP of P, P∗ → minm {b λ | A λ ≥ c}, λ∈R

(2.3)

has a natural analogue for integration, the Laplace transform, and thus the inverse Laplace transform problem (that we call I∗ ) is the formal analogue of P∗ and provides a nice duality for integration (although not usually presented in these terms). Finally, the Z-transform is the obvious analogue for summation of the Laplace transform for integration. We will see that in the light of recent results in counting problems, it is possible to establish a nice duality for Id in the same vein as the duality for (continuous) integration and by (2.2), it also provides a powerful tool for analyzing the integer program Pd .

2.1.2 Summary of content (a) We first review the duality principles that are available for P, I and Id and underline the parallels and connections between them. In particular, a fundamental difference between the continuous and discrete cases is that in the former, the data appear as coefficients of the dual variables whereas in the latter, the same data appear as exponents of the dual variables. Consequently, the (discrete) Z-transform has many more poles than the Laplace transform. Whereas the Laplace transform has only real poles, the Z-transform has additional complex poles associated with each real pole, which induces some periodic behavior, a well-known phenomenon in number theory where the Z-transform (or generating function) is a standard tool (see, for example, Iosevich [11], Mitrinov´ıc et al. [18]). So, if the procedure of inverting the Laplace transform or the Z-transform (that is, solving the dual problems I∗ and I∗d ) is basically of the same nature, that is, a complex integral, it is significantly more complicated in the discrete case, due to the presence of these additional complex poles. (b) Then we use results from (a) to analyze the discrete optimization problem Pd . Central to the analysis is Brion and Vergne’s inverse formula [7] for counting problems. In particular, we provide a closed-form expression for the optimal value fd (b, c) which highlights the special role played by the so-called reduced costs of the linear program P and the complex poles of the Z-transform associated with each basis of the linear program P. We also show that each basis B of the linear program P provides exactly det(B)

18

J.B. Lasserre

complex dual vectors in Cm , the complex (periodic) analogues for Pd of the unique dual vector in Rm for P, associated with the basis B. As in linear programming (but in a more complicated way), the optimal value fd (b, c) of Pd can be found by inspection of (certain sums of) reduced costs associated with each vertex of Ω(b). (c) We also provide a discrete Farkas lemma for the existence of nonnegative integral solutions x ∈ Nn to Ax = b. Its form also confirms the special role of the Z-transform described earlier. Moreover, it allows us to check the existence of a nonnegative integral solution by solving a related linear program.

2.2 Duality for the continuous problems P and I With A ∈ Rm×n and b ∈ Rm , let Ω(b)Rn be the convex polyhedron Ω(b) := {x ∈ Rn | Ax = b;

x ≥ 0},

(2.4)

and consider the standard linear program (LP) P:

f (b, c) := max{c x | Ax = b;

x ≥ 0}

with c ∈ Rn , and its associated integration version

 I : f (b, c) := εc x ds

(2.5)

(2.6)

Ω(b)

where ds is the Lebesgue measure on the affine variety {x ∈ Rn | Ax = b} that contains the convex polyhedron Ω(b). For a vector c and a matrix A we denote by c and A their respective transposes. We also use both the notation c x and c, x for the usual scalar product of two vectors c and x. We assume that both A ∈ Rm×n and b ∈ Rm have rational entries.

2.2.1 Duality for P It is well known that the standard duality for (2.5) is obtained from the Legendre-Fenchel transform F (., c) : Rm → R of the value function f (b, c) with respect to b, that is, here (as y → f (y, c) is concave) λ → F (λ, c) := infm λ, y − f (y, c), y∈R

(2.7)

which yields the usual dual LP problem P∗ →

inf λ, b − F (λ, c) = minm {b λ | A λ ≥ c}.

λ∈Rm

λ∈R

(2.8)

2

Duality and a Farkas lemma for integer programs

19

2.2.2 Duality for integration Similarly, the analogue for integration of the Fenchel transform is the twosided Laplace transform F (., c) : Cm → C of f (b, c), given by

λ → F (λ, c) := ε−λ,y f (y, c) dy. (2.9) Rm

It turns out that developing (2.9) yields F (λ, c) =

n  k=1

1 (A λ − c)k

whenever Re(A λ − c) > 0,

(2.10)

(see for example [7, p. 798] or [13]). Thus F (λ, c) is well-defined provided Re(A λ − c) > 0,

(2.11)

and f (b, c) can be computed by solving the inverse Laplace transform problem, which we call the (integration) dual problem I∗ of (2.12), that is, I∗ → f (b, c) :=

1 (2iπ)m

1 = (2iπ)m

γ+i∞

εb,λ F (λ, c) dλ

γ−i∞ γ+i∞

γ−i∞

εb,λ n 

dλ,

(2.12)



(A λ − c)k

k=1

where γ ∈ Rm is fixed and satisfies A γ −c > 0. Incidentally, observe that the domain of definition (2.11) of F (., c) is precisely the interior of the feasible set of the dual problem P∗ in (2.8). We will comment more on this and the link with the logarithmic barrier function for linear programming (see Section 2.2.5 below). We may indeed call I∗ a dual problem of I as it is defined on the space Cm of variables {λk } associated with the nontrivial constraints Ax = b; notice that we also retrieve the standard “ingredients” of the dual optimization problem P∗ , namely b λ and A λ − c.

2.2.3 Comparing P, P∗ and I, I∗ One may compute f (b, c) directly using Cauchy residue techniques. That is, one may compute the integral (2.12) by successive one-dimensional complex integrals with respect to one variable λk at a time (for example starting with λ1 , λ2 , . . .) and by repeated application of Cauchy’s Residue Theorem

20

J.B. Lasserre

[8]. This is possible because the integrand is a rational fraction, and after application of Cauchy’s Residue Theorem at step k with respect to λk , the ouput is still a rational fraction of the remaining variables λk+1 , . . . , λm . For more details the reader is referred to Lasserre and Zeron [13]. It is not difficult to see that the whole procedure is a summation of partial results, each of them ∈ Rm that annihilates m terms of corresponding to a (multi-pole) vector λ n products in the denominator of the integrand. This is formalized in the nice formula of Brion and Vergne [7, Proposition 3.3 p. 820] that we describe below. For the interested reader, there are several other nice closed-form formulae for f (b, c), notably by Barvinok [4], Barvinok and Pommersheim [5], and Khovanskii and Pukhlikov [12].

2.2.4 The continuous Brion and Vergne formula The material in this section is taken from [7]. To explain the closed-form formula of Brion and Vergne we need some notation. Write the matrix A ∈ Rm×n as A = [A1 | . . . |An ] where Aj ∈ Rm denotes the j-th column of A for all j = 1, . . . , n. With Δ := (A1 , . . . , An ) let C(Δ) ⊂ Rm be the closed convex cone generated by Δ. Let Λ ⊆ Zm be a lattice. A subset σ of {1, . . . , n} is called a basis of Δ if the sequence {Aj }j∈σ is a basis of Rm , and the set of bases of Δ is denoted by B(Δ). For σ ∈ B(Δ) let C(σ) be the cone generated by {Aj }j∈σ . With any y ∈ C(Δ) associate the intersection of all cones C(σ) which contain y. This defines a subdivision of C(Δ) into polyhedral cones. The interiors of the maximal cones in this subdivision are called chambers in Brion and Vergne [7]. For every y ∈ γ, the convex polyhedron Ω(y) in (2.4) is simple. Next, for a chamber γ (whose closure is denoted by γ), let B(Δ, γ) be the set of bases σ such that γ is contained in C(σ), and let μ(σ) denote the volume of the convex polytope  m { j∈σ tj Aj | 0 ≤ tj ≤ 1} (normalized so that  vol(R /Λ) = 1). Observe that for b ∈ γ and σ ∈ B(Δ, γ) we have b = j∈σ xj (σ)Aj for some xj (σ) ≥ 0. Therefore the vector x(σ) ∈ Rn+ , with xj (σ) = 0 whenever j ∈ σ, is a vertex of the polytope Ω(b). In linear programming terminology, the bases σ ∈ B(Δ, γ) correspond to the feasible bases of the linear program P. Denote by V the subspace {x ∈ Rn | Ax = 0}. Finally, given σ ∈ B(Δ), let π σ ∈ Rm be the row vector that solves π σ Aj = cj for all j ∈ σ. A vector c ∈ Rn is said to be regular if cj − π σ Aj = 0 for all σ ∈ B(Δ) and all j ∈ σ. Let c ∈ Rn be regular with −c in the interior of the dual cone (Rn+ ∩ V )∗ (which is the case if A u > c for some u ∈ Rm ). Then, with Λ = Zm , Brion and Vergne’s formula [7, Proposition 3.3, p. 820] states that f (b, c) =

 σ∈B(Δ,γ)

μ(σ)



εc,x(σ) σ k ∈σ (−ck + π Ak )

∀ b ∈ γ.

(2.13)

2

Duality and a Farkas lemma for integer programs

21

Notice that in linear programming terminology, ck − π σ Ak is simply the socalled reduced cost of the variable xk , with respect to the basis {Aj }j∈σ . Equivalently, we can rewrite (2.13) as 

f (b, c) = x(σ):

vertex of



μ(σ) Ω(b)

εc,x(σ) . σ k ∈σ (−ck + π Ak )

(2.14)

Thus f (b, c) is a weighted summation over the vertices of Ω(b) whereas f (b, c) is a maximization over the vertices (or a summation with ⊕ ≡ max). So, if c is replaced by rc and x(σ ∗ ) denotes the vertex of Ω(b) at which  c x is maximized, we obtain ⎤ r1

⎡ 

∗

⎥ εrc,x(σ)−x(σ ) ⎥  ⎦ σ n−m μ(σ) r (−c + π A ) k k x(σ):vertex of Ω(b)

⎢ ∗ f (b, rc)1/r = εc,x(σ ) ⎢ ⎣

k ∈σ

from which it easily follows that lim ln f (b, rc)1/r = c, x(σ ∗ ) = max c, x = f (b, c),

r→∞

x∈Ω(b)

as indicated in (2.2).

2.2.5 The logarithmic barrier function It is also worth noticing that f (b, rc) =

1 (2iπ)m

1 = (2iπ)m

γr −i∞

εb,λ

γr +i∞ n 

dλ



(A λ − rc)k

k=1 γ+i∞ m−n rb,λ

γ−i∞

r n 

ε

dλ

(A λ − c)k

k=1

with γr = rγ and we can see that (up to the constant (m − n) ln r) the logarithm of the integrand is simply the well-known logarithmic barrier function −1

λ → φμ (λ, b) = μ

b, λ −

n  j=1

ln (A λ − c)j ,

22

J.B. Lasserre

with parameter μ := 1/r, of the dual problem P∗ (see for example den Hertog [9]). This should not come as a surprise as a self-concordant barrier function of a cone K ⊂ Rn is given by the logarithm of the Laplace transform φK (x)−x,s ε ds of its dual cone K ∗ (see for example G¨ uler [10], Truong and K∗ Tun¸cel [20]). Thus, when r → ∞, minimizing the exponential logarithmic barrier function on its domain in Rm yields the same result as taking its residues.

2.2.6 Summary The parallel between P, P∗ and I, I∗ is summarized below. Fenchel-duality f (b, c) :=

max

Ax=b; x≥0

c x

F (λ, c) := infm {λ y − f (y, c)} y∈R

Laplace-duality

f (b, c) :=



εc x ds

Ax=b; x≥0

F (λ, c) := =

1 n 

 ε−λ y f (y, c) dy

Rm

(A λ − c)k

k=1

with : A λ − c ≥ 0 

f (b, c) = minm {λ b − F (λ, c)} λ∈R 



= minm {b λ | A λ ≥ c} λ∈R

with : Re(A λ − c) > 0 f (b, c) = 1 = (2iπ)m

1 (2iπ)m

Γ

n 

ελ b F (λ, c) dλ 

Γ



ελ b

dλ



(A λ − c)k

k=1

Simplex algorithm → vertices of Ω(b) → max c x over vertices.

Cauchy’ s Residue → poles of F (λ, c)  c x → ε over vertices.

2.3 Duality for the discrete problems Id and Pd In the respective discrete analogues Pd and Id of (2.5) and (2.6) one replaces the positive cone Rn+ by Nn (or Rn+ ∩ Zn ), that is, (2.5) becomes the integer program (2.15) Pd : fd (b, c) := max {c x | Ax = b; x ∈ Nn }

2

Duality and a Farkas lemma for integer programs

23

whereas (2.6) becomes a summation over Nn ∩ Ω(b), that is,   {εc x | Ax = b; x ∈ Nn }. Id : f d (b, c) :=

(2.16)

We here assume that A ∈ Zm×n and b ∈ Zm , which implies in particular that the lattice Λ := A(Zn ) is a sublattice of Zm (Λ ⊆ Zm ). Note that b in (2.15) and (2.16) is necessarily in Λ. In this section we are concerned with what we call the “dual” problem I∗d of Id , the discrete analogue of the dual I∗ of I, and its link with the discrete optimization problem Pd .

2.3.1 The Z-transform The natural discrete analogue of the Laplace transform is the so-called Ztransform. Therefore with f d (b, c) we associate its (two-sided) Z-transform F d (., c) : Cm → C defined by  z −y f d (y, c), (2.17) z → F d (z, c) := y∈Zm ym . Applying this where the notation z y with y ∈ Zm stands for z1y1 · · · zm definition yields  F d (z, c) = z −y f d (y, c) y∈Zm

=



⎡ z −y ⎣

y∈Zm

=



=



εc x ⎦

x∈Nn ; Ax=y

⎡ 

εc x ⎣

x∈Nn



⎤





⎤

−ym ⎦ z1−y1 · · · zm

y=Ax 

−(Ax)1 εc x z 1

−(Ax)m · · · zm

x∈Nn

= =

n  k=1 n  k=1

1 (1 −

εck z1−A1k z2−A2k

−Amk · · · zm )

1 , (1 − εck z −Ak )

(2.18)

which is well-defined provided Amk |z1A1k · · · zm | (= |z Ak |) > εck

∀k = 1, . . . , n.

(2.19)

24

J.B. Lasserre

Observe that the domain of definition (2.19) of F d (., c) is the exponential version of (2.11) for F (., c). Indeed, taking the real part of the logarithm in (2.19) yields (2.11).

2.3.2 The dual problem I∗d Therefore the value f d (b, c) is obtained by solving the inverse Z-transform problem I∗d (that we call the dual of Id )

1 F d (z) z b−em dzm · · · dz1 , · · · (2.20) f d (b, c) = (2iπ)m |z1 |=γ1 |zm |=γm where em is the unit vector of Rm and γ ∈ Rm is a (fixed) vector that Amk > εck for all k = 1, . . . , n. We may indeed call I∗d satisfies γ1A1k γ2A2k · · · γm the dual problem of Id as it is defined on the space Zm of dual variables zk associated with the nontrivial constraints Ax = b of the primal problem Id . We also have the following parallel. Continuous

Laplace-duality  εc x ds f (b, c) :=

Discrete Z-duality   f d (b, c) := εc x

Ax=b; x∈Rn +

F (λ, c) :=

 ε−λ y f (y, c)dy

Ax=b; x∈Nn

F d (z, c) :=

Rm

=

n  k=1

1 (A λ − c)k

with Re(A λ − c) > 0.

=

n  k=1



z −y f d (y, c)

y∈Zm

1 1 − εck z −Ak

with |z Ak | > εck ,

k = 1, . . . , n.

2.3.3 Comparing I∗ and I∗d Observe that the dual problem I∗d in (2.20) is of the same nature as I∗ in (2.12) because both reduce to computing a complex integral whose integrand is a rational function. In particular, as I∗ , the problem I∗d can be solved by Cauchy residue techniques (see for example [14]). However, there is an important difference between I∗ and I∗d . Whereas the data {Ajk } appears in I∗ as coefficients of the dual variables λk in F (λ, c), it now appears as exponents of the dual variables zk in F d (z, c). And an immediate consequence of this fact is that the rational function F d (., c) has many more poles than F (., c) (by considering one variable at a time), and in

2

Duality and a Farkas lemma for integer programs

25

particular, many of them are complex, whereas F (., c) has only real poles. As a result, the integration of F d (z, c) is more complicated than that of F (λ, c), which is reflected in the discrete (or periodic) Brion and Vergne formula described below. However, we will see that the poles of F d (z, c) are simply related to those of F (λ, c).

2.3.4 The “discrete” Brion and Vergne formula Brion and Vergne [7] consider the generating function H : Cm → C defined by  λ → H(λ, c) := f d (y, c)ε−λ,y , y∈Zm

which, after the change of variable zi = ελi for all i = 1, . . . , m, reduces to F d (z, c) in (2.20). They obtain the nice formula (2.21) below. Namely, and with the same notation used in Section 2.2.4, let c ∈ Rn be regular with −c in the interior of (Rn+ ∩V )∗ , and let γ be a chamber. Then for all b ∈ Λ ∩ γ (recall Λ = A(Zn )), f d (b, c) =





σ∈B(Δ,γ)

εc x(σ) Uσ (b, c) μ(σ)

(2.21)

for some coefficients Uσ (b, c) ∈ R, a detailed expression for which can be found in [7, Theorem 3.4, p. 821]. In particular, due to the occurrence of complex the term Uσ (b, c) in (2.21) is the periodic analogue  poles in F (z, c), of ( k ∈σ (ck − πσ Ak ))−1 in (2.14). Again, as for f (b, c), (2.21) can be re-written as 

f d (b, c) = x(σ):



vertex of

εc x(σ) Uσ (b, c), μ(σ)

(2.22)

Ω(b)

to compare with (2.14). To be more precise, by inspection of Brion and Vergne’s formula in [7, p. 821] in our current context, one may see that Uσ (b, c) =

 ε2iπb (g) , Vσ (g, c)

(2.23)

g∈G(σ)

where G(σ) := (⊕j∈σ ZAj )∗ /Λ∗ (where ∗ denotes the dual lattice); it is a 2iπb finite abelian group of order μ(σ) and with  (finitely many) characters ε for all b ∈ Λ. In particular, writing Ak = j∈σ ujk Aj for all k ∈ σ, ε2iπAk (g) = ε2iπ

 j∈σ

ujk gj

k ∈ σ.

26

J.B. Lasserre

Moreover, Vσ (g, c) =



1 − ε−2iπAk (g)εck −π

σ

Ak

 ,

(2.24)

k ∈σ

with Ak , π σ as in (2.13) (and π σ rational). Again note the importance of the reduced costs ck − π σ Ak in the expression for F d (z, c).

2.3.5 The discrete optimization problem Pd We are now in a position to see how I∗d provides some nice information about the optimal value fd (b, c) of the discrete optimization problem Pd . Theorem 1. Let A ∈ Zm×n , b ∈ Zm and let c ∈ Zn be regular with −c in the interior of (Rn+ ∩ V )∗ . Let b ∈ γ ∩ A(Zn ) and let q ∈ N be the least common multiple (l.c.m.) of {μ(σ)}σ∈B(Δ,γ) . If Ax = b has no solution x ∈ Nn then fd (b, c) = −∞, else assume that   1 max c x(σ) + lim ln Uσ (b, rc) , r→∞ r x(σ): vertex of Ω(b) is attained at a unique vertex x(σ) of Ω(b). Then   1  max c x(σ) + lim ln Uσ (b, rc) fd (b, c) = r→∞ r x(σ): vertex of Ω(b)   1 = max c x(σ) + (deg(Pσb ) − deg(Qσb )) q x(σ): vertex of Ω(b) (2.25) for some real-valued univariate polynomials Pσb and Qσb . Moreover, the term limr→∞ ln Uσ (b, rc)/r or (deg(Pσb ) − deg(Qσb ))/q in (2.25) is a sum of certain reduced costs ck − π σ Ak (with k ∈ σ). For a proof see Section 2.6.1. Remark 1. Of course, (2.25) is not easy to obtain but it shows that the optimal value fd (b, c) of Pd is strongly related to the various complex poles of F d (z, c). It is also interesting to note the crucial role played by the reduced costs ck − π σ Ak in linear programming. Indeed, from the proof of Theorem 1 the optimal value fd (b, c) is the value of c x at some vertex x(σ) plus a sum of certain reduced costs (see (2.50) and the form of the coefficients αj (σ, c)). Thus, as for the LP problem P, the optimal value fd (b, c) of Pd can be found by inspection of (certain sums of) reduced costs associated with each vertex of Ω(b).

2

Duality and a Farkas lemma for integer programs

27

We next derive an asymptotic result that relates the respective optimal values fd (b, c) and f (b, c) of Pd and P. Corollary 1. Let A ∈ Zm×n , b ∈ Zm and let c ∈ Rn be regular with −c in the interior of (Rn+ ∩ V )∗ . Let b ∈ γ ∩ Λ and let x∗ ∈ Ω(b) be an optimal vertex of P, that is, f (b, c) = c x∗ = c x(σ ∗ ) for σ ∗ ∈ B(Δ, γ), the unique optimal basis of P. Then for t ∈ N sufficiently large,   1 ln Uσ∗ (tb, rc) . (2.26) fd (tb, c) − f (tb, c) = lim r→∞ r In particular, for t ∈ N sufficently large, the function t → f (tb, c) − fd (tb, c) is periodic (constant) with period μ(σ ∗ ). For a proof see Section 2.6.2. Thus, when b ∈ γ ∩ Λ is sufficiently large, say b = tb0 with b0 ∈ Λ and t ∈ N, the “max” in (2.25) is attained at the unique optimal basis σ ∗ of the LP (2.5) (see details in Section 2.6.2). From Remark 1 it also follows that for sufficiently large t ∈ N, the optimal ∗ value fd (tb, c) is equal to f (tb, c) plus a certain sum of reduced costs ck −π σ Ak ∗ ∗ (with k ∈ σ ) with respect to the optimal basis σ .

2.3.6 A dual comparison of P and Pd We now provide an alternative formulation of Brion and Vergne’s discrete formula (2.22), which explicitly relates dual variables of P and Pd . Recall that a feasible basis of the linear program P is a basis σ ∈ B(Δ) for which A−1 σ b ≥ 0. Thus let σ ∈ B(Δ) be a feasible basis of the linear program P and consider the system of m equations in Cm : A

Amj z1 1j · · · zm = ε cj ,

j ∈ σ.

(2.27)

Recall that Aσ is the nonsingular matrix [Aj1 | · · · |Ajm ], with jk ∈ σ for all k = 1, . . . , m. The above system (2.27) has ρ(σ) (= det(Aσ )) solutions ρ(σ) {z(k)}k=1 , written as z(k) = ελ ε2iπθ(k) ,

k = 1, . . . , ρ(σ)

(2.28)

for ρ(σ) vectors {θ(k)} in Cm . m Indeed, writing z = ελ ε2iπθ (that is, the vector {eλj ε2iπθj }m j=1 in C ) and passing to the logarithm in (2.27) yields Aσ λ + 2iπ Aσ θ = cσ ,

(2.29)

where cσ ∈ Rm is the vector {cj }j∈σ . Thus λ ∈ Rm is the unique solution of Aσ λ = cσ and θ satisfies

28

J.B. Lasserre

Aσ θ ∈ Zm .

(2.30)

Equivalently, θ belongs to (⊕j∈σ Aj Z)∗ , the dual lattice of ⊕j∈σ Aj Z. Thus there is a one-to-one correspondence between the ρ(σ) solutions {θ(k)} and the finite group G (σ) = (⊕j∈σ Aj Z)∗ /Zm , where G(σ) is a subgroup of G (σ). Thus, with G(σ) = {g1 , . . . , gs } and s := μ(σ), we can write (Aσ )−1 gk = θgk = θ(k), so that for every character ε2iπy of G(σ), y ∈ Λ, we have 

y ∈ Λ, g ∈ G(σ)

ε2iπy (g) = ε2iπy θg ,

(2.31)

and 

ε2iπAj (g) = ε2iπAj θg = 1,

j ∈ σ.

(2.32)

So, for every σ ∈ B(Δ), denote by {zg }g∈G(σ) these μ(σ) solutions of (2.28), that is, g ∈ G(σ), (2.33) zg = ελ ε2iπθg ∈ Cm , with λ = (Aσ )−1 cσ , and where ελ ∈ Rm is the vector {ελi }m i=1 . So, in the linear program P we have a dual vector λ ∈ Rm associated with each basis σ. In the integer program P, with each (same) basis σ there are now associated μ(σ) “dual” (complex) vectors λ + 2iπθg , g ∈ G(σ). Hence, with a basis σ in linear programming, the “dual variables” in integer programming are obtained from (a), the corrresponding dual variables λ ∈ Rm in linear programming, and (b), a periodic correction term 2iπθg ∈ Cm , g ∈ G(σ). We next introduce what we call the vertex residue function. Definition 1. Let b ∈ Λ and let c ∈ Rn be regular. Let σ ∈ B(Δ) be a feasible basis of the linear program P and for every r ∈ N, let {zgr }g∈G(σ) be as in (2.33), with rc in lieu of c, that is, zgr = εrλ ε2iπθg ∈ Cm ;

g ∈ G(σ),

with λ = (Aσ )−1 cσ .

The vertex residue function associated with a basis σ of the linear program P is the function Rσ (zg , .) : N → R defined by r → Rσ (zg , r) :=

b  zgr 1  , −Ak rck μ(σ) (1 − zgr ε ) g∈G(σ)

(2.34)

j ∈σ

which is well defined because when c is regular, |zgr |Ak = εrck for all k ∈ σ. The name vertex residue is now clear because in the integration (2.20), Rσ (zg , r) is to be interpreted as a generalized Cauchy residue, with respect to the μ(σ) “poles” {zgr } of the generating function F d (z, rc). Recall from Corollary 1 that when b ∈ γ ∩Λ is sufficiently large, say b = tb0 with b0 ∈ Λ and some large t ∈ N, the “max” in (2.25) is attained at the unique optimal basis σ ∗ of the linear program P.

2

Duality and a Farkas lemma for integer programs

29

Proposition 1. Let c be regular with −c ∈ (Rn+ ∩ V )∗ and let b ∈ γ ∩ Λ be sufficiently large so that the max in (2.25) is attained at the unique optimal basis σ ∗ of the linear program P. Let {zg }g∈G(σ∗ ) be as in (2.33) with σ = σ ∗ . Then the optimal value of Pd satisfies ⎤ ⎡ b  zgr 1 ⎣ 1 ⎦ fd (b, c) = lim ln  −Ak rck r→∞ r μ(σ ∗ ) ) ∗ k ∈σ ∗ (1 − zgr ε g∈G(σ )

1 = lim ln Rσ∗ (zg , r) r→∞ r

(2.35)

and the optimal value of P satisfies ⎡ ⎤ b  1 1 |zgr | ⎦  f (b, c) = lim ln ⎣ r→∞ r μ(σ ∗ ) (1 − |zgr |−Ak εrck ) ∗ k ∈σ ∗ g∈G(σ )

= lim

r→∞

1 ln Rσ∗ (|zg |, r). r

(2.36)

For a proof see Section 2.6.3. Proposition 1 shows that there is indeed a strong relationship between the integer program Pd and its continuous analogue, the linear program P. Both optimal values obey exactly the same formula (2.35), but for the continuous version, the complex vector zg ∈ Cm is replaced by the vector ∗ |zg | = ελ ∈ Rm of its component moduli, where λ∗ ∈ Rm is the optimal solution of the LP dual of P. In summary, when c ∈ Rn is regular and b ∈ γ ∩ Λ is sufficiently large, we have the following correspondence. Linear program P

Integer program Pd

unique optimal basis σ ∗

unique optimal basis σ ∗

1 optimal dual vector λ∗ ∈ Rm

μ(σ ∗ ) dual vectors zg ∈ Cm , g ∈ G(σ ∗ ) ln zg = λ∗ + 2iπθg

f (b, c) = lim

1

r→∞ r

ln Rσ∗ (|zg |, r)

fd (b, c) = lim

1

r→∞ r

ln Rσ∗ (zg , r)

2.4 A discrete Farkas lemma In this section we are interested in a discrete analogue of the continuous Farkas lemma. That is, with A ∈ Zm×n and b ∈ Zm , consider the issue of the existence of a nonnegative integral solution x ∈ Nn to the system of linear equations Ax = b .

30

J.B. Lasserre

The (continuous) Farkas lemma, which states that given A ∈ Rm×n and b ∈ Rm , {x ∈ Rn | Ax = b, x ≥ 0} = ∅ ⇔ [A λ ≥ 0] ⇒ b λ ≥ 0,

(2.37)

has no discrete analogue in an explicit form. For instance, the Gomory functions used in Blair and Jeroslow [6] (see also Schrijver [19, Corollary 23.4b]) are implicitly and iteratively defined, and are not directly defined in terms of the data A, b. On the other hand, for various characterizations of feasibility of the linear diophantine equations Ax = b, where x ∈ Zn , the interested reader is referred to Schrijver [19, Section 4]. Before proceeding to the general case when A ∈ Zm×n , we first consider the case A ∈ Nm×n , where A (and thus b) has only nonnegative entries.

2.4.1 The case when A ∈ Nm×n In this section we assume that A ∈ Nm×n and thus necessarily b ∈ Nm , since otherwise {x ∈ Nn | Ax = b} = ∅. Theorem 2. Let A ∈ Nm×n and b ∈ Nm . Then the following two propositions (i) and (ii) are equivalent: (i) The linear system Ax = b has a solution x ∈ Nn . bm − 1 can be written (ii) The real-valued polynomial z → z b − 1 := z1b1 · · · zm zb − 1 =

n 

Qj (z)(z Aj − 1),

(2.38)

j=1

for some real-valued polynomials Qj ∈ R[z1 , . . . , zm ], j = 1, . . . , n, all of which have nonnegative coefficients. In addition, the degree of the Qj in (2.38) is bounded by b∗ :=

m  j=1

bj − min k

m 

Ajk .

(2.39)

j=1

For a proof see Section 2.6.4. Hence Theorem 2 reduces the issue of existence of a solution x ∈ Nn to a particular ideal membership problem, that is, Ax = b has a solution x ∈ Nn if and only if the polynomial z b − 1 belongs to the binomial ideal I = z Aj − 1j=1,...,n ⊂ R[z1 , . . . , zm ] for some weights Qj with nonnegative coefficients. Interestingly, consider the ideal J ⊂ R[z1 , . . . , zm , y1 , . . . , yn ] generated by obner basis of J. Using the binomials z Aj − yj , j = 1, . . . , n, and let G be a Gr¨ the algebraic approach described in Adams and Loustaunau [2, Section 2.8], it is known that Ax = b has a solution x ∈ Nn if and only if the monomial

2

Duality and a Farkas lemma for integer programs

31

z b can be reduced (with respect to G) to some monomial y α , in which case, α ∈ Nn is a feasible solution. Observe that in this case, we do not know α ∈ Nn in advance (we look for it!) to test whether z b − y α ∈ J. One has to apply Buchberger’s algorithm to (i) find a reduced Gr¨ obner basis G of J, and (ii) reduce z b with respect to G and check whether the final result is a monomial y α . Moreover, in the latter approach one uses polynomials in n + m (primal) variables y and (dual) variables z, in contrast with the (only) m dual variables z in Theorem 2. ∗

the dimension Remark 2. (a) With b∗ as in (2.39) denote by s(b∗ ) := m+b b∗ of the vector space of polynomials of degree b∗ in m variables. In view of Theorem 2, and given b ∈ Nm , checking the existence of a solution x ∈ Nn to Ax = b reduces to checking whether or not there exists a nonnegative solution y to a system of linear equations with: • n × s(b∗ ) variables, the nonnegative coefficients of the Qj ; n  • s(b∗ + max Ajk ) equations to identify the terms of the same powers on k

j=1

both sides of (2.38). This in turn  reduces to solving an LP problem with ns(b∗ ) variables and s(b∗ + maxk j Ajk ) equality constraints. Observe that in view of (2.38), this LP has a matrix of constraints with coefficients made up only of 0’s and ±1’s. (b) From the proof of Theorem 2 in Section 2.6.4, it easily follows that one may even constrain the weights Qj in (2.38) to be polynomials in Z[z1 , . . . , zm ] (instead of R[z1 , . . . , zm ]) with nonnegative coefficients. However, (a) shows that the strength of Theorem 2 is precisely allowing Qj ∈ R[z1 , . . . , zm ] while enabling us to check feasibility by solving a (continuous) linear program. By enforcing Qj ∈ Z[z1 , . . . , zm ] one would end up with an integer linear system whose size was larger than that of the original problem.

2.4.2 The general case In this section we consider the general case where A ∈ Zm×n so that A may have negative entries, and we assume that the convex polyhedron Ω := {x ∈ Rn+ | Ax = b} is compact. The above arguments cannot be repeated because of the occurrence of negative powers. However, let α ∈ Nn and β ∈ N be such that jk := Ajk + αk ≥ 0, A

k = 1, . . . , n;

bj := bj + β ≥ 0,

for all j = 1, . . . , m. Moreover, as Ω is compact, we have that

(2.40)

32

J.B. Lasserre

max

x∈Nn

⎧ n ⎨ ⎩

αj xj | Ax = b

j=1

⎫ ⎬ ⎭

≤ maxn

x∈R+

⎧ n ⎨ ⎩

αj xj | Ax = b

j=1

⎫ ⎬ ⎭

=: ρ∗ (α) < ∞. (2.41)

∗

Observe that given α ∈ N , the scalar ρ (α) is easily calculated by solving ∈ Nm×n and b ∈ Nm be an LP problem. Choose N  β ≥ ρ∗ (α), and let A as in (2.40). Then the existence of solutions x ∈ Nn to Ax = b is equivalent to the existence of solutions (x, u) ∈ Nn ×N to the system of linear equations ⎧ ⎪ ⎨ n Ax + uem = b Q  (2.42) αj xj + u = β. ⎪ ⎩ n

j=1

Indeed, if Ax = b with x ∈ Nn then Ax + em

n 

αj xj − em

j=1

or equivalently,

n 

αj xj = b + em β − em β,

j=1

⎛ + ⎝β − Ax

n 

⎞ αj xj ⎠ em = b,

j=1

n

and thus, as β ≥ ρ∗ (α) ≥ j=1 αj xj (see, for example, (2.41)), we see that n (x, u) with β − j=1 αj xj =: u ∈ N is a solution of (2.42). Conversely, let and b, it (x, u) ∈ Nn × N be a solution of (2.42). Using the definitions of A then follows immediately that Ax + em

n  j=1

αj xj + uem = b + βem ;

n 

αj xj + u = β,

j=1

so that Ax = b. The system of linear equations (2.42) can be cast in the form ⎤ ⎡     | em A b x (2.43) B = with B := ⎣ − − ⎦ , u β α | 1 and as B only has entries in N, we are back to the case analyzed in Section 2.4.1. Corollary 2. Let A ∈ Zm×n and b ∈ Zm and assume that Ω := {x ∈ Rn+ | Ax = b} is compact. Let α ∈ Nn and β ∈ N be as in (2.40) with β ≥ ρ∗ (α) (see, for example, (2.41)). Then the following two propositions (i) and (ii) are equivalent: (i) The system of linear equations Ax = b has a solution x ∈ Nn ;

2

Duality and a Farkas lemma for integer programs

33

(ii) The real-valued polynomial z → z b (zy)β − 1 ∈ R[z1 , . . . , zm , y] can be written z b (zy)β − 1 = Q0 (z, y)(zy − 1) +

n 

Qj (z, y)(z Aj (zy)αj − 1)

(2.44)

j=1

for some real-valued polynomials {Qj }nj=0 in R[z1 , . . . , zm , y], all of which have nonnegative coefficients. The degree of the Qj in (2.44) is bounded by ⎡ ⎤⎤ ⎡ m m   (m + 1)β + bj − min ⎣m + 1, min ⎣(m + 1)αk + Ajk ⎦⎦ . k=1,...,n

j=1

j=1

∈ Nm×n , b ∈ Nm , α ∈ Nn and β ∈ N be as in (2.40) with Proof. Let A ∗ β ≥ ρ (α). Then apply Theorem 2 to the equivalent form (2.43) of the system Q in (2.42), where B and ( b, β) only have entries in N, and use the definitions and b. of A  Indeed Theorem 2 and Corollary 2 have the flavor of a Farkas lemma as it is stated with the transpose A of A and involving the dual variables zk associated with the constraints Ax = b. In addition, and as expected, it implies the continuous Farkas lemma because if {x ∈ Nn | Ax = b} = ∅, then from (2.44), and with z := ελ and y := (z1 · · · zm )−1 , 

εb λ − 1 =

m 

Qj (eλ1 , . . . eλm , e−

 i

λi



)(ε(A λ)j − 1).

(2.45)

j=1 

Therefore A λ ≥ 0 ⇒ ε(A λ)j − 1 ≥ 0 for all j = 1, . . . , n, and as the Qj have  nonnegative coefficients, we have eb λ − 1 ≥ 0, which in turn implies b λ ≥ 0. Equivalently, evaluating the partial derivatives n of both sides of (2.45) with respect to λj , at the point λ = 0, yields bj = k=1 Ajk xk for all j = 1, . . . , n, with xk := Qk (1, . . . , 1) ≥ 0. Thus Ax = b for some x ∈ Rn+ .

2.5 Conclusion We have proposed what we think is a natural duality framework for the integer program Pd . It essentially relies on the Z-transform of the associated counting problem Id , for which the important Brion and Vergne inverse formula appears to be an important tool for analyzing Pd . In particular, it shows that the usual reduced costs in linear programming, combined with the periodicities phenomena associated with the complex poles of F d (z, c), also play an essential role for analyzing Pd . Moreover, for the standard dual

34

J.B. Lasserre

vector λ ∈ Rm associated with each basis B of the linear program P, there are det(B) corresponding dual vectors z ∈ Cm for the discrete problem Pd . Moreover, for b sufficiently large, the optimal value of Pd is a function of these dual vectors associated with the optimal basis of the linear program P. A topic of further research is to establish an explicit dual optimization problem P∗d in these dual variables. We hope that the above results will stimulate further research in this direction.

2.6 Proofs A proof in French of Theorem 1 can be found in Lasserre [15]. The English proof in [16] is reproduced below.

2.6.1 Proof of Theorem 1 Proof. Use (2.1) and (2.22) to obtain ⎡



εfd (b,c) = lim ⎣ r→∞

x(σ):

⎡

vertex of

x(σ):

⎡

rc x(σ)

μ(σ) Ω(b)



x(σ):

⎤1/r Uσ (b, rc)⎦

Ω(b)

ε

vertex of

= lim ⎣ r→∞

μ(σ)



= lim ⎣ r→∞

ε

rc x(σ)

2iπb

g∈G(σ)

ε (g) ⎦ Vσ (g, rc)

⎤1/r Hσ (b, rc)⎦

vertex of

⎤1/r



.

(2.46)

Ω(b)

Next, from the expression of Vσ (b, c) in (2.24), and with rc in lieu of c, we see that Vσ (g, rc) is a function of y := er , which in turn implies that Hσ (b, rc) is also a function of εr , of the form 

Hσ (b, rc) = (εr )c x(σ)

 g∈G(σ)

 j

ε2iπb (g)

, δj (σ, g, A) × (er )αj (σ,c)

(2.47)

for finitely many coefficients {δj (σ, g, A), αj (σ, c)}. Note that the coefficients αj (σ, c) are sums of some reduced costs ck − π σ Ak (with k ∈ σ). In addition, the (complex) coefficients {δj (σ, g, A)} do not depend on b. Let y := εr/q , where q is the l.c.m. of {μ(σ)}σ∈B(Δ,γ) . As q(ck −π σ Ak ) ∈ Z for all k ∈ σ,  Pσb (y) (2.48) Hσ (b, rc) = y qc x(σ) × Qσb (y)

2

Duality and a Farkas lemma for integer programs

35

for some polynomials Pσb , Qσb ∈ R[y]. In view of (2.47), the degree of Pσb and Qσb , which depends on b but not on the magnitude of b, is uniformly bounded in b. Therefore, as r → ∞, 

Hσ (b, rc) ≈ (εr/q )qc x(σ)+deg(Pσb )−deg(Qσb ) ,

(2.49)



so that the limit in (2.46), which is given by max εc x(σ) lim Uσ (b, rc)1/r (as σ

r→∞

we have assumed unicity of the maximizer σ), is also max x(σ): vertex of



εc x(σ)+(deg(Pσb )−deg(Qσb ))/q . Ω(b)

Therefore fd (b, c) = −∞ if Ax = b has no solution x ∈ Nn , else   1  max c x(σ) + (deg(Pσb ) − deg(Qσb )) , (2.50) fd (b, c) = q x(σ): vertex of Ω(b) from which (2.25) follows easily.

2.6.2 Proof of Corollary 1 Proof. Let t ∈ N and note that f (tb, rc) = trf (b, c) = trc x∗ = trc x(σ ∗ ). As in the proof of Theorem 1, and with tb in lieu of b, we have ⎡ U (tb, rc) + f d (tb, rc) = εtc x ⎣ μ(σ ∗ ) 1 r



%



σ∗

∗

vertex x(σ) =x∗

rc x(σ)

ε εrc x(σ∗ )

&t

⎤ r1 Uσ (tb, rc) ⎦ μ(σ)

and from (2.47)–(2.48), setting δσ := c x∗ − c x(σ) > 0 and y := εr/q , ⎡ f d (tb, rc)1/r = εtc x 

∗

∗ ⎣ Uσ (tb, rc) + μ(σ ∗ )



⎤1/r P (y) σtb ⎦ . y −tqδσ Q σtb (y) ∗

vertex x(σ) =x

Observe that c x(σ ∗ )−c x(σ) > 0 whenever σ = σ ∗ because Ω(y) is simple if y ∈ γ, and c is regular. Indeed, as x∗ is an optimal vertex of the LP problem ∗ P, the reduced costs ck − π σ Ak (k ∈ σ ∗ ) with respect to the optimal basis σ ∗ are all nonpositive, and in fact, strictly negative because c is regular (see Section 2.2.4). Therefore the term  vertex x(σ) =x∗

y −tqδσ

Pσtb (y) Qσtb (y)

36

J.B. Lasserre

is negligible for t sufficiently large, when compared with Uσ∗ (tb, rc). This is because the degrees of Pσtb and Qσtb depend on tb but not on the magnitude of tb (see (2.47)–(2.48)), and they are uniformly bounded in tb. Hence taking the limit as r → ∞ yields % ε

fd (tb,c)

= lim

r→∞

&1/r  ∗  ∗ εrtc x(σ ) Uσ∗ (tb, rc) = εtc x(σ ) lim Uσ∗ (tb, rc)1/r , r→∞ μ(σ ∗ )

from which (2.26) follows easily. Finally, the periodicity comes from the term ε2iπtb (g) in (2.23) for g ∈  G(σ ∗ ). The period is then, of the order G(σ ∗ ).

2.6.3 Proof of Proposition 3.1 ∗

Proof. Let Uσ∗ (b, c) be as in (2.23)–(2.24). It follows immediately that π σ = (λ∗ ) and so ε−π

σ∗

Ak −2iπAk

ε



∗



(g) = ε−Ak λ ε−2iπAk θg = zg−Ak ,

g ∈ G(σ ∗ ).

Next, using c x(σ ∗ ) = b λ∗ , 

∗



∗



εc x(σ ) ε2iπb (g) = εb λ ε2iπb θg = zgb ,

g ∈ G(σ ∗ ).

Therefore  1 1 εc x(σ) Uσ∗ (b, c) = ∗ μ(σ ) μ(σ ∗ )



zgb

g∈G(σ ∗ )

(1 − zg−Ak εck )

= Rσ∗ (zg , 1), and (2.35) follows from (2.25) because, with rc in lieu of c, zg becomes zgr = ∗ εrλ ε2iπθg (only the modulus changes). ∗ Next, as only the modulus of zg is involved in (2.36), we have |zgr | = εrλ for all g ∈ G(σ ∗ ), so that 1 μ(σ ∗ )

 g∈G(σ ∗ )





∗

εrb λ |zgr |b , =  −A rc r(ck −Ak λ∗ ) ) kε k) k ∈σ ∗ (1 − |zgr | k ∈σ ∗ (1 − ε

and, as r → ∞ ,





∗

 ∗ εrb λ ≈ εrb λ , r(ck −Ak λ∗ ) ) (1 − ε k ∈σ ∗

because (ck − Ak λ∗ ) < 0 for all k ∈ σ ∗ . Therefore

2

Duality and a Farkas lemma for integer programs

%



∗

εrb λ 1 ln  r(ck −Ak λ∗ ) r→∞ r ) k ∈σ ∗ (1 − ε

37

&

lim

= b λ∗ = f (b, c),

the desired result.

2.6.4 Proof of Theorem 2 Proof. (ii) ⇒ (i). Assume that z b − 1 can be written as in (2.38) for some coefficients {Qjα }, that is, {Qj(z) } = polynomials {Qj } with  nonnegative  α1 α αm Q z = Q z · · · zm , for finitely many nonzero (and m m jα jα 1 α∈N α∈N nonnegative) coefficients Qjα . Using the notation of Section 2.3, the function f d (b, 0), which (as c = 0) counts the nonnegative integral solutions x ∈ Nn to the equation Ax = b, is given by

z b−em 1 n dz, · · · f d (b, 0) = m −Ak ) (2πi) |z1 |=γ1 |zm |=γm j=1 (1 − z where γ ∈ Rm satisfies A γ > 0 (see (2.18) and (2.20)). Writing z b−em as z −em (z b − 1 + 1) we obtain f d (b, 0) = B1 + B2 , with B1 =

1 (2πi)m

|z1 |=γ1

···

|zm |=γm

z −em dz −Ak ) j=1 (1 − z

n

and 1 B2 := (2πi)m =

=

n  j=1 n 



|z1 |=γ1

1 (2πi)m 

j=1 α∈Nm

···

|zm |=γm



|z1 |=γ1

Qjα (2πi)m

···

z −em (z b − 1) n dz −Ak ) j=1 (1 − z

|zm |=γm

z Aj −em Qj (z)  dz −Ak ) k =j (1 − z



|z1 |=γ1

···

|zm |=γm



z Aj +α−em dz. −Ak ) k =j (1 − z

From (2.20) (with b := 0) we recognize in B1 the number of solutions x ∈ Nn to the linear system Ax = 0, so that B1 = 1. Next, again from (2.20) (now with b := Aj + α), each term Cjα

Qjα := (2πi)m



|z1 |=γ1

···

|zm |=γm



z Aj +α−em dz −Ak ) k =j (1 − z

38

J.B. Lasserre

is equal to Qjα × the number of integral solutions x ∈ Nn−1 (j) x = Aj + α, where A (j) is the matrix in Nm×(n−1) of the linear system A obtained from A by deleting its j-th column Aj . As by hypothesis each Qjα is nonnegative, it follows that B2 =

n  

Cjα ≥ 0,

j=1 α∈Nm

so that f d (b, 0) = B1 + B2 ≥ 1. In other words, the sytem Ax = b has at least one solution x ∈ Nn . (i) ⇒ (ii). Let x ∈ Nn be a solution of Ax = b, and write z b − 1 = z A1 x1 − 1 + z A1 x1 (z A2 x2 − 1) + · · · + z and

 n−1 j=1

Aj xj

( ' z Aj xj − 1 = (z Aj − 1) 1 + z Aj + · · · + z Aj (xj −1) ,

(z An xn − 1)

j = 1, . . . , n,

to obtain (2.38) with z → Qj (z) := z

 j−1 k=1

Ak xk

'

( 1 + z Aj + · · · + z Aj (xj −1) ,

j = 1, . . . , n.

We immediately see that each Qj has all its coefficients nonnegative (and even in {0, 1}). Finally, the bound on the degree follows immediately from the proof for (i) ⇒ (ii). 

References 1. K. Aardal, R. Weismantel and L. A. Wolsey, Non-standard approaches to integer programming, Discrete Appl. Math. 123 (2002), 5–74. 2. W. W. Adams and P. Loustaunau, An Introduction to Gr¨ obner Bases (American Mathematical Society, Providence, RI, 1994). 3. F. Bacelli, G. Cohen, G. J. Olsder and J.-P. Quadrat, Synchronization and Linearity (John Wiley & Sons, Chichester, 1992). 4. A. I. Barvinok, Computing the volume, counting integral points and exponential sums, Discrete Comp. Geom. 10 (1993), 123–141. 5. A. I. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, in New Perspectives in Algebraic Combinatorics, MSRI Publications 38 (1999), 91–147. 6. C. E. Blair and R. G. Jeroslow, The value function of an integer program, Math. Programming 23 (1982), 237–273.

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Duality and a Farkas lemma for integer programs

39

7. M. Brion and M. Vergne, Residue formulae, vector partition functions and lattice points in rational polytopes, J. Amer. Math. Soc. 10 (1997), 797–833. 8. J. B. Conway, Functions of a Complex Variable I, 2nd ed. (Springer, New York, 1978). 9. D. den Hertog, Interior Point Approach to Linear, Quadratic and Convex Programming (Kluwer Academic Publishers, Dordrecht, 1994). 10. O. G¨ uler, Barrier functions in interior point methods, Math. Oper. Res. 21 (1996), 860–885. 11. A. Iosevich, Curvature, combinatorics, and the Fourier transform, Notices Amer. Math. Soc. 48 (2001), 577–583. 12. A. Khovanskii and A. Pukhlikov, A Riemann-Roch theorem for integrals and sums of quasipolynomials over virtual polytopes, St. Petersburg Math. J. 4 (1993), 789–812. 13. J. B. Lasserre and E. S. Zeron, A Laplace transform algorithm for the volume of a convex polytope, JACM 48 (2001), 1126–1140. 14. J. B. Lasserre and E. S. Zeron, An alternative algorithm for counting integral points in a convex polytope, Math. Oper. Res. 30 (2005), 597–614. 15. J. B. Lasserre, La valeur optimale des programmes entiers, C. R. Acad. Sci. Paris Ser. I Math. 335 (2002), 863–866. 16. J. B. Lasserre, Generating functions and duality for integer programs, Discrete Optim. 1 (2004), 167–187. 17. G. L. Litvinov, V. P. Maslov and G. B. Shpiz, Linear functionals on idempotent spaces: An algebraic approach, Dokl. Akad. Nauk. 58 (1998), 389–391. 18. D. S. Mitrinovi´c, J. S´ andor and B. Crstici, Handbook of Number Theory (Kluwer Academic Publishers, Dordrecht, 1996). 19. A. Schrijver, Theory of Linear and Integer Programming (John Wiley & Sons, Chichester, 1986). 20. V. A. Truong and L. Tun¸cel, Geometry of homogeneous convex cones, duality mapping, and optimal self-concordant barriers, Research report COOR #2002-15 (2002), University of Waterloo, Waterloo, Canada. 21. L. A. Wolsey, Integer programming duality: Price functions and sensitivity analysis, Math. Programming 20 (1981), 173–195.

Chapter 3

Some nonlinear Lagrange and penalty functions for problems with a single constraint J. S. Giri and A. M. Rubinov†

Abstract We study connections between generalized Lagrangians and generalized penalty functions, which are formed by increasing positively homogeneous functions. In particular we show that the least exact penalty parameter is equal to the least Lagrange multiplier. We also prove, under some natural assumptions, that the natural generalization of a Lagrangian cannot improve it. Key words: Generalized Lagrangians, generalized penalty functions, single constraint, IPR convolutions, IPH functions

3.1 Introduction Consider the following constrained optimization problem P (f0 , f1 ): min f0 (x) subject to x ∈ X, f1 (x) ≤ 0,

(3.1)

where X ⊆ IRn and f0 (x), f1 (x) are real-valued, continuous functions. (We shall assume that these functions are directionally differentiable in Section 3.4.) Note that a general mathematical programming problem: min f0 (x) subject to x ∈ X, gi (x) ≤ 0, (i ∈ I),

hj (x) = 0 (j ∈ J),

J. S. Giri School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria, AUSTRALIA e-mail: [email protected] A. M. Rubinov† School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria, AUSTRALIA e-mail: [email protected] C. Pearce, E. Hunt (eds.), Structure and Applications, Springer Optimization and Its Applications 32, DOI 10.1007/978-0-387-98096-6 3, c Springer Science+Business Media, LLC 2009 

41

42

J.S. Giri and A.M. Rubinov

where I and J are finite sets, can be reformulated as (3.1) with f1 (x) = max(max gi (x), max |hj (x)|). i∈I

j∈J

(3.2)

Note that the function f1 defined by (3.2) is directionally differentiable if the functions gi (i ∈ I) and hj (j ∈ J) possess this property. The traditional approach to problems of this type has been to employ a Lagrange function of the form L(x; λ) = f0 (x) + λf1 (x). The function q(λ) = inf x∈X (f0 (x) + λf1 (x)) is called the dual function and the problem max q(λ) subject to λ > 0 is called the dual problem. The equality sup inf L(§, λ) = inf{{ (§) : § ∈ X , {∞ (§) ≤ } λ>0 x∈X

¯ > 0 such that is called the zero duality gap property. The number λ ¯ = inf{f0 (x) : x ∈ X, f1 (x) ≤ 0} inf L(x, λ)

x∈X

is called the Lagrange multiplier. Let f1+ (x) = max(f1 (x), 0). Then the Lagrange function for the problem P (f0 , f1+ ) is called the penalty function for the initial problem P (f0 , f1 ). The traditional Lagrange function may be considered to be a linear convolution of the objective and constraint functions. That is, L(x; λ) ≡ p(f0 (x), λf1 (x)), where p(u, v) = u + v. It has been shown in [4, 5] that for penalty functions, increasing positively homogeneous (IPH) convolutions provide exact penalization for a large class of objective functions. The question thus arises “are there nonlinear convolution functions for which Lagrange multipliers exist?” The most interesting example of a nonlinear IPH convolution func 1/k . These convolutions also oftion is the function sk (u, v) = uk + v k ten provide a smaller exact penalty parameter than does the traditional linear convolution. (See Section 3.3 for the definition of an exact penalty parameter.) We will show in this chapter that for problems where a Lagrange multiplier exists, an exact penalty parameter also exists, and the smallest exact penalty parameter is equal to the smallest Lagrange multiplier. We also show that whereas a generalized penalty function can often improve the classical situation (for example, provide exact penalization with a smaller parameter than that of the traditional function), this is not true

3

Some nonlinear Lagrange and penalty functions

43

for generalized Lagrange functions. Namely, we prove, under some natural assumptions, that among all functions sk the Lagrange multiplier may exist only for the k = 1 case. So generalized Lagrangians cannot improve the classical situation.

3.2 Preliminaries Let us present some results and definitions which we will make use of later in this chapter. We will refer to the solution of the general problem, P (f0 , f1 ), as M (f0 , f1 ). We will also make use of the sets, X0 = {x ∈ X : f1 (x) ≤ 0} and X1 = {x ∈ X : f1 (x) > 0}. It will be convenient to talk about Increasing Positively Homogeneous (IPH) functions. These are defined as functions which are increasing, that is, if (δ, γ) ≥ (δ  , γ  ) then p(δ, γ) ≥ p(δ  , γ  ), and positively homogeneous of the first degree, that is, p(α(δ, γ)) = αp(δ, γ), α > 0 . We shall consider only continuous IPH functions defined on either the halfplane {(u, v) : u ≥ 0} or on the quadrant IR2+ = {(u, v) ∈ IR2 : u ≥ 0, v ≥ 0}. In the latter case we consider only IPH functions p : IR2++ → IR, which possess the following properties: p(1, 0) = 1,

lim p(1, v) = +∞.

v→+∞

We shall denote by P1 the class of all such functions. The simplest example of functions from P1 is the following function sk (0 < k < +∞), defined on IR2+ :

1 (3.3) sk (u, v) = uk + v k k . 2l + 1 with k, l ∈ N then the function sk is well defined and IPH 2m + 1 on the half-plane {(u, v) : u ≥ 0}. (Here N is the set of positive integers.) A perturbation function plays an important part in the study of extended penalty functions and is defined on IR+ = {y ∈ IR : y ≥ 0} by If k =

β(y) = inf{f0 (x) : x ∈ X, f1 (x) ≤ y}. We denote by CX the set of all problems (f0 , f1 ) such that: 1. inf x∈X f0 (x) > 0; 2. there exists a sequence xk ∈ X1 such that f1 (xk ) → 0 and f0 (xk ) → M (f0 , f1 ); 3. there exists a point x ∈ X such that f1 (x) ≤ 0; 4. the perturbation function of the problem (f0 , f1 ) is l.s.c. at the point y = 0.

44

J.S. Giri and A.M. Rubinov

An important result which follows from the study of perturbation functions is as follows. Theorem 1. Let P (f0 , f1 ) ∈ CX . Let k > 0 and let p = pk be defined as 1 ¯ = pk (δ, γ) = (δ k + γ k ) k . There exists a number d¯ > 0 such that qp+ (d) M (f0 , f1 ) if and only if β is calm of degree k at the origin. That is, lim inf

y→+0

β(y) − β(0) > −∞. yk

A proof of this is presented in [4] and [5].

3.3 The relationship between extended penalty functions and extended Lagrange functions Let (f0 , f1 ) ∈ CX and let p be an IPH function defined on the half-plane IR2∗ = {(u, v) : u ≥ 0}. Recall the following definitions. The Lagrange-type function with respect to p is defined by Lp (x, d) = p(f0 (x), df1 (x)). (Here d is a real number and df does not mean the differential of f .) The dual function qp (d) with respect to p is defined by qp (d) = inf p(f0 (x), df1 (x)), x∈X

d > 0.

Let p+ be the restriction of p to IR2+ . Consider the penalty function L+ p and the dual function qp+ corresponding to p+ : + + L+ p (x, d) = p (f0 (x), df1 (x)), (x ∈ X, d ≥ 0),

qp+ (d) = inf L+ p (x, d), x∈X

(d ≥ 0).

Note that if f1 (x) = 0 for x ∈ X0 then qp = qp+ . Let tp (d) = inf p(f0 (x), df1 (x)). x∈X0

(3.4)

Then qp (d) = min(tp (d), rp+ (1, d)),

(3.5)

rp+ (d0 , d) = inf p+ (d0 f0 (x), df1 (x)).

(3.6)

where rp+ is defined by x∈X1

3

Some nonlinear Lagrange and penalty functions

45

(The function rp+ was introduced and studied in [4] and [5].) If the restriction p+ of p on IR2+ belongs to P1 then rp+ (1, d) = qp+ (d) (see ([4, 5]), so qp (d) = min(tp (d), qp+ (d)). Note that the function tp is decreasing and tp (d) ≤ tp (0) = M (f0 , f1 ),

(d > 0).

The function qp+ (d) = rp+ (1, d) is increasing. It is known (see [4, 5]) that qp+ (d) ≡ rp+ (1, d) ≤ lim rp (1, u) = M (f0 , f1 ). u→+∞

(3.7)

Recall that a positive number d¯ is called a Lagrange multiplier of P (f0 , f1 ) ¯ = M (f0 , f1 ). A positive number d¯ is called an exact with respect to p if qp (d) ¯ = M (f0 , f1 ). We penalty parameter of P (f0 , f1 ) with respect to p+ , if qp+ (d) will now show that the following statement holds. Theorem 2. Consider (f0 , f1 ) ∈ CX and an IPH function p defined on IR2∗ . Assume that the restriction p+ of p to IR2+ belongs to P1 . Then the following assertions are equivalent: 1) there exists a Lagrange multiplier d¯ of P (f0 , f1 ) with respect to p. 2) there exists an exact penalty parameter d¯ of P (f0 , f1 ) with respect to p+ and max(tp (d), rp+ (d)) = M (f0 , f1 ) for all d ≥ 0.

(3.8)

Proof. 1) =⇒ 2). Let d¯ be a Lagrange multiplier of P (f0 , f1 ). Then ¯ 1 (x)) = M (f0 , f1 ). inf p(f0 (x), df

x∈X

¯ + (x) ≥ df ¯ 1 (x) for all x ∈ X, we have Since p is an increasing function and df 1 ¯ = inf p+ (f0 (x), df ¯ + (x)) qp+ (d) 1 x∈X

¯ + (x)) = inf p(f0 (x), df 1 x∈X

¯ 1 (x)) = M (f0 , f1 ). ≥ inf p(f0 (x), df x∈X

On the other hand, due to (3.7) we have qp+ (d) ≤ M (f0 , f1 ) for all d. Thus ¯ = M (f0 , f1 ), that is, d¯ is an exact penalty parameter of P (f0 , f1 ) with qp+ (d) respect to p+ . Due to (3.5) we have ¯ rp+ (1, d)) ¯ = M (f0 , f1 ). min(tp (d), Since tp (d) ≤ M (f0 , f1 ) and rp+ (1, d) ≤ M (f0 , f1 ), it follows that ¯ = M (f0 , f1 ) and rp+ (1, d) ¯ = M (f0 , f1 ). tp (d)

(3.9)

46

J.S. Giri and A.M. Rubinov

Since tp (d) is decreasing and rp+ (1, d) is increasing, (3.9) implies the equalities tp (d) = M (f0 , f1 ), rp+ (1, d) = M (f0 , f1 ),

¯ (0 ≤ d ≤ d), (d¯ ≤ d < +∞),

which, in turn, implies (3.8). 2) Assume now that (3.8) holds. Let Ds = {d : tp (d) = M (f0 , f1 )},

Dr = {d : rp+ (1, d) = M (f0 , f1 )}.

Since p is a continuous function it follows that tp is upper semicontinuous. Also M (f0 , f1 ) is the greatest value of tp and this function is decreasing, therefore it follows that the set Ds is a closed segment with the left end-point equal to zero. It should also be noted that the set Dr is nonempty. Indeed, since p+ ∈ P1 it follows that Dr contains a penalty parameter of P (f0 , f1 ) with respect to p+ . It is observed that the function rp+ (1, ·) is increasing and upper semicontinuous and since M (f0 , f1 ) is the greatest value of this function, it follows that Dr is a closed segment. Due to (3.8) we can say that Ds ∪ Dr = [0, +∞). Since both Ds and Dr are closed segments, we conclude ¯ = M (f0 , f1 ) that the set Dl := Ds ∩ Dr = ∅. Let d¯ ∈ Dl and therefore tp (d) ¯ = M (f0 , f1 ). Due to (3.5) we have qp (d) ¯ = M (f0 , f1 ). and rp+ (1, d)  Remark 1. Assume that p+ ∈ P1 and an exact penalty parameter exists. It easily follows from the second part of the proof of Proposition 2 that the set of Lagrange multipliers coincides with the closed segment Dl = Ds ∩ Dr . Note that for penalty parameters the following assertion (Apen ) holds: a number which is greater than an exact penalty parameter is also an exact penalty parameter. The corresponding assertion, Alag : a number, which is greater than a Lagrange multiplier is also a Lagrange multiplier, does not hold in general. Assume that a Lagrange multiplier exists. Then according to Proposition 2 an exact penalty parameter also exists. It follows from Remark 1 that (Alag ) holds if and only if Ds = [0, +∞), that is, inf p(f0 (x), df1 (x)) = M (f0 , f1 ) for all d ≥ 0.

x∈X0

(3.10)

We now point out two cases where (3.10) holds. One of them is closely related to penalization. Let p be an arbitrary IPH function, such that p(1, 0) = 1 and f1 (x) = 0 for all x ∈ X0 (in other words, f1+ = f1 ), then (3.10) holds.

3

Some nonlinear Lagrange and penalty functions

47

We now remove condition f1+ = f1 and consider very special IPH functions, for which (3.10) holds without this condition. Namely, we consider a class P∗ of IPH functions defined on the half-plane IR2∗ = {(u, v) : u ≥ 0} such that (3.10) holds for each problem (f0 , f1 ) ∈ CX . The class P∗ consists of functions p : IR2∗ → IR, such that the restriction of p on the cone IR2+ belongs to P1 and p(u, v) = u for (u, v) ∈ IR2∗ with v ≤ 0. It is clear that each p ∈ P∗ is positively homogeneous of the first degree. Let us now describe some further properties of p. Let (u, v) ≥ (u , v  ). Assuming without loss of generality that v ≥ 0, v  ≤ 0 we have p(u, v) ≥ p(u , 0) ≥ u = p(u , v  ) so p is increasing. Since p(u, 0) = u, it follows that p is continuous. Thus P∗ consists of IPH continuous functions. The simplest example of a function p ∈ P∗ is p(u, v) = max(u, av) with a > 0. Clearly the function 1

pk (u, v) = max((uk + av k ) k , u) 2l+1 with k = 2m+1 , l, m ∈ N belongs to P∗ as well. Let us check that (3.10) holds for each (f0 , f1 ) ∈ CX . Indeed, since f0 (x) > 0 for all x ∈ X, we have

inf p(f0 (x), df1 (x)) = inf f0 (x) = M (f0 , f1 ) for all d ≥ 0.

x∈X0

x∈X0

3.4 Generalized Lagrange functions In this section we consider problems P (f0 , f1 ) such that both f0 and f1 are directionally differentiable functions defined on a set X ⊆ IRn . Recall that a function f defined on X is called directionally differentiable at a point x ∈ intX if for each z ∈ IRn there exists the derivative f  (x, z) at the point x in the direction z: f  (x, z) = lim

α→+0

1 (f (x + αz) − f (x)). α

Usually only directionally differentiable functions with a finite derivative are considered. We also accept functions whose directional derivative can attain the values ±∞. It is well known (see, for example, [1]) that the maximum of two directionally differentiable functions is also directionally differentiable. In particular the function f + is directionally differentiable, if f is directionally differentiable. Let f (x) = 0. Then (f + ) (x, z) = max(f  (x, z), 0) = (f  (x, z))+ .

48

J.S. Giri and A.M. Rubinov

Let sk , k > 0 be a function defined on IR2+ by (3.3). Assume that there exists an exact penalty parameter for a problem P (f0 , f1 ) with (f0 , f1 ) ∈ CX . It easily follows from results in [5, 6] that an exact penalty parameter with respect to k  < k also exists and that the smallest exact penalty parameter d¯k with respect to sk is smaller than the smallest exact penalty parameter d¯k with respect to sk . The question then arises, does this property hold for 2l + 1 with Lagrange multipliers? (This question makes sense only for k = 2m + 1 k, l ∈ N and functions sk defined by (3.3) on the half-plane IR2∗ .) We provide a proof that the answer to this question is, in general, negative. Let f be a directionally differentiable function defined on a set X and let x ∈ intX. We say that x is a min-stationary point of f on X if for each direction z either f  (x, z) = 0 or f  (x, z) = +∞. We now present a simple example. Example 1. Let X = IR, )√ x if x > 0, f1 (x) = −x if x ≤ 0, ) √ − x if x > 0, f3 (x) = −x if x ≤ 0.

)√ f2 (x) =

x if x > 0, x if x ≤ 0,

Then the point x = 0 is a min-stationary point for f1 and f2 , but this point is not min-stationary for f3 . Proposition 1. (Necessary condition for a local minimum). Let x ∈ intX be a local minimizer of a directionally differentiable function f . Then x is a min-stationary point of f . Proof. Indeed, for all z ∈ IRn and sufficiently small α > 0 we have (1/α)(f (x + αu) − f (x)) ≥ 0. Thus the result follows.  Consider a problem P (f0 , f1 ) where (f0 , f1 ) ∈ CX are functions with finite directional derivatives. Consider the IPH function sk defined by (3.3). Let us define the corresponding Lagrange-type function Lsk : Lsk (x, λ) = f0 (x)k + λf1 (x)k .

(3.11)

We have for x ∈ X such that f1 (x) = 0 that Lsk (x, z; λ) = kf0 (x)k−1 (f0 ) (x, z) + λkf1 (x)k−1 (f1 ) (x, z).

(3.12)

Assume now that f1 (x) = 0. We consider the following cases separately. 1) k > 1. Then Lsk (x, z; λ) = kf0 (x)k−1 (f0 ) (x, z).

(3.13)

3

Some nonlinear Lagrange and penalty functions

2) k = 1. Then

49

Lsk (x, z; λ) = (f0 ) (x, z).

(3.14)

3) k < 1. First we calculate the limit 1 (f1 (x + αz))k α 1 (f1 (x) + αf1 (x, z) + o(α))k = lim α→+0 α 1 (αf1 (x, z) + o(α))k . = lim α→+0 α

A(z) := lim

α→+0

We have

⎧ ⎨ +∞ if f1 (x, z) > 0, 0 if f1 (x, z) = 0, A(z) = ⎩ −∞ if f1 (x, z) < 0.

Hence Lsk (x, z; λ)

⎧ ⎨

+∞ if f1 (x, z) > 0,  (f0 ) (x, z) if f1 (x, z) = 0, = kf0 (x) ⎩ −∞ if f1 (x, z) < 0. k−1

(3.15)

Note that for problems P (f0 , f1 ) with (f0 , f1 ) ∈ CX a minimizer is located on the boundary of the the set of feasible elements {x : f1 (x) ≤ 0}. Proposition 2. Let k > 1. Let (f0 , f1 ) ∈ CX . Assume that the functions ¯ ∈ intX, which is a f0 and f1 have finite directional derivatives at a point x minimizer of the problem P (f0 , f1 ). Assume that there exists u ∈ IRn such that (f0 ) (¯ x, u) < 0,

(3.16)

(that is, x ¯ is a not a min-stationary point for the function f0 over X). Then the point x ¯ is not a min-stationary point of the function Lk for each λ > 0. Proof. Assume that x ¯ is a min-stationary point of the function Lsk (x; λ) over X. Then combining Proposition 1 and (3.13) we have f0 (¯ x)k−1 (f0 ) (¯ x, z) ≥ 0,

z ∈ IRn .

Since f0 (¯ x) > 0 it follows that (f0 ) (¯ x, z) ≥ 0 for all z, which contradicts (3.16). It follows from this proposition that the Lagrange multiplier with respect to Lsk (k > 1) does not exist for a problem P (f0 , f1 ) if (3.16) holds. Condition (3.16) means that the constraint f1 (x) ≤ 0 is essential, that is, a minimum under this constraint does not remain a minimum without it. Remark 2. Consider a problem P (f0 , f1 ) with (f0 , f1 ) ∈ CX . Then under some mild assumptions there exists a number k > 1 such that the zero duality

50

J.S. Giri and A.M. Rubinov

gap property holds for the problem P (f0k , f1k ) with respect to the classical Lagrange function (see [2]). This means that sup inf (f0k (x) + λf1k (x)) = λ>0 x∈X

inf

f0k (x).

inf

f0 (x),

x∈X:f1 (x)≤0

Clearly this is equivalent to sup inf sk (f0 (x), λf1 (x)) = λ>0 x∈X

x∈X:f1 (x)≤0

that is, the zero duality gap property with respect to sk holds. It follows from Proposition 2 that a Lagrange multiplier with respect to sk does not exist. Hence there is no a Lagrange multiplier for P (f0 , f1 ) with respect to the classical Lagrange function. Remark 3. Let g(x) = f1+ (x). Then the penalty-type function for P (f0 , f1 ) with respect to sk coincides with the Lagrange-type function for P (f0 , g) with respect to sk . Hence an exact penalty parameter with respect to this penalty function does not exist if (3.16) holds. Proposition 3. Let k < 1 and let (f0 , f1 ) ∈ CX . Assume that the functions ¯ ∈ intX, which is a f0 and f1 have finite directional derivatives at a point x minimizer for the problem P (f0 , f1 ). Assume that there exists u ∈ IRn such that (f1 ) (¯ x, u) < 0,

(3.17)

(that is, x ¯ is not a min-stationary point for the function f0 over X). Then the point x ¯ is not a min-stationary point of the function Lsk for each λ > 0. Proof. Assume that a min-stationary point exists. Then combining Proposition 1, (3.15) and (3.17) we get a contradiction.  It follows from this proposition that a Lagrange multiplier with respect to Lsk , k < 1, does not exist if condition (3.17) holds. We now give the simplest example, when (3.17) is valid. Let f1 be a differentiable function and ∇f (¯ x) = 0. Then (3.17) holds. Consider now a more complicated and interesting example. Let f1 (x) = maxi∈I gi (x), where gi are differentiable functions. Then f1 is a directionally x, u) = maxi∈I(¯x) [∇gi (x), u], where I(¯ x) = {i ∈ differentiable function and f  (¯ I : gi (x) = f1 (x)} and [x, y] stands for the inner product of vectors x and y. Thus (3.17) holds in this case if and only if there exists a vector u such that x), u] < 0 for all i ∈ I(¯ x). To understand the essence of this result, let us [∇gi (¯ consider the following mathematical programming problem with m inequality constraints: min f0 (x) subject to gi (x) ≤ 0, i ∈ I = {1, . . . , m}.

(3.18)

We can present (3.18) as the problem P (f0 , f1 ) with f1 (x) = maxi∈I gi (x). Recall the well-known Mangasarian–Fromovitz (MF) constraint qualification

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for (3.18) (see, for example, [3]): (MF) holds at a point x ¯ if there exists a x), u] < 0 for all i ∈ I such that gi (¯ x) = 0. Thus vector u ∈ IRn such that [∇gi (¯ (3.17) for P (f, f1 ) is equivalent to (MF) constraint qualification for (3.18). In other words, if (MF) constraint qualification holds then a Lagrange multiplier for Lsk with k < 1 does not exist. (It is known that (MF) implies the existence of a Lagrange multiplier with k = 1.) Let (f, f1 ) ∈ CX , where f0 , f1 are functions with finite directional derivatives. Let g = f1+ and x be a point such that f1 (x) = 0. Then g  (x, z) = max(f  (x, z), 0) ≥ 0 for all z, hence (3.17) does not hold for the problem P (f0 , g). This means that Proposition 3 could not be applied to a penalty function for P (f, f1 ) with respect to sk . Simple examples show that exact penalty parameters with respect to sk with k < 1 can exist. We now present an example from [5]. We do not provide any details. (These can be found in [5], Example 4.6.) Example 2. Let 0 < b < c < a be real numbers and X = [0, c]. Let f (x) = (a − x)2 , f1 (x) = x − b, so P (f, f1 ) coincides with the following problem: minimize (a − x)2 subject to x ≤ b, x ∈ X. Let k = 1. Then an exact penalty parameter exists and the least exact penalty d¯1 is equal to 2(a − b). Let k = 1/2. Then an exact penalty parameter also exists and the least exact penalty parameter d¯1/2 coincides with c − b. We indicate the following two points: 1) d¯1 does not depend on the set X; d¯1/2 depends on this set. 2) d¯1 depends on the parameter a, that is on the turning point of the parabola; d¯1/2 does not depend on this parameter.

3.5 Example Consider the following one-dimensional optimization problem: 7x 9x2 + + 5, 2 2 subject to f1 (x) = x − 2 ≤ 0, x ∈ X = [0, 4].

min f0 (x) = x3 −

(3.19)

A graphical representation of this problem is given in Figure 3.1 where the shaded area represents the product of the feasible region and the axis {(0, y) : y ∈ R}. ¯ = 2. It can easily be shown that for this problem M (f0 , f1 ) = 2 at x

3.5.1 The Lagrange function approach The corresponding extended Lagrangian for (3.20) is Lsk (x, λ) = sk (f0 (x), λf1 (x)) = ((x3 −

1 7x 9x2 + + 5)k + λk (x − 2)k ) k , 2 2

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8

objective function (f–0)

6

4

2

x

0 –6

–4

–2

0

2

4

6

8

10

12

Fig. 3.1 P (f0 , f1 ).

recalling that k = Now consider

2l + 1 . 2m + 1 dL ∂L  ∂L  = f0 (¯ x) + λ f (¯ x). d¯ x ∂f0 ∂f1 1

(3.20)

An easy calculation shows that ⎧ 5 ⎪ k > 1, ⎨−2, dL 5 (¯ x) = − + λ, k = 1, ⎪ dx ⎩ ∞,2 k < 1. ¯ = From this it is clear that an exact Lagrange multiplier λ only for the case k = 1.

(3.21)

5 2

may exist

¯ = 5 provides a Remark 4. In fact Figure 3.2 shows that in this example λ 2 local minimum at x ¯ for k = 1 but not a global minimum, therefore it follows that no exact Lagrange multiplier exists for this problem.

3.5.2 Penalty function approach The corresponding penalty function for (3.20) is 1

+ k k k L+ sk (x; λ) = (f0 + (λf1 ) )  2 1 k k k k ((x3 − 9x2 + 7x 2 + 5) + λ (x − 2) ) , for x ≥ 2, = 2 x3 − 9x2 + 7x for x ≤ 2. 2 + 5,

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y 10

8

6

4

2 x

0 –8

–6

–4

–2

0

2

4

6

8

10

12

Fig. 3.2 L(x; 52 ).

By Theorem 13.6 it can easily be shown that an exact penalty parameter exists when k < 1. This is shown in Figure 3.3 where an exact penalty parameter, d¯ = 1, is used.

Fig. 3.3 L+ s 1 (x; 1). 3

From these results we have shown that whereas the adoption of extended penalty functions of the form sk yields an improvement to the traditional penalty function approach, this cannot be generalized to improve the Lagrange approach.

References 1. V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis (Peter Lang, Frankfurt on Main, 1995). 2. D. Li, Zero duality gap for a class of nonconvex optimization problems, J. Optim. Theory Appl., 85 (1995), 309–324.

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3. Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programming with Equilibrium Constraints (Cambridge University Press, Cambridge, 1996). 4. A. M. Rubinov, B. M. Glover and X. Q. Yang, Decreasing functions with applications to penalization, SIAM J. Optim., 10(1) (1999), 289–313. 5. Rubinov A. M., Abstract Convexity and Global Optimization (Kluwer Academic Publishers, Dordrecht, 2000). 6. A. M. Rubiniv, X. Q. Yang and A. M. Bagirov, Penalty functions with a small penalty parameter, Optim. Methods Softw. 17 (2002), 931–964.

Chapter 4

Convergence of truncates in l1 optimal feedback control Robert Wenczel, Andrew Eberhard and Robin Hill

Abstract Existing design methodologies based on infinite-dimensional linear programming generally require an iterative process often involving progressive increase of truncation length, in order to achieve a desired accuracy. In this chapter we consider the fundamental problem of determining a priori estimates of the truncation length sufficient for attainment of a given accuracy in the optimal objective value of certain infinite-dimensional linear programs arising in optimal feedback control. The treatment here also allows us to consider objective functions lacking interiority of domain, a problem which often arises in practice. Key words: l1 -feedback control, epi-distance convergence, truncated convex programs

4.1 Introduction In the literature on feedback control there exist a number of papers addressing the problem of designing a controller to optimize the response of a system to a fixed input. In the discrete-time context there are many compelling Robert Wenczel Department of Mathematics, Royal Melbourne University of Technology, Melbourne 3001, AUSTRALIA Andrew Eberhard Department of Mathematics, Royal Melbourne University of Technology, Melbourne 3001, AUSTRALIA e-mail: [email protected] Robin Hill Department of Mathematics, Royal Melbourne University of Technology, Melbourne 3001, AUSTRALIA C. Pearce, E. Hunt (eds.), Structure and Applications, Springer Optimization and Its Applications 32, DOI 10.1007/978-0-387-98096-6 4, c Springer Science+Business Media, LLC 2009 

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reasons for using error-sequences in the space l1 as the basic variable, and using various measures of performance (see, for example, [15]) which includes the l1 -norm (see, for example, [11]). The formulation of l1 -problems leads to the computation of inf M f (and determination of optimal elements, if any), where M is an affine subspace of a suitable product X of l1 with itself. This subspace is generated by the YJBK (or Youla) parameterization [24] for the set of all stabilizing feedback-controllers for a given linear, timeinvariant system. The objective f will typically be the l1 -norm, (see [6], [11]). As is now standard in the literature on convex optimization ( [20], [21]), we will use convex discontinuous extended-real-valued functions, of the form f = · 1 + δC , where δC is the indicator function (identically zero on C, identically +∞ elsewhere) of some closed convex set in l1 . This formalism is very flexible, encompassing many problem formats, including the case of timedomain-template constraints ([10, Chapter 14], [14]). These represent bounds on signal size, of the form Bi ≤ ei ≤ Ai (for all i), where e = {ei }∞ i=0 denotes the error signal. Often there are also similar bounds on the control signal u. As the Youla parameterization generates controllers having rational z-transform, the variables in M should also be taken to be rational in this sense, if the above infimum is to equal the performance limit [6] for physically realizable controllers. (If this condition is relaxed, then inf M f provides merely a lower bound for the physical performance limit.) The set M may be recharacterized by a set of linear constraints and thus forms an affine subspace of X. The approach in many of the above references is to evaluate inf M f by use of classical Lagrangian-type duality theory for such minimization problems. The common assumption is that the underlying space X is l1 , or a product thereof, and that M is closed, forcing it to contain elements with non-rational ztransform, counter to the “physical” model. Consequently, inf M f may not coincide with the performance limit for physically realizable (that is, rational) controllers. However, in the context of most of the works cited above, such an equality is actually easily established (as was first noted in [27]). Indeed, whenever it is assumed that C has nonempty interior, on combining this with the density in M of the subset consisting of its rational members [19], we may deduce (see Lemma 15 below) that the rational members of C ∩ M are l1 -dense in C ∩ M . This yields the claimed equality for any continuous objective function (such as the l1 -norm). Use of the more modern results of conjugate duality permits the extension of the above approach to a more general class of (−∞, +∞]-valued objective functions f . They are applicable even when int C may vanish. In this case, the question of whether inf M f equals the physical limit becomes nontrivial (in contrast to the case when C has interior). Indeed, if inf M f is strictly less than the physical limit, any result obtained by this use of duality is arguably of questionable engineering significance. It is therefore important to know when inf M f is precisely the performance limit for physically realizable controllers, to ensure that results obtained via the duality approaches described above are physically meaningful. Note that

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this question is posed in the primal space, and may be analyzed purely in the primal space. This question will be the concern of this chapter. In this paper, we derive conditions on the system, and on the time-domain template set C, that ensure inf M (f0 + δC ) = inf C∩M f0 is indeed the performance limit for physically realizable controllers, for various convex lowersemicontinuous performance measures f0 . Here we only treat a class of 1input/2-output problems (referred to as “two-block” in the control literature), although the success of these methods for this case strongly suggests the possibility of a satisfactory extension to multivariable systems. Existing results on this question (see, for example, [14, Theorem 5.4]) rely on Lagrangian duality theory, and thereby demand that the time-domain template C has interior. Here, for the class of two-block problems treated, we remove this interiority requirement. Our result will be obtained by demonstrating the convergence of a sequence of truncated (primal) problems. Moreover, this procedure will allow the explicit calculation of convergence estimates, unlike all prior works with the exception of [23]. (This latter paper estimates bounds on the truncation length for a problem with H∞ -norm constraints and uses state-space techniques, whereas our techniques are quite distinct.) The approach followed in our chapter has two chief outcomes. First, it validates the duality-based approach in an extended context, by ensuring that the primal problem posed in the duality recipe truly represents the limit-of-performance for realizable controllers. Secondly, it provides a computational alternative to duality itself, by exhibiting a convergent sequence of finite-dimensional primal approximations with explicit error estimates along the sequence. (This contrasts with traditional “primal–dual” approximation schemes, which generally do not yield explicit convergence rates.) This will be achieved by the use of some recently developed tools in optimization theory (the relevant results of which are catalogued in Section 4.5)— namely, the notion of epi-distance (or Attouch–Wets) convergence for convex sets and functions [2, 3, 4, 5]. Epi-distance convergence has the feature that if fn converges to f in this sense, then subject to some mild conditions, * * * * *inf fn − inf f * ≤ d(f, fn ), X

X

where d(f, g) is a metric describing this mode of convergence. Since the l1 control problem is expressible as inf X (f + δM ), this leads naturally to the question of the Attouch–Wets convergence of sums fn + δMn of sequences of functions (where fn and Mn are approximations to f and M respectively). A result from [5], which estimates the “epi-distances” d(fn + gn , f + g) between sums of functions, in terms of sums of d(fn , f ) and d(gn , g), is restated and modified to suit our purpose. This result requires that the objective and constraints satisfy a so-called “constraint qualification” (CQ). In Section 4.6 some conditions on the template C and on M are derived that ensure that the CQ holds. Also, the truncated problems will be defined, and some fundamental limitations on the truncation scheme (in relation to

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satisfiability of the CQ) will be discussed. Specifically, the basic optimization will be formulated over two variables e and u in l1 . The truncated approximate problems will be formed by truncating in the e variable only, since the CQ will be seen to fail under any attempt to truncate in both variables. Thus in these approximate problems, the set of admissible (e, u) pairs will contain a u of infinite length. Despite this, it will be noted that the set of such (e, u)’s will be generated by a finite basis (implied by the e’s), so these approximates are truly finite-dimensional. Finally, in Section 4.7, the results of Section 4.5 will be applied to deduce convergence of a sequence of finite-dimensional approximating minimizations. This will follow since the appropriate distances d(Cn , C) can be almost trivially calculated. Also, we observe that for a sufficiently restricted class of systems, our truncation scheme is equivalent to a simultaneous truncation in both variables e and u. In fact, this equivalence is satisfied precisely when the system has no minimum-phase zeros. The motivation for this work arose from a deficiency in current computational practices which use simultaneous primal and dual truncations. These yield upper and lower bounds for the original primal problem, but have the disadvantage of not providing upper estimates on the order of truncation sufficient for attainment of a prescribed accuracy.

4.2 Mathematical preliminaries We let R stand for the extended reals [−∞, +∞]. For a Banach space X, balls in X centered at 0 will be written as B(0, ρ) = { x ∈ X | x < ρ } and ¯ ρ) = { x ∈ X | x ≤ ρ }. Corresponding balls in the dual space X ∗ will B(0, ¯ ∗ (0, ρ) respectively. The indicator function of a set be denoted B ∗ (0, ρ) and B A ⊆ X will be denoted δA . We will use u.s.c. to denote upper-semicontinuity and l.s.c. to denote lower-semicontinuity. Recall that a function f : X → R is called proper if never equal to −∞ and not identically +∞, and proper closed if it is also l.s.c. For a function f : X → R, the epigraph of f , denoted epi f , is the set {(x, α) ∈ X × R | f (x) ≤ α}. The domain, denoted dom f, is the set {x ∈ X | f (x) < +∞}. The (sub-)level set {x ∈ X | f (x) ≤ α} (where α > inf X f ) will be given the abbreviation {f ≤ α}. For  > 0, and if inf X f is finite, -argmin f = {x ∈ X | f (x) ≤ inf X f + } is the set of approximate minimizers of f . Any product X×Y of normed spaces will always be understood to be endowed with the box norm (x, y) = max{ x , y }; any balls in such product spaces will always be with respect to the box norm. ∞ Here l1 (C) denotes ∞the Banach space of all complex sequences a = {an }n=0 such that a 1 := n=0 |an | is finite; l1 denotes the Banach space of all real sequences in l1 (C); and l∞ denotes the Banach space of all real sequences a = {an }∞ n=0 such that a ∞ := supn |an | is finite.

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For two sequences a, b their convolution a ∗ b is the sequence (a ∗ b)i = i j=0 aj bi−j . The length of a sequence a, denoted l(a), is the smallest integer n such that ai = 0 for all i ≥ n. ¯ to be the closed unit disk {z ∈ C | |z| ≤ 1} in the complex We define D plane; D is the open unit disk {z ∈ C | |z| < 1}. ∞ ˆ(z) = n=0 an z n for The z-transform of a = {an }∞ n=0 is the function a complex z wherever it is defined. The inverse z-transform of a ˆ will be written a). as Z −1 (ˆ Also l 1 denotes the set of all z-transforms of sequences in l1 . It can be ¯ that regarded as a subset of the collection of all continuous functions on D are analytic on D. We use R∞ to denote the set of all rational functions of a complex variable, ¯ with no poles in D. Definition 1. Let A be a convex set in a topological vector space and x ∈ A. Then 1 cone A = ∪λ>0 λA (the smallest convex cone containing A); 2 The core or algebraic interior of A is characterized as x ∈ core A iff ∀y ∈ X, ∃ > 0 such that ∀λ ∈ [−, ] we have x + λy ∈ A. The following generalized interiority concepts were introduced in [8] and [17, 18] respectively in the context of Fenchel duality theory, to frame a sufficient condition for strong duality that is weaker than the classical Slater condition. Definition 2. Let A be a convex set in a topological vector space and x ∈ A. Then 1 The quasi relative interior of A (qri A) consists of all x in A for which cone (A − x) is a closed subspace of X; 2 The strong quasi relative interior of A (sqri A) consists of all x in X for which cone (A − x) is a closed subspace of X. Note that 0 ∈ core A if and only if cone A = X, and that in general, core A ⊆ sqri A ⊆ qri A. Nearly all modern results in conjugate duality theory use constraint qualifications based on one or other of these generalized interiors. Some applications of such duality results to (discrete-time) feedback optimal control may be found in [11] and [15]. An example of an application to a problem in deterministic optimal control (that is, without feedback) is outlined in [17]. From [16] and [20] we have the following. Recall that a set A in a topological linear space X is ideally convex if for any bounded sequence{xn } ⊆ A ∞ ∞ and {λn } of nonnegative numbers with n=1 λn = 1, the series n=1 λn xn either converges to an element of A, or else does not converge at all. Open or closed convex sets are ideally convex, as is any finite-dimensional convex

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set. In particular, if X is Banach, then such series ∞ always converge, and the definition of ideal convexity only requires that n=1 λn xn be in A. From [16, Section 17E] we have the following proposition. Proposition 1. For a Banach space X, 1 If C ⊆ X is closed convex, it is ideally convex. 2 For ideally convex C, core C = core C = int C = int C. 3 If A and B are ideally convex subsets of X, one of which is bounded, then A − B is ideally convex. Proof. We prove the last assertion only; the rest can be found in the cited reference. Let {a n − bn } ⊆ A − B be a bounded sequence and let λn ≥ ∞ 0 be such that n=1 λn = 1. Then, due to the assumed boundedness of one of A or B, {a } ⊆ A and {bn } ⊆ B are both bounded, yielding the ∞ ∞ n∞ λ a ∈ A and λ b ∈ B. Thus convergent sums n n n n n=1 n=1 n=1 λn (an − ∞ ∞  bn ) = n=1 λn an − n=1 λn bn ∈ A − B. Corollary 1. Let A and C be closed convex subsets of the Banach space X. Then 0 ∈ core (A − C) implies 0 ∈ int (A ∩ B(0, ρ) − C ∩ B(0, ρ)) for some ρ > 0. Proof. Let ρ > inf A∩C · and let x ¯ ∈ A ∩ C ∩ B(0, ρ). Then for x ∈ X, x = λ(a−c) for some λ > 0, a ∈ A, c ∈ C since by assumption, cone (A−C) = X. x < ρ, Then for any t ≥ 1 sufficiently large so that t−1 ( a + c ) + ¯ + + + ,   , , 1 1 1 1 a+ 1− x = tλ x ¯ − c+ 1− x ¯ t t t t ∈ tλ (A ∩ B(0, ρ) − C ∩ B(0, ρ)) ⊆ cone (A ∩ B(0, ρ) − C ∩ B(0, ρ)) . Hence 0 ∈ core (A ∩ B(0, ρ) − C ∩ B(0, ρ)) from the arbitrariness of x ∈ X. The result follows since by Proposition 1 the core and interior of A∩B(0, ρ)−  C ∩ B(0, ρ) coincide.

4.3 System-theoretic preliminaries 4.3.1 Basic system concepts In its most abstract form, a system may be viewed as a map H : XI → XO between a space XI of inputs and a space XO of outputs. In the theory of feedback stabilization of systems, interconnections appear where the output of one system forms the input of another, and for this to make sense, XI

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and XO will, for simplicity, be taken to be the same space X. The system H is said to be linear, if X is a linear space and H a linear operator thereon. Our focus will be on SISO (single-input/single-output) linear discrete-time systems. In this case, X will normally be contained in the space RZ of realvalued sequences. For n ∈ N define a time-shift τn on sequences in X by (τn φ)i := φi−n ,

φ∈X.

If each τn commutes with H, then H is called shift- (or time-) invariant. Our interest will be in linear time-invariant (or LTI) systems. It is well known [12] that H is LTI if and only if it takes the form of the convolution operator h∗ for some h ∈ X. This h is called the impulseresponse of H, since h = H(δ) = h∗δ, where δ is the (Dirac) delta-function in continuous time, or is the unit pulse sequence (1, 0, 0, . . . ), in discrete time. The discrete-time LTI system H = h∗ is causal if the support of h lies in the positive time-axis N = {0, 1, 2, . . .}. The significance of this notion is clarified after  observing the action of H on an input u, which takes the form (Hu)n = k≤n hk un−k , so if h takes any nonzero values for negative time then evaluation of the output at time n would require foreknowledge of input behavior at later times. Note that for LTI systems, a natural (but by no means the only) choice for the “space of signals” X is the space of sequences for which the z-transform exists. From now on we identify a LTI system H with its impulse-response h := H(δ). Definition 3. The LTI system h is BIBO (bounded-input/bounded-output)stable if h ∗ u ∈ l∞ whenever u ∈ l∞ . From the proof of [12, Theorem 7.1.5], it is known that H = h∗ is BIBOstable if and only if h ∈ l1 . Any LTI system H can be characterized by its transfer function, defined ˆ of its impulse-response h. The input–output relation to be the z-transform h ˆ u for appropriate inputs u. Thus convolution equathen takes the form u ˆ → hˆ tions can be solved by attending to an algebraic relation, much simplifying the analysis of such systems. For systems H1 , H2 we shall often omit the convolution symbol ∗ from products H1 H2 , which will be understood to be the system (h1 ∗ h2 )∗. By the commutativity of ∗, H1 H2 = H2 H1 . This notation will be useful in that now formal manipulation of (LTI) systems is identical to that of their transforms (that is, transfer functions). Consequently, we may let the symbol H stand either for the system, or its impulse-response h, or its transfer ˆ function h. We now express stability in terms of rational transfer functions. The algebra R∞ to be defined below shall have a fundamental role in the theory of feedback-stabilization of systems. First, we recall some notation:

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R[z] — the space of polynomials with real coefficients, of the complex variable z; R(z) — the space of rational functions with real coefficients, of the complex variable z. Here R∞ := {h ∈ R(z) | h has no poles in the closed unit disk} . It is readily established that R∞ = l 1 ∩ R(z) , so R∞ forms the set of all rational stable transfer functions.

4.3.2 Feedback stabilization of linear systems Here we summarize the theory of stabilization of (rational) LTI systems by use of feedback connection with other (rational) LTI systems. All definitions and results given in this subsection may be found in [24]. Consider the feedback configuration of Figure 4.1. Here w is the “reference input,” e = w − y is the “(tracking–) error,” representing the gap between the closed-loop output y and the reference input w, and u denotes the control signal (or “input activity,” or “actuator output”).

Fig. 4.1 A closed-loop control system.

Definition 4. A (rational) LTI discrete-time system K is said to (BI-BO-) stabilize the (rational) LTI system P if the closed loop in Figure 4.1 is stable in the sense that for any bounded input w ∈ l∞ and any bounded additivelyapplied disturbance (such as Δ in Figure 4.1) at any point in the loop, all resulting signals in the loop are bounded. Such K is referred to as a stabilizing compensator or controller. It should be noted that the definition as stated applies to general non-LTI, even nonlinear systems, and amounts to the requirement of “internal,” as well as “external,” stability. This is of importance, since real physical systems have

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a finite operating range, and it would not be desirable for, say, the control signal generated by K to become too large and “blow up” the plant P . Writing P = p∗ and K = k∗, with pˆ and kˆ in R(z) and noting that the transfer function between any two points of the loop can be constructed by addition or multiplication from the three transfer functions given in (4.1) below, we obtain the following well-known result. Proposition 2. K stabilizes P if and only if ˆ , k/(1 ˆ ˆ and pˆ/(1 + pˆk) ˆ ∈ R∞ . 1/(1 + pˆk) + pˆk)

(4.1)

We denote by S(P ) the set of rational stabilizing compensators for the plant P . The fundamental result of the theory of feedback stabilization (the YJBK factorization — see Proposition 3) states that S(P ) has the structure of an affine subset of R(z). Before we move to a precise statement of this result, we remind the reader that we only intend to deal with singleinput/single-output systems. In the multi-input/multi-output case, each system would be represented by a matrix of transfer functions, which greatly complicates the analysis. We note, however, that this factorization result does extend to this case (see [24, Chapter 5]). ˆ and dˆ are coprime if Definition 5. Let n ˆ and dˆ be in R∞ . We say that n there exist x ˆ and yˆ in R∞ such that x ˆn ˆ + yˆdˆ ≡ 1 in R(z) , ˆ = 1 except at singularities). This (that is, for all z ∈ C, x ˆ(z)ˆ n(z) + yˆ(z)d(z) can be easily shown to be identical to coprimeness in R∞ considered as an abstract ring. Let pˆ ∈ R(z). It has a coprime factorization pˆ = n ˆ /dˆ where n ˆ and dˆ are in r for polynomials qˆ and rˆ having no common R∞ . Indeed, we can write pˆ = qˆ/ˆ factors, which implies coprimeness in R[z] and hence in R∞ ⊇ R[z]. We can now state the fundamental theorem [24, Chapter 2], also referred to as the Youla (or YJBK) parameterization. Proposition 3. Let the plant P have rational transfer function pˆ ∈ R(z), let n ˆ and dˆ in R∞ form a coprime factorization and let x ˆ and yˆ in R∞ arise ˆ Then from the coprimeness of n ˆ and d.  * ˆq * x ˆ + dˆ * qˆ ∈ R∞ , qˆ = yˆ/ˆ n . (4.2) S(P ) = yˆ − n ˆ qˆ * This result has the following consequence for the stabilized closed-loop mappings. Recall that 1/(1 + pˆcˆ) is the transfer function taking input w to e (that is, eˆ = w/(1 ˆ + pˆcˆ)) and cˆ/(1 + pˆcˆ) maps the input w to u. We now have (see [24]) the following result.

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Corollary 2. The set of all closed-loop maps Φ taking w to (e, u), achieved by some stabilizing compensator C ∈ S(P ), is + , + ,* . ) ˆ ** = dˆ yˆ − qˆdˆ n q ˆ ∈ R Φ , q ˆ = y ˆ /ˆ n , (4.3) ∞ x ˆ dˆ * and has the form of an affine set in R(z) × R(z).

4.4 Formulation of the optimization problem in l1 In the following, we consider SISO (single-input/single-output) rational, linear time-invariant (LTI) systems. For such a system (the ‘plant’) we characterize the set of error-signal and control-signal pairs achievable for some stabilizing compensator, in the one-degree-of-freedom feedback configuration (Figure 4.1). The derivation, based on the Youla parameterization, is standard. Assume that the reference input w = 0 has rational z-transform and the plant P is rational and causal, so it has no pole at 0. For a = eiθ on the unit circle, define a subspace Xa of l∞ by Xa = l1 + span {c, s}, where

∞ c := {cos kθ}∞ k=0 and s := {sin kθ}k=0 .

We shall assume that the error signals (usually denoted by e here) are in l1 , so we are considering only those controllers that make the closed-loop system track the input (in an l1 -sense), and we shall assume also that the associated control signal remains bounded (and in fact resides in Xa ). Definition 6. For any u ∈ Xa , let uc and us denote the (unique) real numbers such that u − uc c − us s is in l1 . Let the plant P have transfer function P (z) with the coprime factorization ˆ where n P (z) = n ˆ (z)/d(z) ˆ and dˆ are members of R[z] (the space of polynomis als, with real coefficients, in the complex variable z). Write n ˆ (z) = i=0 ni z i ,  ˆ = t di z i , n := (n0 , n1 , .., ns , 0, ..) and d := (d0 , .., dt , 0, ..). d(z) i=0 Let x and y be finite-length sequences such that x ˆn ˆ + yˆdˆ = 1 in R[z]. Their existence is a consequence of the coprimeness of n ˆ and dˆ in R[z] (see [24]). If we aim to perform an optimization over a subset of the set S(P ) of stabilizing controllers K for P , such that for each K in this subset, the corresponding error signal φ(K) and control output u(K) are in l1 and Xa respectively, the appropriate feasible set is

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F0 := {(e, u) ∈ l1 × Xa | e = e(K) for some K ∈ S(P ), u = K ∗ e} = {(e, u) ∈ l1 × Xa | (e, u) = w ∗ d ∗ (y, x) − q ∗ w ∗ d ∗ (n, −d) for some q such that qˆ ∈ R∞ \{ˆ y /ˆ n}}, where the latter equality follows from the YJBK factorization for S(P ) (use Property 3 or Corollary 2). Henceforth we shall always assume that w ∈ Xa is rational (that is, has rational z-transform), the plant P is rational and has no zero at a. By the lemma to follow (whose proof is deferred to the Appendix) we observe that a single sinusoid suffices to characterize the asymptotic (or steadystate) behavior of u for all feasible signal pairs (e, u) ∈ F0 . Lemma 1. Let w ∈ Xa and P be rational and assume that P (a) = 0. Then for any (e, u) ∈ F0 , 1 1 uc = βc := wc Re P (a) − ws Im P (a) , 1 1 us = βs := ws Re P (a) + wc Im P (a) .

Using this lemma, we can translate F0 in the u variable to obtain the set F := F0 − (0, βc c + βs s), having the form (where we use the notation x ∗ (y, z) := (x ∗ y, x ∗ z) for sequences x, y, z) F =(e, { u) ∈ l1 × l1 | (e, u) = w ∗ d ∗ (y, x) − (0, βc c + βs s) − q ∗ w ∗ d ∗ (n, −d) for some q such that qˆ ∈ R∞ \{ˆ y /ˆ n}}. We need to recast F into a form to which the tools of optimization theory can be applied. As a first step, formally define the sets M := ⎧ * ⎫ * eˆ(¯ pi ) = 0 ( p¯i pole of P : |¯ pi | ≤ 1) i = 1,.., m1⎪ ⎪ * ⎪ ⎪ ⎨ ⎬ * eˆ(¯ z ) = w(¯ ˆ zj )(¯ zj zero of P : |¯ zj | ≤ 1) j = 1,.., m2 (e, u) ∈ (l1 )2 ** j , vk ) = 0 (¯ vk zero of w ˆ : |¯ vk | ≤ 1) k = 1,.., m3⎪ ⎪ * eˆ(¯ ⎪ ⎪ ⎩ ⎭ d ∗ e*+ n ∗ u = w ∗ d − n ∗ (βc c + βs s) Mr := {(e, u) ∈ M | eˆ, u ˆ rational, e = 0} and M (0) :=* ⎫ ⎧ * eˆ(¯ pi ) = 0 (¯ pi a pole of P with |¯ pi | ≤ 1) i = 1,.., m1 ⎬ ⎨ * zj ) = w(¯ ˆ zj )(¯ zj a zero of P with |¯ zj | ≤ 1) j = 1,.., m2 e ∈ l1 ** eˆ(¯ ⎭ ⎩ * eˆ(¯ vk ) = 0 (¯ vk a zero of w ˆ with |¯ vk | ≤ 1) k = 1,.., m3 with the understanding that in the above sets, whenever P and w ˆ have a common pole at a (and hence at a ¯), the constraint eˆ(a) = 0 is absent.

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Moreover, note that in the definition of M , the constraints on eˆ at the zeros z¯ (if z¯ = a, a ¯) are redundant, as follows from the closed-loop equation d ∗ e + n ∗ u = w ∗ d − n ∗ (βc c + βs s). The above constraint system can also be obtained from the general multivariable formalism of [9], [14] etc., on removal of the redundancies in the latter. The content of the following remark will be used repeatedly in our work. Remark 1. We note the following relation between M (0) and M . Let (¯ e, u ¯) be e, u ¯) consist of elements satisfying any element of M . Then M (0) − e¯ and M −(¯ the corresponding constraints with right-hand sides set to zero. Assuming that P (equivalently, n ˆ ) has no zeros on the unit circle, and that all its zeros ˆe/ˆ n) maps in D are simple, then the map T on M (0) − e¯ taking e to −Z −1 (dˆ 1 e, u ¯), into l (by Lemma 4 below) with T ≤ κ d 1 . Then (e, T e) ∈ M − (¯ since d ∗ e + n ∗ T e = 0, which follows on taking z-transforms. The next two lemmas give simple sufficient conditions for the feasible set F to be fully recharacterized as an affine subspace defined by a set of linear constraints, and for when elements of finite length exist. Since they are only minor modifications of standard arguments, we omit the proofs here, relegating them to the Appendix. Lemma 2. Assume that either ¯ the poles/zeros of P and w 1 in D ˆ are distinct and simple; or ¯ 2 in D\{a, a ¯} the poles/zeros of P and w ˆ are distinct and simple, and P and w ˆ have a common simple pole at z = a. Then Mr = F. Lemma 3. Let P and w ∈ Xa be rational with P (a) = 0. Assume also the following: 1 either ¯ the poles/zeros of P and w (a) in D ˆ are distinct and simple; or ¯ (b) in D\{a, a ¯} the poles/zeros of P and w ˆ are distinct and simple, and P and w ˆ have a common simple pole at z = a; 2 (w − wc c − ws s) ∗ d has finite length; 3 all zeros (in C) of P are distinct and simple. Then M contains elements (¯ e, u ¯) of finite length. Remark 2. In the above result, one can always choose that e¯ = 0, so that (¯ e, u ¯) ∈ Mr , which coincides with F by Lemma 2. Thus the latter is nonempty. Lemma 4. Let fˆ ∈ l 1 and let p(·) be a polynomial with real coefficients with no zeros on the unit circle. Assume also at each zero of p(·) in the open unit

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disk, that fˆ has a zero also, of no lesser multiplicity. Then fˆ(·)/p(·) ∈ l 1 . Also, if q := Z −1 (fˆ/p), then q 1 ≤ κ f 1 where + ,  1 (1 − |a|) 1 −  1/κ = , |a | a, a zeros of p: |a|<1, |a |>1 and where in the above product, the zeros appear according to multiplicity, and moreover, both members of any conjugate pair of zeros are included. 

Proof. See the Appendix.

Remark 3. Such a result will not hold in general if p(·) has zeros on the unit circle; a counterexample for |a| = 1 is as follows. a−j for j ≥ 1. Then f ∈ l1 (C) and fˆ(a) = 0 since Let f0 = −1 and fj = j(j+1) ∞ fˆ(z) 1 k=1 k(k+1) = 1. If q is the inverse z-transform of [z → z−a ], then for each k, * * * 1  1 j* |qk | = * ak+1 f a j>k j * = k+1 so q ∈ / l1 (C). The hard bounds on the error and on input activity will be represented by the closed convex sets C (e) and C (u) respectively, given by (e)

C (e) := {e ∈ l1 | Bi (e)

where Bi

(e)

≤ 0 ≤ Ai

(e)

≤ ei ≤ Ai for all i},

(4.4)

eventually for large i, and (u)

(u)

(4.5)

C := C (e) × C (u) .

(4.6)

C (u) := {u ∈ l1 | Bi

≤ ui ≤ Ai for all i}, where

We shall also make use of the truncated sets (e)

Ck := {e ∈ C (e) | ei = 0 for i ≥ k}.

(4.7)

(u)

We shall not use the truncated sets Ck , for reasons that will be indicated in Section 4.6.1.

4.5 Convergence tools It is well known that there is no general correspondence between pointwise convergence of functions, and the convergence, if any, of their infima, or of optimal or near optimal elements. Hence approximation schemes for minimization of functions, based on this concept, are generally useless. In contrast,

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schemes based on monotone pointwise convergence, or on uniform convergence, have more desirable consequences in this regard. However, the cases above do not include all approximation schemes of interest. In response to this difficulty, a new class of convergence notions has gained prominence in optimization theory, collectively referred to as “variational convergence” or “epi-convergence.” These are generally characterized as suitable set-convergences of epigraphs. For an extensive account of the theory, the reader is referred to [1, 7]. Such convergences have the desirable property that epi-convergence of functions implies, under minimal assumption, the convergence of the infima to that of the limiting function, and further, for some cases, the convergence of the corresponding optimal, or near-optimal, elements. The convergence notion with the most powerful consequences for the convergence of infima is that of Attouch and Wets, introduced by them in [2, 3, 4]. Propositions 4 and 5 (see below) will form the basis for the calculation of explicit error estimates for our truncation scheme. Definition 7. For sets C and D in a normed space X, and ρ > 0, d(x, D) := inf x − d , d∈D

eρ (C, D) :=

sup

d(x, D),

x∈C∩B(0,ρ)

haus ρ (C, D) := max{eρ (C, D), eρ (D, C)}, (the ρ-Hausdorff distance) . For functions f and g mapping X to R, the “ρ-epi-distance” between f and g is defined by dρ (f, g) := haus ρ (epi f, epi g). It is easily shown that haus ρ (C, D) = dρ (δC , δD ). Definition 8. Let Cn and C be subsets of X. Then Cn Attouch–Wets converges to C iff for each ρ > 0, we have lim haus ρ (Cn , C) = 0.

n→∞

If fn and f are R-valued functions on X, then fn Attouch–Wets (or epidistance) converges to f if for each ρ > 0, lim dρ (fn , f ) = 0,

n→∞

(that is, if their epigraphs Attouch–Wets converge). This convergence concept is well suited to the treatment of approximation schemes in optimization, since the epi-distances dρ provide a measure of the

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difference between the infima of two functions, as indicated by the following result of Attouch and Wets. For ϕ : X → R with inf X ϕ finite, we denote by -argmin ϕ the set {x ∈ X | ϕ(x) ≤ inf X ϕ + } of -approximate minimizers of ϕ on X. Proposition 4. [4, Theorem 4.3] Let X be normed and let ϕ and ψ be proper R-valued functions on X, such that 1 inf X ϕ and inf X ψ are both finite; 2 there exists ρ0 > 0 such that for all  > 0, (-argmin ϕ) ∩ B(0, ρ0 ) = ∅ and (-argmin ψ) ∩ B(0, ρ0 ) = ∅. * * * * *inf ϕ − inf ψ * ≤ dα(ρ0 ) (ϕ, ψ),

Then

X

X

where α(ρ0 ) := max{ρ0 , 1 + | inf X ϕ|, 1 + | inf X ψ|}. Since our optimization problems are expressible as the infimum of a sum of two (convex) functions, we will find useful a result on the Attouch–Wets convergence of the sum of two Attouch–Wets convergent families of (l.s.c. convex) functions. The following result by Az´e and Penot may be found in full generality in [5, Corollary 2.9]. For simplicity, we only quote the form this result takes when the limit functions are both nonnegative on X. Proposition 5. Let X be a Banach space, let fn , f , gn and g be proper closed convex R-valued functions on X, with f ≥ 0 and g ≥ 0. Assume that we have the Attouch–Wets convergence fn → f and gn → g, and that for some s > 0, t ≥ 0 and r ≥ 0, B(0, s)2 ⊆ Δ(X) ∩ B(0, r)2 − {f ≤ r} × {g ≤ r} ∩ B(0, t)2 ,

(4.8)

where Δ(X) := {(x, x) | x ∈ X} and B(0, s)2 denotes a ball in the box norm in X × X (that is, B(0, s)2 = B(0, s) × B(0, s)). Then for each ρ ≥ 2r + t and all n ∈ N such that dρ+s (fn , f ) + dρ+s (gn , g) < s , we have dρ (fn + gn , f + g) ≤

2r + s + ρ [dρ+s (fn , f ) + dρ+s (gn , g)] . s

In particular, fn + gn Attouch–Wets converges to f + g. Corollary 3. Let Cn , C and M be closed convex subsets of a Banach space X and let fn and f be proper closed convex R-valued functions on X with f ≥ 0 and Attouch–Wets convergence fn → f and Cn → C. Suppose that for some s > 0, t ≥ 0 and r ≥ 0, B(0, s)2 ⊆ Δ(X) ∩ B(0, r)2 − (C × M ) ∩ B(0, t)2 ;

(4.9)

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B(0, s)2 ⊆ Δ(X) ∩ B(0, r)2 − [{f ≤ r} × (C ∩ M )] ∩ B(0, t)2 .

(4.10)

Assume further that fn and f satisfy ∃n0 ∈ N, α ∈ R such that max{ sup

inf fn , inf f } ≤ α and

n≥n0 Cn ∩M

C∩M

(4.11)

there exists ρ0 > 0 such that / {fn ≤ α + 1} ∩ Cn ∩ M and B(0, ρ0 ) ⊇ {f ≤ α + 1} ∩ C ∩ M . B(0, ρ0 ) ⊇ n≥n0

(4.12) Then for any fixed ρ ≥ max{2r + t, ρ0 , α + 1}, and all n ≥ n0 for which dρ+s (fn , f ) +

2r + 2s + ρ dρ+2s (Cn , C) < s , and dρ+2s (Cn , C) < s , (4.13) s

we have * * * * * inf fn − inf f * ≤ dρ (fn + δC ∩M , f + δC∩M ) n *Cn ∩M C∩M *   2r + 2s + ρ 2r + s + ρ dρ+2s (Cn , C) . ≤ dρ+s (fn , f ) + s s Proof. For n ≥ n0 , ϕ := f + δC∩M and ϕn := fn + δCn ∩M satisfy the hypotheses of Proposition 4 and so * * * * * inf fn − inf f * ≤ dα(ρ ) (ϕ, ϕn ) ≤ dρ (ϕ, ϕn ) .  0 * * Cn ∩M

C∩M

The estimates for dρ (ϕ, ϕn ) will now follow from two applications of Proposition 5. Indeed, from (4.9) and Proposition 5, whenever ρ ≥ 2r+t and n is such dρ+2s (Cn , C). that dρ+2s (Cn , C) < s, then dρ+s (Cn ∩M, C ∩M ) ≤ 2r+s+(ρ+s) s Taking n to be such that (4.13) holds, we find that dρ+s (fn , f ) + dρ+s (Cn ∩ M, C ∩ M ) < s, so from (4.10) and Proposition 5 again, 2r + s + ρ dρ (ϕ, ϕn ) ≤ [dρ+s (fn , f ) + dρ+s (Cn ∩ M, C ∩ M )] s   2r + 2s + ρ 2r + s + ρ dρ+2s (Cn , C) . ≤ dρ+s (fn , f ) + s s If we keep fn fixed in this process, then only one iteration of Proposition 5 is required, which will lead to a better coefficient for the rate of convergence than that obtained from taking dρ (fn , f ) ≡ 0 in Corollary 3. Corollary 4. Let Cn , C and M be closed convex subsets of a Banach space X and let f be a proper closed convex R-valued functions on X with f ≥ 0 and Attouch–Wets convergence Cn → C. Suppose that for some s > 0, t ≥ 0 and r ≥ 0,

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Convergence of truncates in l1 optimal feedback control

B(0, s)2 ⊆ Δ(X) ∩ B(0, r)2 − [({f ≤ r} ∩ M ) × C] ∩ B(0, t)2 .

71

(4.14)

Assume further that for some n0 ∈ N, α ∈ R and ρ0 > 0, inf

Cn0 ∩M

f ≤ α and B(0, ρ0 ) ⊇ {f ≤ α + 1} ∩ C ∩ M .

(4.15)

Then for any fixed ρ ≥ max{2r + t, ρ0 , α + 1}, and all n ≥ n0 for which dρ+s (Cn , C) < s , we have

(4.16)

* * * * * inf f − inf f * ≤ dρ (f + δC ∩M , f + δC∩M ) n *Cn ∩M C∩M * ≤

2r + s + ρ dρ+s (Cn , C) . s

Proof. Similar to that of Corollary 3, but with only one appeal to Proposition 5. 

4.6 Verification of the constraint qualification Our intention will be to apply the results of the preceding section with X = l1 ×l1 (with the box norm), so the norm on X ×X will be the four-fold product of the norm on l1 . From now on we will not notationally distinguish balls in the various product spaces, the dimensionality being clear from context. Before we can apply the convergence theory of Section 4.5, we need to check that the constraint qualifications (4.9) and (4.10) (or (4.14)) are satisfied. This will be the main concern in this section. First, we consider some more readily verifiable sufficient conditions for (4.9) and (4.10) to hold. Lemma 5. Suppose that for sets C and M we have B(0, s) ⊆ C ∩ B(0, σ) − M

(4.17)

for some s and σ. Then it follows that B(0, s/2)2 ⊆ Δ(X) ∩ B(0, σ + 3s/2)2 − (C × M ) ∩ B(0, σ + s)2

(4.18)

(of the form (4.9)). Next, suppose we have B(0, μ) ⊆ {f ≤ ν} ∩ B(0, λ) − (C ∩ M ) ∩ B(0, λ) for some λ, μ and ν. Then it follows that

(4.19)

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B(0, μ/2)2 ⊆ Δ(X) ∩ B(0, μ/2 + λ)2 − ({f ≤ ν} × (C ∩ M )) ∩ B(0, λ)2 (4.20) (of the form (4.10)). Also, B(0, s) ⊆ C ∩ B(0, σ) − {f ≤ r} ∩ M ∩ B(0, σ + s)

(4.21)

implies B(0, s/2)2 ⊆ Δ(X) ∩ B(0, σ + 3s/2)2 − (C × ({f ≤ r} ∩ M )) ∩ B(0, σ + s)2 (4.22) (which is of the form of (4.14)). Proof. Now (4.17) implies B(0, s) ⊆ C ∩ B(0, σ) − M ∩ B(0, σ + s). Place D := Δ(X)−(C ×M )∩B(0, σ +s)2 . Then, if P denotes the subtraction map taking (x, y) to y − x, we have P (D) = C ∩ B(0, σ + s) − M ∩ B(0, σ + s) ⊇ C ∩ B(0, σ) − M ∩ B(0, σ + s) ⊇ B(0, s), and hence B(0, s/2)2 ⊆ P −1 (B(0, s)) ⊆ P −1 P (D) = D + P −1 (0) = D + Δ(X) = D since Δ(X) + Δ(X) = Δ(X). Thus B(0, s/2)2 ⊆ Δ(X) − (C × M ) ∩ B(0, σ + s)2 , from which (4.18) clearly follows (and is of the form (4.9) for suitable r, s and t). Next, suppose we have (4.19). If we define D := Δ(X) − ({f ≤ ν} × (C ∩ M ))∩B(0, λ)2 , then similarly P (D) = {f ≤ ν}∩B(0, λ)−(C ∩M )∩B(0, λ) ⊇ B(0, μ) so, proceeding as above, we obtain (4.20) (which is of the form (4.10)). The last assertion follows from the first on substituting {f ≤ r} ∩ M for M therein. In this chapter, the objective functions f will always be chosen such that (4.17) will imply (4.19) or (4.21). Hence in this section we focus on verification of (4.17). The constraints of M (0) have the form Ae = b, where m = 2m1 +2m2 +2m3 and b = (Re w(¯ ˆ z1 ), Im w(¯ ˆ z1 ),.., Re w(¯ ˆ zm1 ), Im w(¯ ˆ zm1 ), 0,.., 0,.., 0)T ∈ Rm with A : l1 → Rm given by

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Ae := (.., Re eˆ(¯ zi ), Im eˆ(¯ zi ),.., Re eˆ(¯ pj ), Im eˆ(¯ pj ),.., Re eˆ(¯ vk ), Im eˆ(¯ vk ),..)T , (4.23) where z¯i , p¯j and v¯k denote distinct elements of the unit disk, and i, j and k range over {1, . . . , m1 }, {1, . . . , m2 } and {1, . . . , m3 } respectively. Then A is expressible as a matrix operator of the form ⎛ ⎞ 1 Re z¯1 Re z¯12 · · · ⎜ 0 Im z¯1 Im z¯12 · · · ⎟ ⎜ ⎟ ⎜ .. ⎟ .. .. ⎜. ⎟ . . ⎜ ⎟ 2 ⎜ 1 Re z¯m1 Re z¯m ⎟ · · · 1 ⎜ ⎟ 2 ⎜ 0 Im z¯m1 Im z¯m ⎟ · · · 1 ⎜ ⎟ ⎜ 1 Re p¯1 Re p¯21 · · · ⎟ ⎜ ⎟ ⎜ 0 Im p¯1 Im p¯21 · · · ⎟ ⎜ ⎟ ⎜ ⎟ .. .. (aij )1≤i≤m; 0≤j<∞ = ⎜ ... ⎟, . . ⎜ ⎟ ⎜ 1 Re p¯m Re p¯2 · · · ⎟ m2 2 ⎜ ⎟ ⎜ 0 Im p¯m Im p¯2 · · · ⎟ m2 2 ⎜ ⎟ ⎜ 1 Re v¯1 Re v¯2 · · · ⎟ 1 ⎜ ⎟ ⎜ 0 Im v¯1 Im v¯2 · · · ⎟ 1 ⎜ ⎟ ⎜. ⎟ .. .. ⎜ .. ⎟ . . ⎜ ⎟ ⎝ 1 Re v¯m Re v¯2 · · · ⎠ m3 3 2 ··· 0 Im v¯m3 Im v¯m 3 where rows of imaginary parts of the matrix (aij ) and of b are omitted whenever the associated z¯i , p¯j or v¯k is real. For integer K, define A(K) to be the truncated operator taking RK into m R given by the matrix (aij )1≤i≤m,0≤j
β ∞ = 1 A(K) ξ = β

is finite. In particular, αK ≤ αm for all K ≥ m and 2 2' 2 (m) (−1 2 2, A αm = 2 2 2 the norm being taken relative to the 1-norm on the range of the inverse 3 (m) 4−1 A and the ∞-norm on its domain. Note that αK satisfies (∀β ∈ Rm )(∃ξ ∈ RK )(A(K) ξ = βand ξ 1 ≤ αK β ∞ ).

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Lemma 6. Let C (e) := {e ∈ l1 | Bi ≤ ei ≤ Ai ; i = 0, 1, 2, . . . } where the bounds Bi , Ai satisfy: ∃ξ ∈ RK such that Aξ = b and Bi < ξi < Ai for i = 0, 1, . . . , K − 1; and Bi ≤ 0 ≤ Ai for i ≥ K. Let K ≥ m, and let αK be as in (4.24). Also, let  :=

min

0≤i≤K−1

{|Ai − ξi |, |Bi − ξi |}

(so  > 0). Then −1 −1 B(0, αK ) ⊆ C (e) ∩ B(0,  + ξ 1 ) − M (0) ∩ B(0, (1 + αK ) + ξ 1 ) .

Proof. Place s := /αK and let η ∈ B(0, s). Set e := {ξ, 0, 0, ..} ∈ C (e) ∩M (0) . As A(K) maps onto Rm , there exists p ∈ RK ⊆ l1 such that Ap = A(K) p = Aη with p 1 ≤ αK Aη ∞ ≤ αK max {|ˆ η (¯ zi )|, |ˆ η (¯ pj )|, |ˆ η (¯ vk )|} ≤ αK η 1 <  . z¯i ,p¯j ,¯ vk

Consequently, p ∈ C (e) − e since for each i ≤ K − 1, |pi | <  ≤ mini κ d 1 (ρ + ¯ e 1 ), where κ 3 B(0, μ) ⊆ C (u) − u is given in Lemma 4. e 1 )}, we have Then, if s := min{τ, μ − κ d 1 (ρ + ¯ B(0, s) ⊆ C ∩ B(0, max{σ, μ + ¯ u 1 }) − M.

(4.25)

Proof. Let (ξ, η) ∈ B(0, s). From Assumption 7, ξ = v − e ∈ C (e) ∩ B(0, σ) − M (0) ∩ B(0, ρ) so ξ = v  − e where v  = v − e¯ ∈ (C (e) − e¯) ∩ B(0, σ + ¯ e 1 ) e 1 ). Place u = η + T e , then and e = e − e¯ ∈ (M (0) − e¯) ∩ B(0, ρ + ¯ u 1 ≤ η 1 + T e 1 ≤ η 1 + T ( ¯ e 1 + ρ) < s + T ( ¯ e 1 + ρ) ≤ μ,

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where the last inequality follows by Remark 1, and so u ∈ C (u) − u ¯ from Asu 1 ). Also, v  + e¯ = v ∈ C (e) ∩B(0, σ) sumption 7 and u+ u ¯ ∈ C (u) ∩B(0, μ+ ¯ and e, u ¯)) (ξ, η) = (v  , u) − (e , T e ) ∈ (v  , u) − (M − (¯ = (v  + e¯, u + u ¯) − M = (v, u + u ¯) − M ⊆ C ∩ B(0, max{σ, μ + ¯ u 1 }) − M.



Remark 4. The existence of (¯ e, u ¯) in M of finite length is ensured, for in(e) (e) stance, under the conditions of Lemma 3. If the bounds Ai and Bi satisfy (e) (e) Bi < e¯i < Ai for i ≤ l(¯ e) (where l(·) denotes length), then Condition 7 of Lemma 7 follows, for suitable constants, from Lemma 6. (Note again, that (e) (e) this holds for arbitrary Ai and Bi for i > l(¯ e), so by making these bounds decay to zero sufficiently rapidly, as will be shown in Lemma 13 to come, we can enforce compactness of C (e) , which will be essential for the Attouch–Wets (e) convergence of Cn to C (e) ). (u) (u) If, furthermore, the bounds Ai and Bi are chosen to envelop u ¯ (∀i, (u) (u) Bi < u ¯i < Ai ), and to be bounded away from zero by sufficient distance, for all i, Condition 7 of Lemma 7 is also satisfied. Remark 5. Note that 0 ∈ int (C − M ) iff (4.9) holds for some r, s and t. Indeed, if 0 ∈ int (C−M ) then by Corollary 1 we obtain (4.17) for some s and σ, which implies (4.18), which is of the form (4.9) for suitable constants. Conversely, (4.9) implies (where P (x, y) := x−y) 0 ∈ int P (Δ(X) ∩ B(0, r)2 − (C × M ) ∩ B(0, t)2 ) ⊆ int (C ∩ B(0, t) − M ∩ B(0, t)) ⊆ int (C − M ). If we are interested only in knowing that Cn ∩ M Attouch–Wets converges to C ∩ M , and not in the actual rate of such convergence, then 0 ∈ int (C − M ) certainly suffices for the applicability of the results of the theory to this end. Note however that in this case, the error bounds obtained in Section 4.7 are now not “computable” since we do not have explicit values for the constants r, s and t appearing in the constraint qualification. A sufficient condition for 0 ∈ int (C − M ) may be obtained on modification of Lemma 7. Lemma 8. Let C (e) , C (u) , C and M be as above, and assume that: 1 there exists (¯ e, u ¯) ∈ C ∩ M such that 0 ∈ core (C (u) − u ¯); (e) 2 0 ∈ core (C − M (0) ); and 3n ˆ has no zeros on the unit circle. Then 0 ∈ int (C − M ).

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Proof. Note first that   cone (C (e) − e¯) × (C (u) − u ¯) = cone (C (e) − e¯) × cone (C (u) − u ¯) ¯ are convex sets containing 0. Then (where T since C (e) − e¯ and C (u) − u denotes the mapping introduced in Remark 1) cone (C − M ) = cone (C − (¯ e, u ¯)) − (M − (¯ e, u ¯)) = cone (C (e) − e¯) × cone (C (u) − u ¯) − (M − (¯ e, u ¯)) (e) 1 = cone (C − e¯) × l −{(e, u) | e ∈ M (0) − e¯, d ∗ e + n ∗ u = 0} = cone (C (e) − e¯) × l1 − {(e, T e) | e ∈ M (0) − e¯} ⊆ cone (C (e) − M (0) ) × l1 , where the final inclusion is in fact an equality, as follows by noting that if (ξ, η) ∈ cone (C (e) −M (0) )×l1 , then ξ ∈ cone (C (e) −M (0) ) = cone (C (e) −¯ e)− (M (0) − e¯) so ξ = v − e for some v ∈ cone (C (e) − e¯) and e ∈ M (0) − e¯. Setting e)×l1 −{(e, T e) | u := η+T e ∈ l1 yields (ξ, η) = (v, u)−(e, T e) ∈ cone (C (e) −¯ e ∈ M (0) − e¯}. Thus cone (C − M ) = cone (C (e) − M (0) ) × l1 , which by the assumptions yields 0 ∈ core (C − M ) and the result then follows from Corollary 1. 

4.6.1 Limitations on the truncation scheme In Section 4.7, we will apply Corollary 3 to deduce various convergence re(e) (u) sults. For this, it will be necessary that Ck := Ck × Ck Attouch–Wets (u) converges to C = C (e) × C (u) , where Ck denote the corresponding trun(u) cations of C . Recall that we need a condition of the form (4.17) for the application of the convergence theory of Section 4.5. This has an untoward consequence in relation to convergence of truncations of C (u) . From Lemma 9 below, we see that Attouch–Wets convergence of Ck to C is impossible unless (u) we keep Ck = C (u) for all k; indeed, if truncations of C (u) are included, then Attouch–Wets convergence will occur if and only if C, and hence C (u) , is locally compact (in the sense of having compact intersections with all closed balls), which is incompatible with the constraint qualification (4.17), as we shall observe in Lemma 10. Further, if instead we try to truncate the space M to form an expanding family of finite-dimensional subspaces Mn , then similarly, any Attouch–Wets convergence of Mn to M demands local compactness of M , which is an impossibility since the latter has infinite dimension. We therefore use truncations in the e-variable only, yielding the form Ck := (e) Ck ×C (u) . Thus our truncations will generally not consist purely of elements (e, u) of fixed finite length. It will be shown currently however (see the end

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of this section) that each Cn ∩ M is in fact contained in a finite-dimensional subspace, but the basis thereof may consist of infinite-length members (in the u-variable). If we wish for these truncations to contain only (e, u) of some fixed finite length dependent only on n, then further assumptions on the plant will be required (see Lemma 16). Lemma 9. Let X be a Banach space, let Cn and C be closed convex subsets, with Cn ⊆ C for all n, and Cn Attouch–Wets convergent to C. Assume also ¯ ρ) is compact. Then C ∩ B(0, ¯ ρ) is for each n ∈ N and ρ > 0 that Cn ∩ B(0, compact whenever C ∩ B(0, ρ) is nonempty. ¯ ρ) is totally bounded. Now 0 ∈ int (C− Proof. It suffices to show that C∩B(0, ¯ ρ)) and hence B(0, s) ⊆ C∩B(0, ¯ ρ+s)−B(0, ¯ ρ) for some s > 0 (see (4.17) B(0, ¯ ρ)). On comparing (4.17) and (4.18) we find the indicator with M := B(0, functions fn = δCn , f = δC and g = δB(0,ρ) satisfy a condition of the form ¯ ¯ ρ) Attouch–Wets converges to C∩B(0, ¯ ρ). (4.8), so by Proposition 5, Cn ∩B(0, ¯ ρ) ⊆ Let  > 0. By this convergence, there exists n such that C ∩ B(0, ¯ ρ) + B(0, /2). From the compactness of Cn ∩ B(0, ¯ ρ), there exist Cn ∩ B(0, 5N ¯ ¯ x1 , . . . , xN in Cn ∩ B(0, ρ) ⊆ C ∩ B(0, ρ) such that i=1 B(xi , /2) contains ¯ ρ). Hence C ∩ B(0, ¯ ρ) ⊆ 5N B(xi , ). Cn ∩ B(0, i=1 Lemma 10. Suppose that C = C (e) × C (u) is locally compact in the sense of Lemma 9 and n ˆ has no zeros on the unit circle. Then 0∈ / int (C − M ) . Proof. Supposing the contrary, Corollary 1 yields ρ > 0 satisfying cone (C ∩ B(0, ρ) − M ) = l1 × l1 , which in turn implies that (∀(ξ, η) ∈ l1 × l1 )(∃e ∈ M (0) − e¯)

(4.26)

with ξ + e ∈ cone (C (e) ∩ B(0, ρ) − e¯) and η + T e ∈ cone (C (u) ∩ B(0, ρ) − u ¯), where T is as in Remark 1, and (¯ e, u ¯) is a fixed member of C ∩ B(0, ρ) ∩ M . Let χ ∈ l1 . By the surjectivity of A : l1 → Rm given in (4.23), there exists ˆ z ) = χ(¯ ˆ z ). Place η := ˆ , ξ(¯ ˆ z )/d(¯ ξ ∈ l'1 such that(for each zero z¯ in D for n ˆ n . Since χ ˆ − dˆξ)/ˆ ˆ − dˆξˆ now must have a zero at each (simple) zero Z −1 (χ in D for n ˆ , and the latter has no zeros on the unit circle, we have η ∈ l1 by Lemma 4. Thus χ = d ∗ ξ + n ∗ η. With e ∈ M (0) − e¯ as in (4.26), it follows (on noting that d ∗ e + n ∗ T e = 0 from the definition of T ) that χ = d∗ξ+n∗η ∈ d ∗ cone (C (e) ∩ B(0, ρ) − e¯) + n ∗ cone (C (u) ∩ B(0, ρ) − u ¯) −(d ∗ e + n ∗ T e) = cone (d ∗ (C ∩ B(0, ρ) − e¯)) + cone (n ∗ (C (u) ∩ B(0, ρ) − u ¯)) ' ( ⊆ cone d ∗ (C (e) ∩ B(0, ρ) − e¯) + n ∗ (C (u) ∩ B(0, ρ) − u ¯) , (e)

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where the latter inclusion follows since both C (e) ∩ B(0, ρ) − e¯ and C (u) ∩ B(0, ρ) − u ¯ are convex sets containing 0. Since χ ∈ l1 is arbitrary, ' ( cone d ∗ (C (e) ∩ B(0, ρ) − e¯) + n ∗ (C (u) ∩ B(0, ρ) − u ¯) = l 1 , ¯ ρ) − e¯) + n ∗ (C (u) ∩ B(0, ¯ ρ) − u ¯) so the compact convex set d ∗ (C (e) ∩ B(0, has a nonempty core (Definition 1) and hence by Proposition 1, a nonempty interior. However, this latter property is forbidden for any compact subset of an infinite-dimensional normed space. Thus we arrive at a contradiction.  We end this section with the promised verification that each truncation (e) Cn ∩ M = (Cn × C (u) ) ∩ M is indeed finite-dimensional. Lemma 11. Under the assumptions of this section, Ck ∩ M is of finite dimension for each k. Proof. Assume that C ∩ M has a member (¯ e, u ¯) with e¯ of finite length. Now e)}. Since d∗(e− e¯)+n∗(u− u ¯) = 0 let (e, u) ∈ Ck ∩M , and let K := max{k, l(¯ ¯ = T (e − e¯). Since e − e¯ ∈ (M (0) − e¯) ∩ RK , and e − e¯ ∈ M (0) − e¯, then u − u K we can write e − e¯ = i=1 αi e(i) where the {e(i) }K i=0 is some spanning set for (M (0) − e¯) ∩ RK . Placing u(i) := T e(i) ∈ l1 , we obtain K 

K  αi u(i) = T ( αi e(i) ) = T (e − e¯) = u − u ¯

i=1

i=1

K so that u ∈ u ¯ +span {u(i) }K u +span {u(i) }K i=1 , and hence Ck ∩M ⊆ R ×(¯ i=1 ), a subspace of finite dimension. Note again that there is no guarantee that any of the u(i) has finite length.

4.7 Convergence of approximates As asserted in the opening paragraph of Section 4.6.1, if we wish to apply convergence theory, we cannot simultaneously truncate in both e and u. (e) Accordingly, our truncations will always be of the form Cn = Cn × C (u) . The following two lemmas show that compactness of C (e) is essential for the (e) Attouch–Wets convergence of the truncations Cn and Cn . Lemma 12. Let C, C (e) and C (u) be as usual, with also ∞ (e) (e) i=0 max{|Ai |, |Bi |} < +∞. Then dρ (Cn , C) =



(e)

(e)

max{|Ai |, |Bi |}for any ρ ≥

i≥n

and Cn Attouch–Wets converges to C.

∞  i=0

(e)

(e)

max{|Ai |, |Bi |}

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(e)

Proof. Since Cn ⊆ C (e) for all n, we have dρ (Cn , C) = dρ (Cn(e) × C (u) , C (e) × C (u) ) = dρ (C (e) , Cn(e) ) = eρ (C (e) , Cn(e) ) , (e)

(e) (e) and we compute  the latter. Let e ∈ C = C ∩ B(0, ρ). Then d(e, Cn ) = e − e(n) = i≥n |ei | where e(n) denotes the truncation to length n (that is, e(n) = (e0 , ..., en−1 , 0, 0, ..)), and hence as n → ∞   (e) (e) eρ (C (e) , Cn(e) ) = sup |ei | = max{|Ai |, |Bi |} → 0 .  e∈C (e) i≥n

i≥n

Lemma 13. The set C (e) is compact in l1 if and only if the bounds satisfy ∞ 

(e)

(e)

max{|Ai |, |Bi |} < +∞ .

i=0 (e)

(e)

Proof. Let i0 be such that Bi ≤ 0 ≤ Ai for all i ≥ i0 . If C (e) is compact, (e) (e) then xn := (A0 , .., An , 0, 0, ..) ∈ C (e) (n ≥ i0 ) must have a convergent subsequence, along which we then have the uniform boundedness of the norms nk ∞ (e) (e) |Ai |, and since these increase with k, i=0 |Ai | is finite. xnk = i=0 ∞ Similarly, i=0 |Bi | is finite. ∞ (e) (e) (e) Conversely, if i=0 max{|Ai |, |Bi |} < +∞, the compactness of C (e) follows from Lemma 9 since its truncations Cn are all compact and Attouch– (e) Wets converges to C by Lemma 12.  The next lemma shows that C ∩M is always bounded whenever the bounds on e define sequences in l1 . This ensures that condition (4.12) will always be satisfied for any objective f . Lemma 14. Suppose (as usual here) that n ˆ has no zeros on the unit circle and that all its zeros in the unit disk are simple. Then C ∩ M ⊆ B(0, ρ0 ), where ∞ 6 7  (e) (e) ρ0 = max max |Ai |, |Bi | , i=0

 κ b 1 + d 1

∞ 

6 7 (e) (e) max |Ai |, |Bi |

i=0

where κ is as in Lemma 4 and b := w ∗ d − n ∗ (βc c + βs s). ˆe Proof. Let (e, u) ∈ C ∩ M , then from the relation d ∗ e +n ∗ u = b, ˆb − dˆ ˆ ˆ ¯ of n has zeros at each zero in D ˆ , and since u = Z −1 b−dˆe , Lemma 4 yields n ˆ

that u 1 ≤ κ b − d ∗ e 1 ≤ κ( b 1 + d 1 e 1 ). From this, and the relation ∞ (e) (e) e 1 ≤ i=0 max{|Ai |, |Bi |}, the result follows. 

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Assembling all the parts we obtain our main result. Theorem 1. Let fn and f be proper closed convex R-valued functions, with fn Attouch–Wets convergent to f and f ≥ 0. Also, assume the following for C and M : 1n ˆ has no zeros on the unit circle, and all its zeros in the unit disk are simple; (e) (e) (e) 2 The bounds {Bi , Ai }∞ form sequences in l1 , and i=0 characterizing C also satisfy the requirement that for some K ≥ m, ∃ξ ∈ RK such that (e) (e) (e) (e) Aξ = b with Bi < ξi < Ai for i = 0, 1, . . . , K − 1; and Bi ≤ 0 ≤ Ai for i ≥ K; ¯ 3 There exists (¯ e, u ¯) ∈ C ∩ M of finite length such that B(0, μ) ⊆ C (u) − u −1 ) + ξ 1 + ¯ e 1 ], where κ is for some positive μ with μ > κ d 1 [(1 + αK given in Lemma 4 and αK in (4.24); 4 γ := max{supn≥n0 inf Cn ∩M fn , inf C∩M f } is finite; 5 B(0, μ) ⊆ {f ≤ ν} ∩ B(0, λ) − (C ∩ M ) ∩ B(0, λ) for some λ, μ and ν (that is, (4.19) holds). Define the constants  :=

min

0≤i≤K−1

{|Ai − ξi |, |Bi − ξi |}

(4.27)

8 −1 9 1 −1 min αK , μ − κ d 1 [(1 + αK ) + ξ 1 + ¯ e 1 ] 2 s := min{s , μ} r := max{ + ξ 1 + 3s , μ + ¯ u 1 + 3s , μ/2 + λ, ν} t := max{ + ξ 1 + 2s , μ + ¯ u 1 + 2s , λ} . s :=

(4.28) (4.29) (4.30) (4.31)

Let ρ0 be as in Lemma 14. Then for any fixed ρ satisfying  ∞ 6 7  (e) (e) max |Ai |, |Bi | ρ > max 2r + t, ρ0 , γ + 1, , i=0

and all n ≥ n0 for which dρ+s (fn , f ) +



(e)

i≥n

(e)

max{|Ai |, |Bi |} < s and

2r + 2s + ρ  (e) (e) max{|Ai |, |Bi |} < s , s i≥n

it follows that * * 6 7  * * (e) (e) * inf fn − inf f * ≤ (2r + s + ρ)(2r + 2s + ρ) max |A |, |B | i i * Cn ∩M C∩M * s2 i≥n

2r + s + ρ dρ+s (fn , f ) . + s

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Proof. By assumption 1 and Lemma 6, we obtain an inclusion of the form of (4.17): −1 −1 B(0, αK ) ⊆ C (e) ∩ B(0,  + ξ 1 ) − M (0) ∩ B(0, (1 + αK ) + ξ 1 ) .

This, along with assumptions 1 and 1, may be inserted into Lemma 7 to yield B(0, 2s ) ⊆ C ∩ B(0, max{ + ξ 1 , μ + ¯ u 1 }) − M . Lemma 5 then gives (where r := max{ + ξ 1 , μ + ¯ u 1 }) B(0, s ) ⊆ Δ ∩ B(0, r + 3s ) − (C × M ) ∩ B(0, r + 2s ) , which, along with (4.20) (a consequence of assumption 1 via Lemma 5) yields (4.9) and (4.10) (in Corollary 3) for r, s and t as above. Further, assumption 1 gives (4.11), and (4.12) follows from Lemma 14 with the indicated value for ρ0 . Noting the explicit form for dρ (Cn , C) in ∞ (e) (e) Lemma 12 (for ρ ≥ i=0 max{|Ai |, |Bi |}), the result may now be read from Corollary 3.  In particular, if fn = f for all n, we see that * * * * *7 6*  * * * (e) * * (e) * * inf f − inf f * ≤ (2r + s + ρ)(2r + 2s + ρ) max *Ai * , *Bi * . *Cn ∩M * 2 C∩M s i≥n

However, in this case, we can obtain better constants by using Corollary 4 in place of Corollary 3, which will require the condition (4.21) (or (4.14)). To illustrate, if f is like a norm, say, f (e, u) := e 1 + ζ u 1 (ζ > 0), then for any r > 0, {f ≤ r} ⊇ B(0, r/(2 max{1, ζ})) so if (4.17) holds (for some s, σ) then taking r = 2(s + σ) max{1, ζ} gives C ∩ B(0, σ) − {f ≤ r} ∩ M ∩ B(0, σ + s) ⊇ C ∩ B(0, σ) − M ∩ B(0, σ + s) ⊇ B(0, s)

by (4.17)

so (4.21), and hence (4.14), holds for suitable constants. Accordingly, we arrive at the following result. Theorem 2. Let f : l1 × l1 → R have the form f (e, u) := e 1 + ζ u 1 for some ζ > 0. Assume Conditions 1, 1 and 1 of Theorem 1 hold, and Condition 1 is replaced by γ  :=

inf

Cn0 ∩M

f <∞

for some n0 .

Define  as in (4.27), and s := s by (4.28), with r := 2 max{1, ζ} (max{ + ξ 1 , μ + ¯ u 1 } + 2s) and

(4.32)

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t := max{ + ξ 1 + 2s, μ + ¯ u 1 + 2s} . Then, for any fixed ρ satisfying  

ρ > max 2r + t, ρ0 , γ + 1,

∞ 

(4.33)

6 7 (e) (e) max |Ai |, |Bi |

i=0

(where ρ0 appears in Lemma 14) and all n ≥ n0 for which  (e) (e) max{|Ai |, |Bi |} < s , i≥n

we have

* * 6 7  * * (e) (e) * inf f − inf f * ≤ 2r + s + ρ max |Ai |, |Bi | . *Cn ∩M * C∩M s i≥n

Proof. This follows along similar lines to that for Theorem 1, but uses the last displayed relation before Theorem 2 to obtain (4.21) from (4.17), so that (4.14) is obtained for the above r, s and t and we may apply Corollary 4.  In summary: we have obtained • that inf C∩M f provides the exact lower bound for the performance of rational controllers for the l1 control problem, and • computable convergence estimates for the approximating truncated problems. Note further that these results are obtained for the case where the hardbound set C (or time-domain template) has no interior (since C (e) is assumed compact and hence has an empty interior). This then extends a result of [14] on such approximations (in the particular two-block control problem we consider) to cases where int C is empty. We note however that the cited result from [14] in fact has an alternate short proof by an elementary convexity argument (see Lemma 15 below) once the density in M of the subset of members of finite length is demonstrated. (This density is established, in the general “multi-block” case, in [19]. For the special two-block setup we consider, this property is proved in [26].) The above convergence results should be readily extendible to the multi-block formalism of [14]. Lemma 15. Let C, M and M0 be convex sets, with M = M0 and (int C) ∩ M = ∅. Then C ∩ M = C ∩ M0 . Proof. Let x ∈ C ∩ M and x0 ∈ (int C) ∩ M . Then for each 0 < λ < 1, xλ := λx0 + (1 − λ)x ∈ (int C) ∩ M , and xλ → x for λ → 0. For each λ, the density of M0 yields approximates to xλ from M0 which, if sufficiently close,  must be in (int C) ∩ M0 .

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This argument in fact leads to a very quick proof of [14, Theorem 5.4] (whose original proof contains a flaw, detailed in [27]) which asserts the equality inf C∩M · 1 = inf C∩Mr · 1 under a slightly stronger assumption, which in fact implies nonemptiness of (int C) ∩ M . To see the applicability of Lemma 15 to the cited result from [14], we temporarily adopt the notation thereof. Assign C and M as follows: C := {Φ ∈ l1nz ×nw | Atemp Φ ≤ btemp } and M := {Φ ∈ l1nz ×nw | Afeas Φ = bfeas } , where btemp ∈ l∞ , bfeas ∈ Rcz × l1nz ×nw , Atemp : l1nz ×nw → l∞ and Afeas : l1nz ×nw → Rcz × l1nz ×nw are bounded linear, and where the symbol ≤ stands for the partial order on l∞ induced by its standard positive cone P + . The assumption of [14, Theorem 5.4] is that btemp − Atemp Φ0 ∈ int P + for some Φ0 ∈ M . However, the continuity of Atemp implies that Φ0 ∈ int C and hence Φ0 ∈ (int C) ∩ M , which is the assumption of Lemma 15. This, coupled with the density of M0 := Mr in M , gives the result. As was discussed in Section 4.6.1, the approximating problems are constructed by truncating in the e-variable only, otherwise the method fails. However, the truncated constraint-sets Ck ∩ M satisfy dim span Ck ∩ M ≤ 2k

(for large k)

by Lemma 11, whereby the approximating minimizations are indeed over finite-dimensional sets. Since in general (e, u) ∈ Ck ∩ M does not imply that u has finite length, one may ask under what conditions it may be possible for there to exist, for each k, a uniform bound m(k) to the length of u whenever (e, u) ∈ Ck ∩ M . In this case, the truncated sets Ck ∩ M would resemble those obtained by the more natural method of simultaneously truncating both e and u (a strategy that fails to yield convergence, in general, by the results of Section 4.6.1). From the lemma to follow, such a property can hold only when the plant has no minimum-phase zeros (that is, no zeros outside the unit circle). Hence, except for some highly restrictive cases, Ck ∩ M will always contain some u of infinite length. Lemma 16. Suppose that the assumptions of Theorem 1 hold, and assume further that n ˆ has no zeros outside the open unit disk. (As usual, all zeros are assumed simple). Let (¯ e, u ¯) be in C ∩ M with both e¯ and u ¯ having finite length. Then for any k and any (e, u) ∈ Ck ∩ M , l(u) ≤ max{l(¯ u), l(d) − l(n) + max{l(¯ e), k}}, where l(·) denotes length. (e) Moreover, assuming that the conditions of Remark 4 hold (with Ai > (e) 0 > Bi for i > l(¯ e)), then if for all k, each (e, u) ∈ Ck ∩ M has u of finite length, it follows that n ˆ cannot have zeros outside the unit disk.

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Proof. Let (e, u) ∈ Ck ∩ M and let (e1 , u1 ) := (e, u) − (¯ e, u ¯). Then l(e1 ) ≤ e1 /ˆ n) from the max{l(e), l(¯ e)} and similarly for u1 . Also u1 = −d ∗ Z −1 (ˆ corresponding convolution relation defining M . Since eˆ1 is a polynomial having zeros at each zero of n ˆ in the unit disk and hence the whole plane (reˆ1 is a polynomial of degree at most call e1 ∈ M (0) − e¯), we have that u e)} − 1, and so, as u = u1 + u ¯, l(d) − l(n) + l(e1 ) − 1 ≤ l(d) − l(n) + max{l(e), l(¯ the result follows. e, u ¯) For the second assertion, suppose n ˆ (z0 ) = 0 for some |z0 | > 1. With (¯ as above, it follows from the closed–loop equation d∗¯ e +n∗¯ u = w∗d−n∗β =: ˜b ˆ 0 ) is finite and eˆ¯(z0 ) = w(z ˆ 0 ). Let k exceed (where β := βc c + βs s) that w(z (0) and the length of e¯, and let both the number of constraints defining M  

e ∈ M (0) ∩ Rk . If u := Z −1

ˆ ˜ ˆe b−dˆ n ˆ

, the interpolation constraints on e ensure

that u ˆ has no poles in the closed unit disk (so u ∈ l1 ) and hence (e, u) ∈ M . (e) Also, since from the assumptions, cone (Ck − e¯) ⊇ Rk and cone (C (u) − u ¯) = 1 l , we have (e, u) − (¯ e, u ¯) ∈ (M − (¯ e, u ¯)) ∩ (Rk × l1 )   (e) ⊆ (M − (¯ e, u ¯)) ∩ cone (Ck − e¯) × cone (C (u) − u ¯) e, u ¯)) = (M − (¯ e, u ¯)) ∩ cone (Ck − (¯ (e)

(recalling that Ck := Ck × C (u) ) = cone (Ck ∩ M − (¯ e, u ¯)) , whence λ((e, u) − (¯ e, u ¯)) ∈ Ck ∩ M − (¯ e, u ¯) for some positive λ. Since now λ(e, u) + (1 − λ)(¯ e, u ¯) ∈ Ck ∩ M , the hypothesized finiteness of length of λu + (1 − λ)¯ u implies, via the equation d ∗ (λe + (1 − λ)¯ e) + n ∗ (λu + (1 − λ)¯ u) = w ∗ d − n ∗ β , that (λe + (1 − λ)¯ e) (z0 ) = w(z ˆ 0 ) and hence eˆ(z0 ) = w(z ˆ 0 ). Since k is arbitrary, we have shown that every finite-length e ∈ M (0) satisfies an additional ˆ 0 ) at z0 , which yields a contradiction.  interpolation constraint eˆ(z0 ) = w(z Remark 6. Under the assumptions of the preceding lemma, note that for k ≥ max{l(¯ e), l(¯ u) − l(d) + l(n)}, and (e, u) ∈ Ck ∩ M , we have l(u) ≤ k + l(d) − l(n) := k + l, so for such k, Ck ∩ M consists precisely of those elements (e, u) of C ∩ M with e of length k and u of length k + l. If l ≤ 0, then Cn ∩ M = {(e, u) ∈ C ∩ M | l(e) ≤ n, l(u) ≤ n} := Qn . If l ≥ 0, then for all n ≥ max{l(¯ e), l(¯ u) − l(d) + l(n)}, Cn ∩ M ⊆ Qn+l ⊆ Cn+l ∩ M,

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85

and hence Qn Attouch–Wets converges to C ∩ M , so from Corollary 3, inf fn → inf f Qn

C∩M

as n → ∞. Observe that the sets Qn represent truncations of C ∩ M to the same length n for both e and u.

4.7.1 Some extensions So far we have considered CQ of the form of an interiority 0 ∈ int (C − M ), leading, via Proposition 5, to the determination of rates of convergence for our truncation scheme. If the CQ is weakened to the strong quasi relative interiority 0 ∈ sqri (C − M ) (meaning cone (C − M ) is a closed subspace), it is not immediately possible to apply Proposition 5. In this case, then, explicit convergence estimates may not be obtainable. We may still, however, derive limit statements of the form inf Cn ∩M f → inf C∩M f for reasonable f . To achieve this, we use an alternate result [13] on the Attouch–Wets convergence of sums of functions, based on a sqri-type CQ, but unfortunately, this result will not provide the estimates obtainable through Proposition 5. We proceed by first establishing the Attouch–Wets convergence of Cn ∩ M → C ∩ M using [13, Theorem 4.9]. In this context, the required CQ is: (1) 0 ∈ sqri (C − M ) and cone (C − M ) has closed algebraic complement Y ; and (2) that Y ∩ span (Cn − M ) = {0} for all n. Note that (1) implies (2) since Cn ⊆ C for all n, so we need only consider (1). The following two lemmas provide a sufficient condition for 0 ∈ sqri (C − M ). Lemma 17. Suppose that: 1 0 ∈ sqri (C (e) − M (0) ); ¯) for some (¯ e, u ¯) ∈ C ∩ M ; and 2 0 ∈ core (C (u) − u 3n ˆ has no zeros on the unit circle. Then 0 ∈ sqri (C − M ) and cone (C − M ) has a closed algebraic complement. Proof. By arguing as in Lemma 8, cone (C − M ) = cone (C (e) − M (0) ) × l1 . Thus it forms a closed subspace, so strong quasi relative interiority is established. Since M (0) − e¯ is a subspace of finite codimension in l1 , the complementary space Y0 to cone (C (e) − M (0) ) is finite dimensional and hence closed. Clearly then cone (C − M ) has the closed complement Y :=  Y0 × {0}. From Lemma 17 the problem is reduced to finding conditions under which 0 ∈ sqri (C (e) − M (0) ). (e)

Lemma 18. Let e¯ ∈ l1 and C (e) = {e ∈ l1 | Bi (e) (e) 0, 1, 2, . . . }, where the bounds Bi and Ai satisfy:

(e)

≤ ei ≤ Ai ; i =

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(e)

1 Bi ≤ e¯i ≤ Ai for all i ∈ N;   (e) (e) 2 i |Ai | and i |Bi | < ∞; (e) (e) ¯ik = Aik , and for all i not in 3 for a subsequence {ik }∞ k=0 we have Bik = e (e)

this subsequence, Bi

(e)

< e¯i < Ai .

Then 0 ∈ sqri (C (e) − M (0) ). Proof. After projecting into a suitable subspace of l1 , we shall follow the argument of Lemma 6. Let P denote the projection on l1 onto the subspace consisting of sequences vanishing on {ik }∞ k=0 . That is, (Pe)i := 0 for i ∈ and (Pe) := e otherwise. Evidently P is continuous and maps closed {ik }∞ i i k=0 sets to closed sets. Next, observe that (I − P)cone (C (e) − e¯) = cone (I − P)(C (e) − e¯) = {0}, since if e ∈ (C (e) − e¯) then for all k, we have eik ∈ (e) (e) [Bik − e¯ik , Aik − e¯ik ] = {0} so eik = 0, yielding (I − P)e = 0 by the definition of P. Thus we obtain cone (C (e) − M (0) ) = cone P(C (e) − M (0) ) + (I − P)(M (0) − e¯) .

(4.34)

If we can show that cone P(C (e) − M (0) ) = Pl1 then (4.34) would give cone (C (e) − M (0) ) = Pl1 + (I − P)(M (0) − e¯) = Pl1 + P(M (0) − e¯) + (M (0) − e¯) = Pl1 + (M (0) − e¯) which must be closed since Pl1 is closed and the linear subspace M (0) − e¯ has finite codimension. We now verify that cone P(C (e) −M (0) ) = Pl1 . (This part of the argument parallels that of Lemma 6.) Let ξ ∈ Pl1 , with −1 −1  := αK ξ 1 < αK

min

(e)

(e)

{|Ai − ξi |, |Bi

0≤i≤K−1; i∈{i / k}

− ξi |} > 0 .

There exists η ∈ RK ⊆ l1 such that η 1 ≤ αK ξ 1 <  with Aη = Aξ. Then (e) (e) for i ∈ / {ik }, we have Bi < e¯i + ηi < Ai whenever i < K, and for i ≥ K, (e) (e) e¯i + ηi = e¯i ∈ [Bi , Ai ], whence P(η + e¯) ∈ PC (e) (or Pη ∈ P(C (e) − e¯)). (0) Also, η − ξ ∈ M − e¯ as A(η − ξ) = 0, implying Pη − ξ = Pη − Pξ ∈ P(M (0) − e¯). Thus ξ = Pη − (Pη − ξ) ∈ P(C (e) − e¯) − P(M (0) − e¯) = P(C (e) − M (0) ). −1 This shows that B(0, αK ) ∩ Pl1 ⊆ P(C (e) − M (0) ) whence (e) (0) cone P(C − M ) = Pl1 as required.



Corollary 5. Under the conditions of Lemmas 17 and 18, and for any convex closed real-valued f : l1 × l1 → R,

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Convergence of truncates in l1 optimal feedback control

lim

87

inf f = inf f .

n→∞ Cn ∩M

C∩M

Proof. By the lemmas (and the cited result from [13]) the indicator functions δCn ∩M Attouch–Wets converge to δC∩M , and (since dom f has an interior) by any of the cited sum theorems, f + δCn ∩M → f + δC∩M also. The result then follows from Proposition 4.  Our formulation of the control problem (see Section 4.4) was chosen to permit its re-expression as an optimization over l1 × l1 . This choice resulted from a wish to compare with the other methods we described in the introduction, which used duality theory. Accordingly, we sought to formulate minimizations over a space such as l1 , which has a nice dual (namely, l∞ ). However, this is not the only problem formulation that can be treated by the methods of this paper. Recall from Section 4.4 that we considered only stabilizing controllers K ∈ S(P ) for which the resulting control signal u was restricted to the subspace Xa ⊆ l∞ . This requirement will now be relaxed to u ∈ l∞ . This will be seen to entail only trivial changes to the main results of this section. (Note that in this case the resulting optimization is taken over l1 × l∞ , so duality methods would not be readily applicable, since the dual (l∞ )∗ has a complicated characterization.) The basic feasible set is now of the form F0 := {(e, u) ∈ l1 × l∞ | e = e(K)for some K ∈ S(P ) u = K ∗ e} where u is free to range through l∞ instead of its subspace Xa . If we define (where w ∈ l∞ and has rational z-transform) the altered sets ⎧ ⎫ (e, u) ∈ l1 × l∞ | ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e ˆ (¯ p ) = 0 (¯ p pole of P : |¯ p | ≤ 1) i = 1,.., m ⎪ i i i 1 ⎪ ⎨ ⎬ M := eˆ(¯ zj ) = w(¯ ˆ zj ) (¯ zj zero of P : |¯ zj | ≤ 1) j = 1,.., m2 , ⎪ ⎪ ⎪ ⎪ ⎪ eˆ(¯ vk ) = 0 (¯ vk zero of w ˆ : |¯ vk | ≤ 1) k = 1,.., m3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ d∗e+n∗u=w∗d Mr := {(e, u) ∈ M | eˆ, u ˆ rational, e = 0}, the argument of Lemma 2 again yields Mr = F0 . Accordingly, we may now frame a minimization problem over the subset C ∩ M of l1 × l∞ where C = C (e) × C (u) but now C (u) = {u ∈ l∞ | Bi

(u)

(u)

≤ ui ≤ Ai

for all i ∈ N} ⊆ l∞ .

For simplicity, we consider only the required changes to Theorem 2, since this result has a simpler statement than Theorem 1. In fact, with the assumptions of Theorem 2, but with f : l1 ×l∞ → R given by f (e, u) = e 1 +ζ u ∞ , and u ∞ , the form of Theorem 2 is unaltered, all references to ¯ u 1 converted to ¯

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except that the ρ0 of Lemma 14 is not available, but another value may be used (namely, ρ0 = (γ  + 1) max{1, ζ −1 }). Theorem 3. Let f : l1 × l∞ → R have the form f (e, u) := e 1 + ζ u ∞ for some ζ > 0. Assume Conditions 1, 1 and 1 of Theorem 1 hold (where the interiority in Condition 1 is relative to the l∞ -norm) and Condition 1 is replaced by γ  := inf f < ∞ for some n0 . Cn0 ∩M

Define  as in (4.27), with 8 −1 9 1 −1 min αK , μ − κ d 1 [(1 + αK ) + ξ 1 + ¯ e 1 ] , 2 u ∞ } + 2s) and r := 2 max{1, ζ} (max{ + ξ 1 , μ + ¯ t := max{ + ξ 1 + 2s, μ + ¯ u ∞ + 2s} . s :=

Then for any fixed ρ satisfying  ρ > max 2r + t, (γ  + 1) max{1, ζ −1 },

∞ 

6 7 (e) (e) max |Ai |, |Bi |

-

i=0

and all n ≥ n0 for which 

(e)

(e)

max{|Ai |, |Bi |} < s ,

i≥n

we obtain * * 6 7  * * (e) (e) * inf f − inf f * ≤ 2r + s + ρ max |Ai |, |Bi | . *Cn ∩M * C∩M s i≥n

Proof. Since the form of the convergence theory of Section 4.5 (and the associated CQs) is insensitive to the particular norm used, all we need to do to obtain a proof in this context is to reinterpret all statements in u-space as referring instead to the norm · ∞ . The only places where changes occur u ∞ ; and instead of are in Lemma 7, where all references to ¯ u 1 become ¯ using ρ0 from Lemma 14, we exploit the form of f to ensure the existence of  some ρ0 for which Condition (4.15) of Proposition 4 is valid.

4.8 Appendix Remark 7. (A comment on z-transforms) ˆ On a small Let w = {wn }∞ n=0 be a sequence with a rational z-transform w. neighborhood of 0 in the complex plane, this is given by a power series

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89

∞

wn z n for such z. However, by rationality, w ˆ has extension to the whole plane by its functional form, and so is extended beyond the domain of convergence of the power series. ∞ For example, if w = (1, 1, ...), then for |z| < 1, w(z) ˆ = n=0 z n = 1/(1−z), but the function taking z to 1/(1 − z) is defined on the whole plane except at 1 and so constitutes an extension in the above sense. We shall always assume that such transforms have been so extended. n=0

Proof of Lemma 1: Let (e, u) ∈ F0 and write u = u ¯ + uc c + us s and 1 c+w s, with u ¯ and w ¯ in l . The ensuing relation d∗e+n∗u = d∗w w = w+w ¯ c s implies that ˆe + n ˆ¯ − w ˆ¯ dˆ ∈ lˆ1 (wc dˆ − uc n ˆ )ˆ c + (ws dˆ − us n ˆ )ˆ s ≡ dˆ ˆu ¯ Since and so cannot have any poles in the closed disk D. cˆ(z) =

z sin θ 1 − z cos θ and sˆ(z) = , (z − a)(z − a ¯) (z − a)(z − a ¯)

it follows that z →

ˆ − uc n ˆ − us n (wc d(z) ˆ (z))(1 − z cos θ) + (ws d(z) ˆ (z))z sin θ (z − a)(z − a ¯)

¯ and hence none at all in C, and must be a polynomial (so has no poles in D that d ∗ (wc c + ws s) − n ∗ (uc c + us s) has finite length). If now θ = 0, so s = 0, the above amounts to stating that z →

ˆ − uc n wc d(z) ˆ (z) 1−z

ˆ is polynomial, from which we obtain uc = wc d(1)/ˆ n(1) = wc /P (1). Simˆ cn wc d−u ˆ ilarly, if θ = π (so again s = 0), we have polynomial, so that 1+· ˆ n(−1) = wc /P (−1). For other values of θ, similar reasonuc = wc d(−1)/ˆ ing implies ˆ − uc n ˆ − us n (wc d(a) ˆ (a))(1 − a cos θ) + (ws d(a) ˆ (a))a sin θ = 0. Rerranging terms and taking real and imaginary parts yields us cos θ sin θ + uc sin2 θ = Re us sin2 θ − uc sin θ cos θ

ws a sin θ+wc (1−a cos θ) P (a)

= Im

and

ws a sin θ+wc (1−a cos θ) P (a)

,

with the right-hand side vanishing if P has a pole at a. Solving this for uc and us gives the desired relation. 

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Proof of Lemma 2: It suffices to show that Mr ⊆ F, since the reverse ˆ n − eˆ/(ˆ nw ˆ d). inclusion is easy to demonstrate. Let (e, u) ∈ Mr . Define R := yˆ/ˆ ˆ ˆ d(ˆ y −Rˆ n) Then R = yˆ/ˆ n since e = 0. If we can prove that R ∈ R∞ then eˆ = w ˆ x + Rd) ˆ and thus ˆ d(ˆ and from the convolution relation, u ˆ = βc cˆ + βs sˆ + w (e, u) ∈ F. ¯ the only candidates for poles of R are the Now, in the closed unit disk D, poles/zeros of P and the zeros of w. ˆ The proof now proceeds by successive elimination of these possibilities. ˆ p) = 0 and n If p¯ is such a pole (so d(¯ ˆ (¯ p) = 0), then, if p¯ is not a pole for w, ˆ w(¯ ˆ p) is finite and nonzero (from the assumptions on pole/zero positioning), ˆ which itself is implied so the nonsingularity of R at p¯ follows from that of eˆ/d, by the interpolation constraint eˆ(¯ p) = 0, whereby eˆ has a zero at p¯ cancelling ˆ If p¯ is also a pole of w the simple zero at p¯ for d. ˆ (so p¯ = a or p¯ = a ¯) then ˆ is finite and nonzero at a and a w ˆ d(·) ¯, so R has no pole there, regardless of the value of eˆ(a). ˆ z ) = 0 and n ¯ for P (so d(¯ If z¯ is a zero in D ˆ (¯ z ) = 0), then again w ˆ is finite and nonzero here. Now R is expressible as nˆ1dˆ(1 − weˆˆ ) − xdˆˆ at least in a punctured neighborhood of z¯, where we used the relation x ˆn ˆ + yˆdˆ = 1. The interpolation constraint eˆ(¯ z ) = w(¯ ˆ z ) = 0 means that 1 − weˆˆ has a zero at z¯, cancelling the simple zero of n ˆ dˆ there. Again the singularity of R here is removable. If v¯ is a zero of w, ˆ it is neither a pole nor a zero of P (by assumption) so ˆ v ) are both nonzero. The constraint eˆ(¯ n ˆ (¯ v ) and d(¯ v ) = 0 then implies that eˆ/w, ˆ and hence R, has a removable singularity there. Thus R has no poles in ¯ as claimed.  D The proof of Lemma 4 will follow from the next two elementary lemmas. 1 (C) and let |a| < 1 be a zero of fˆ. Then fˆ(·)/(· − a) ∈ Lemma 19. Let fˆ ∈ l 1 (C) and is in l 1 if fˆ ∈ l 1 and a ∈ R. l

Proof. The indicated function is analytic on a neighborhood of 0 in the plane, so q := Z −1 [fˆ(·)/(· − a)] exists as a complex sequence. Assume now that a = 0 (the argument for a = 0 is trivial and hence omitted). Then since q = Z −1 [1/(· − a)] ∗ f = − a1 (1, a1 , a12 , . . . ) ∗ f , 1 qk = − ak+1

k j=0

fj aj =

1 ak+1

 j>k

fj aj since fˆ(a) = 0.

Then, where an interchange of the order of summation occurs below, ∞ ∞    1 1  j q 1 ≤ |f ||a| = |fj ||a|j j k+1 |a| |a|k+1 j=1 k=0 j>k k<j + , ∞  1 f 1 j . ≤ |fj ||a| ≤ j (1 − |a|) |a| 1 − |a| j=1



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Convergence of truncates in l1 optimal feedback control

1 (C) and |a| > 1, then Lemma 20. If fˆ ∈ l and a ∈ R.

fˆ(·) ·−a

91

1 (C) and is in l 1 if fˆ ∈ l 1 ∈ l

Proof. As in the preceding lemma, let q be the inverse transform of the function in question. From the expression for qk given there, and an interchange of summations, ∞ ∞   1  1 f 1  = |fj ||a|j = |fj ||a|j  q 1 ≤ 1 . k+1 |a| 1 − |a| |a|j 1 − 1 j=0 j=0 k≥j |a|

Proof of Lemma 4: By Lemma 20, no generality is lostby assuming that ¯i ) p has no zeros outside the open unit disk. Write p(z) = C i (z − ai )(z − a ¯i are the zeros of p, with the understanding that for real ai where ai and a we only include the single factor z − ai in the above product. Also, we allow the possibility that the ai are nondistinct. Now, as fˆ(a1 ) = 0, Lemma 19 1 (C). If a is real, then the function is in l 1 —if implies that fˆ(·)/(· − a1 ) ∈ l 1 ¯1 and since fˆ(¯ a1 ) = 0, so fˆ(·)/(· − a1 ) has a zero at a ¯1 complex, then a1 = a fˆ(·) 1 (C) and hence is also in l 1 since it and by Lemma 19 again, ∈ l (·−a1 )(·−¯ a1 )

is a symmetric function of z. Continue inductively to complete the proof. Proof of Lemma 3: Let b1 , .., bp ∈ R and a1 , .., aq ∈ C\IR denote the ¯ the zeros of w ¯ along with the zeros collection of poles/zeros of P in D, ˆ in D, ¯ of P outside D. From the assumed distinctness of these, it follows that the corresponding square matrix ⎞ ⎛ 1 b1 · · · bp+2q−1 1 ⎟ ⎜. .. .. ⎟ ⎜ .. . . ⎟ ⎜ ⎜1 b p+2q−1 ⎟ · · · bp p ⎟ ⎜ ⎜ p+2q−1 ⎟ ⎟ ⎜ 1 Re a1 · · · Re a1 ⎟ ⎜. .. .. ⎟ ⎜. ⎟ ⎜. . . ⎜ p+2q−1 ⎟ ⎟ ⎜ 1 Re aq · · · Re aq ⎟ ⎜ ⎟ ⎜ 0 Im a1 · · · Im ap+2q−1 1 ⎟ ⎜ .. .. ⎟ ⎜ .. ⎠ ⎝. . . 0 Im aq · · · Im ap+2q−1 q has full rank over R. ¯ is a zero of P , then w(¯ Now, note that if z¯ ∈ / D ˆ z ) is finite. To see this, ˆ note that d(¯ z ) = 0, so by (2), (w − wc c − ws s)ˆ has no pole at z¯, and hence ˆ j) neither has w, ˆ since z¯ equals neither a nor its conjugate. Thus w(a ˆ i ) and w(b are all finite. By the surjectivity of the above matrix, and constructing an ˆ j ), we can find a nonzero appropriate real (p + 2q)-vector from w(a ˆ i ) and w(b e ∈ l1 of finite length such that all the interpolation constraints of M are satisfied, with, furthermore, ¯ of P . eˆ(¯ z ) = w(¯ ˆ z )at each zero z¯ ∈ /D

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We now seek u ∈ l1 of finite length such that u ˆ= =

ˆe w ˆ dˆ − n ˆ (βc cˆ + βs sˆ) − dˆ n ˆ 6 7 ˆ ˆe ˆ ¯ d + d(wc cˆ + ws sˆ) − n w ˆ (βc cˆ + βs sˆ) − dˆ n ˆ

where w ¯ := w − wc c − ws s. From the definition of βc and βs (see Lemma 1), ˆ c cˆ + ws sˆ) − n ˆ (βc cˆ + βs sˆ) has no pole at a (nor at a ¯) and it follows that d(w so must be polynomial, and hence the numerator in the above expression for ¯ dˆ and eˆ are. To show that u u ˆ is polynomial, since w ˆ is polynomial, we only need to see that the numerator has a zero at each zero of n ˆ . (Recall that we assumed all these rational transforms to be extended by their functional form to the whole complex plane, as in Remark 7). Indeed, let z¯ be a zero of n ˆ (that is, of P ). Then z¯ = a, a ¯ and eˆ(¯ z ) = w(¯ ˆ z ), and since cˆ(¯ z ) and sˆ(¯ z ) are both finite, the numerator evaluated at z¯ is ˆ z) ˆ z )−d(¯ ˆ z )(wc cˆ(¯ ¯ z )d(¯ z ) + ws sˆ(¯ z )) + n ˆ (¯ z )(βc cˆ(¯ z ) + βs sˆ(¯ z )) − eˆ(¯ z )d(¯ w(¯ ˆ z ) − d(¯ ˆ z )(wc cˆ(¯ ˆ z )d(¯ ˆ z) ¯ z )d(¯ = w(¯ z ) + ws sˆ(¯ z )) − φ(¯ ˆ z) = 0 = (w(¯ ˆ z ) − eˆ(¯ z ))d(¯ as claimed. Thus u has finite length.

References 1. H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series (Pitman, London, 1984). 2. H. Attouch and R. J.–B. Wets, (1991), Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc., 328 (1991), 695–729. 3. H. Attouch and R. J.–B. Wets, (1993), Quantitative stability of variational systems: II. A framework for nonlinear conditioning, SIAM J. Optim., 3, No. 2 (1993), 359–381. 4. H. Attouch and R. J.–B. Wets, Quantitative stability of variational systems: III. – approximate solutions, Math. Programming, 61 (1993), 197–214. 5. D. Az´ e and J.–P. Penot, Operations on convergent families of sets and functions, Optimization, 21, No. 4 (1990), 521–534. 6. S. P. Boyd and C. H. Barratt, Linear Controller Design: Limits of Performance (Prentice–Hall, Englewood Cliffs, NJ 1991). 7. G. Beer, Topologies on Closed and Closed Convex Sets, Mathematics and its Applications, 268 (Kluwer Academic Publishers, Dordrecht, 1993). 8. J. M. Borwein and A. S. Lewis, Partially–finite convex programming, Part I: Quasi relative interiors and duality theory, Math. Programming, 57 (1992), 15–48. 9. M. A. Dahleh and J. B. Pearson, l1 -Optimal feedback controllers for MIMO discrete– time systems, IEEE Trans. Automat. Control, AC-32 (1987), 314–322. 10. M. A. Dahleh and I. J. Diaz-Bobillo, Control of Uncertain Systems: A Linear Programming Approach (Prentice–Hall, Englewood Cliffs, NJ 1995). 11. G. Deodhare and M. Vidyasagar, Control system design by infinite linear programming, Internat. J. Control, 55, No. 6 (1992), 1351–1380.

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12. C. A. Desoer and M. Vidyasagar, Feedback Systems: Input–Output Properties (Academic Press, New York, 1975). 13. A. C. Eberhard and R. B. Wenczel, Epi–distance convergence of parametrised sums of convex functions in non–reflexive spaces, J. Convex Anal., 7, No. 1 (2000), 47–71. 14. N. Elia and M. A. Dahleh, Controller design with multiple objectives, IEEE Trans. Automat. Control, AC-42, No. 5 (1997), 596–613. 15. R. D. Hill, A. C. Eberhard, R. B. Wenczel and M. E. Halpern, Fundamental limitations on the time–domain shaping of response to a fixed input, IEEE Trans. Automat. Control, AC-47, No. 7 (2002), 1078–1090. 16. R. B. Holmes, Geometric Functional Analysis and its Applications, Graduate Texts in Mathematics, 24 (Springer-Verlag, New York, 1975). 17. V. Jeyakumar, Duality and infinite–dimensional optimization, Nonlinear Analysis: Theory, Methods and Applications, 15, (1990), 1111–1122. 18. V. Jeyakumar and H. Wolkowicz, Generalizations of Slater’s constraint qualification for infinite convex programs, Math. Programming, 57 (1992), 85–102. 19. J. S. McDonald and J. B. Pearson, l1 –Optimal control of multivariable systems with output norm constraints, Automatica J. IFAC, 27, No. 2 (1991), 317–329. 20. R. T. Rockafellar, Conjugate Duality and Optimization (SIAM, Philadelphia, PA, 1974). 21. R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970). 22. R. T. Rockafellar and R. J.–B. Wets, Variational systems, an introduction, Multifunctions and Integrands, G. Salinetti (Ed.), Springer-Verlag Lecture Notes in Mathematics No. 1091 (1984), 1–54. 23. H. Rotstein, Convergence of optimal control problems with an H∞–norm constraint, Automatica J. IFAC, 33, No. 3 (1997), 355–367. 24. M. Vidyasagar, Control System Synthesis (MIT Press, Cambridge MA, 1985). 25. M. Vidyasagar, Optimal rejection of persistent bounded disturbances, IEEE Trans. Automat. Control, AC-31 (1986), 527–534. 26. R. B. Wenczel, PhD Thesis, Department of Mathematics, RMIT, 1999. 27. R. B. Wenczel, A. C. Eberhard and R. D. Hill, Comments on “Controller design with multiple objectives,” IEEE Trans. Automat. Control, 45, No. 11 (2000), 2197–2198.

Chapter 5

Asymptotical stability of optimal paths in nonconvex problems Musa A. Mamedov

Abstract In this chapter we study the turnpike property for the nonconvex optimal control problems described by the differential inclusion x˙ ∈ a(x). We T study the infinite horizon problem of maximizing the functional 0 u(x(t)) dt as T grows to infinity. The purpose of this chapter is to avoid the convexity conditions usually assumed in turnpike theory. A turnpike theorem is proved in which the main conditions are imposed on the mapping a and the function u. It is shown that these conditions may hold for mappings a with nonconvex images and for nonconcave functions u. Key words: Turnpike property, differential inclusion, functional

5.1 Introduction and background Let x ∈ Rn and Ω ⊂ Rn be a given set. Denote by Πc (Rn ) the set of all compact subsets of Rn . We consider the following problem: ·

x ∈ a(x), x(0) = x0 ,

(5.1)

T u(x(t))dt → max .

JT (x(·)) =

(5.2)

0

Here x0 ∈ Ω ⊂ Rn is an assigned initial point. The multivalued mapping a : Ω → Πc (Rn ) has compact images and is continuous in the Hausdorff Musa A. Mamedov School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria 3353, AUSTRALIA e-mail: musa [email protected] C. Pearce, E. Hunt (eds.), Structure and Applications, Springer Optimization and Its Applications 32, DOI 10.1007/978-0-387-98096-6 5, c Springer Science+Business Media, LLC 2009 

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metric. We assume that at every point x ∈ Ω the set a(x) is uniformly locally–connected (see [3]). The function u : Ω → R1 is a given continuous function. In this chapter we study the turnpike property for the problem given by (6.1) and (5.2). The term ‘turnpike property’ was first coined by Samuelson in 1958 [16] when he showed that an efficient expanding economy would spend most of its time in the vicinity of a balanced equilibrium path. This property was further investigated by Radner [13], McKenzie [11], Makarov and Rubinov [7] and others for optimal trajectories of a von Neuman–Gale model with discrete time. In all these studies the turnpike property was established under some convexity assumptions. In [10] and [12] the turnpike property was defined using the notion of statistical convergence (see [4]) and it was proved that all optimal trajectories have the same unique statistical cluster point (which is also a statistical limit point). In these works the turnpike property is proved when the graph of the mapping a does not need to be a convex set. The turnpike property for continuous-time control systems has been studied by Rockafellar [14], [15], Cass and Shell [1], Scheinkman [18], [17] and others who imposed additional conditions on the Hamiltonian. To prove a turnpike theorem without these kinds of additional conditions has become a very important problem. This problem has recently been further investigated by Zaslavsky [19], [21], Mamedov [8], [9] and others. The theorem proved in the current chapter was first given as a short note in [8]. In this work we give the proof of this theorem and explain the assumptions used in the examples.

Definition 1. An absolutely continuous function x(·) is called a trajectory (solution) to the system (6.1) in the interval [0, T ] if x(0) = x0 and almost · everywhere on the interval [0, T ] the inclusion x (t) ∈ a(x(t)) is satisfied. We denote the set of trajectories defined on the interval [0,T] by XT and let JT∗ =

sup JT (x(·)). x(·)∈XT

We assume that the trajectories of system (5.1) are uniformly bounded, that is, there exists a number L < +∞ such that |x(t)| ≤ L for all t ∈ [0, T ], x(·) ∈ XT , T > 0.

(5.3)

Note that in this work we focus our attention on the turnpike property of optimal trajectories. So we do not consider the existence of bounded trajectories defined on [0, ∞]. This issue has been studied for different control problems by Leizarowitz [5], [6], Zaslavsky [19], [20] and others. Definition 2. The trajectory x(·) is called optimal if J(x(·)) = JT∗ and is called ξ-optimal (ξ > 0) if

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97

J(x(·)) ≥ JT∗ − ξ. Definition 3. The point x is called a stationary point if 0 ∈ a(x). Stationary points play an important role in the study of the asymptotical behavior of optimal trajectories. We denote the set of stationary points by M: M = {x ∈ Ω : 0 ∈ a(x)}. We assume that the set M is bounded. This is not a hard restriction, because we consider uniformly bounded trajectories and so the set Ω can be taken as a bounded set. Since the mapping a(·) is continuous the set M is also closed. Then M is a compact set. Definition 4. The point x∗ ∈ M is called an optimal stationary point if u(x∗ ) = max u(x). x∈M

In turnpike theory it is usually assumed that the optimal stationary point x∗ is unique. We also assume that the point x∗ is unique, but the method suggested here can be applied in the case when we have several different optimal stationary points.

5.2 The main conditions of the turnpike theorem Turnpike theorems for the problem (6.1), (5.2) have been proved in [14], [18] and elsewhere, where it was assumed that the graph of the mapping a is a compact convex set and the function u is concave. The main conditions are imposed on the Hamiltonian. In this chapter a turnpike theorem is presented in which the main conditions are imposed on the mapping a and the function u. Here we present a relation between a and u which provides the turnpike property without needing to impose conditions such as convexity of the graph of a and of the function u. On the other hand this relation holds if the graph of a is a convex set and the function u is concave. Condition M There exists b < +∞ such that for every T > 0 there is a trajectory x(·) ∈ XT satisfying the inequality JT (x(·)) ≥ u∗ T − b. Note that satisfaction of this condition depends in an essential way on the initial point x0 , and in a certain sense it can be considered as a condition for the existence of trajectories converging to x∗ . Thus, for example, if there exists a trajectory that hits x∗ in a finite time, then Condition M is satisfied. Set B = {x ∈ Ω : u(x) ≥ u∗ }.

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We fix p ∈ Rn , p = 0, and define a support function c(x) = max py. y∈a(x)

Here the notation py means the scalar product of the vectors p and y. By |c| we will denote the absolute value of c. We also define the function ϕ(x, y) =

u(x)−u∗ |c(x)|

+

u(y)−u∗ c(y) .

Condition H There exists a vector p ∈ Rn such that H1 c(x) < 0 for all x ∈ B, x = x∗ ; x) > 0; H2 there exists a point x ˜ ∈ Ω such that p˜ x = px∗ and c(˜ H3 for all points x, y, for which px = py, c(x) < 0, c(y) > 0, the inequality ϕ(x, y) < 0 is satisfied; and also if xk → x∗ , yk → y  = x∗ , pxk = pyk , c(xk ) < 0 and c(yk ) > 0, then

lim supk→∞ ϕ(xk , yk ) < 0.

Note that if Condition H is satisfied for any vector p then it is also satisfied for all λp, (λ > 0). That is why we assume that ||p|| = 1. Condition H1 means that derivatives of the system (6.1) are directed to one side with respect to p, that is, if x ∈ B, x = x∗ , then py < 0 for all y ∈ a(x). It is also clear that py ≤ 0 for all y ∈ a(x∗ ) and c(x∗ ) = 0. Condition H2 means that there is a point x ˜ on the plan {x ∈ Rn : p(x − ∗ y > 0 for some y˜ ∈ a(˜ x). This is not a restrictive x ) = 0} such that p˜ assumption, but the turnpike property may be not true if this condition does not hold. The main condition here is H3. It can be considered as a relation between the mapping a and the function u which provides the turnpike property. Note that Conditions H1 and H3 hold if the graph of the mapping a is a convex set (in Rn × Rn ) and the function u is strictly concave. In the next example we show that Condition H can hold for mappings a without a convex graph and for functions u that are not strictly concave (in this example the function u is convex). Example 1 Let x = (x1 , x2 ) ∈ R2 and the system (6.1) have the form ·

x1 = λ[x21 + (x22 + 1)2 + w], ·

x2 = f (x1 , x2 , v),

− 1 ≤ w ≤ +1, v ∈ U ⊂ Rm .

Here λ > 0 is a positive number, the function f (x1 , x2 , v) is continuous and f (0, 0, v˜) = 0 for some v˜ ∈ U.

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99

The mapping a can be written as a(x) = { y = (y1 , y2 ) : y1 = λ[x21 + (x22 + 1)2 + w], y2 = f (x1 , x2 , v), x = (x1 , x2 ) ∈ R2 , − 1 ≤ w ≤ +1, v ∈ U }. The function u is given in the form u(x) = cx2 + dx2k 1 , where

d > 0, c ≥ 2d, k ∈ {1, 2, 3, ...}.

We show that Condition H holds. It is not difficult to see that the set of stationary points M contains the point (0, 0) and also M ⊂ B1 (0, −1), where B1 (0, −1) represents the sphere with center (0, −1) and with radius 1. We have u∗ = max u(x) = u(0, 0) = 0. x∈M

Therefore x∗ = (0, 0) is a unique optimal stationary point. We fix the vector p = (−1, 0) and calculate the support function c(x) = maxy∈a(x) py : c(x) = −λ(x21 + x22 + 2x2 ). Take any point x = (x1 , x2 ) ∈ B = {x : u(x) > 0} such that x = / B1 (0, −1) and therefore c(x) < 0. Then Condition x∗ = (0, 0). Clearly x ∈ H1 holds. Condition H2 also holds, because, for example, for the point x ˜= (0, −1) for which p˜ x = 0 we have c(˜ x) = λ > 0. Now we check Condition H3. Take any two points x = (x1 , x2 ), y = (y1 , y2 ) for which px = py, c(x) < 0, c(y) > 0. If u(x) < 0, from the expression of the function φ(x, y) we obtain that φ(x, y) < 0. Consider the case u(x) ≥ 0. From px = py we have x1 = y1 . Denote ξ = x1 = y1 . Since c(y) > 0 and λ > 0 we obtain ξ 2 + (y2 + 1)2 < 1. Therefore 0 < ξ < 1 and y2 + (1/2)ξ2 < 0. On the other hand u(y) − u∗ = cy2 + dξ 2k < cy2 + dξ 2 ≤ c[y2 + (1/2)ξ 2 ]. Since c(x) < 0 and u(x) ≥ 0 then |c(x)| = λ(ξ 2 +x22 +2x2 ), x2 +(1/2)ξ 2 + (1/2)x22 > 0, u(x) − u∗ = cx2 + dξ 2k < cx2 + dξ 2 ≤ c[x2 + (1/2)ξ 2 ]. Thus c(x2 + (1/2)ξ 2 ) c u(x) − u∗ < ≤ , |c(x)| λ(ξ 2 + x22 + 2x2 ) 2λ u(y) − u∗ c(y2 + (1/2)ξ 2 ) c < ≤− . c(y) −λ(ξ 2 + y22 + 2y2 ) 2λ

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From these inequalities we have ϕ(x, y) < 0, that is, the first part of H3 holds. The second part of Condition H3 may also be obtained from these inequalities. Therefore Condition H holds. We now formulate the main result of the current chapter. Theorem 1. Suppose that Conditions M and H are satisfied and that the optimal stationary point x∗ is unique. Then: 1. there exists C < +∞ such that

T

(u(x(t)) − u∗ )dt ≤ C

0

for every T > 0 and every trajectory x(t) ∈ XT ; 2. for every ε > 0 there exists Kε,ξ < +∞ such that meas {t ∈ [0, T ] : ||x(t) − x∗ || ≥ ε} ≤ Kε,ξ for every T > 0 and every ξ-optimal trajectory x(t) ∈ XT ; 3. if x(t) is an optimal trajectory and x(t1 ) = x(t2 ) = x∗ , then x(t) ≡ x∗ for t ∈ [t1 , t2 ]. The proof of this theorem is given in Section 7, and in Sections 3 to 6 we give preliminary results.

5.3 Definition of the set D and some of its properties In this section a set D is introduced. This set will be used in all sections below. Denote M∗ = {x ∈ Ω : c(x) ≥ 0}. Clearly M ⊂ M∗ . We recall that B = {x ∈ Ω : u(x) ≥ u∗ }. Consider a compact set D ⊂ Ω for which the following conditions hold: a) x ∈ int D for all x ∈ B, x = x∗ ; b) c(x) < 0 for all x ∈ D, x = x∗ ; c) D ∩ M∗ = {x∗ } and B ⊂ D. It is not difficult to see that there exists a set D with properties a), b) and c). For example such a set can be constructed as follows. Let x ∈ B, x = x∗ . Then c(x) < 0. Since the mapping a is continuous in the Hausdorff metric the function c(x) is continuous too. Therefore there exists εx > 0 such that c(x ) < 0 for all x ∈ Vεx (x) ∩ Ω. Here Vε (x) represents the open ε-neighborhood of the point x. In this case for the set

5

Asymptotical stability of optimal paths in nonconvex problems

D = cl

⎧ ⎨

/

⎩

x∈B,x =x∗

101

⎫ ⎬ V 21 εx (x) ∩ Ω ⎭

Conditions a) to c) are satisfied. Lemma 1. For every ε > 0 there exists νε > 0 such that u(x) ≤ u∗ − νε for every x ∈ Ω, x ∈ / int D and ||x − x∗ || ≥ ε. Proof. Assume to the contrary that for any ε > 0 there exists a sequence xk such that xk ∈ / int D, ||xk − x∗ || ≥ ε and u(xk ) → u∗ as k → ∞. Since the / int D sequence xk is bounded it has a limit point, say x . Clearly x = x∗ , x ∈ and also u(x ) = u∗ , which implies x ∈ B. This contradicts Condition a) of the set D.  Lemma 2. For every ε > 0 there exists ηε > 0 such that c(x) < −ηε and for all x ∈ D, ||x − x∗ || ≥ ε.

Proof. Assume to the contrary that for any ε > 0 there exists a sequence xk such that xk ∈ D, ||xk − x∗ || ≥ ε and c(xk ) → 0. Let x be a limit point of the sequence xk . Then x ∈ D, x = x∗ and c(x ) = 0. This contradicts Property b) of the set D. 

5.4 Transformation of Condition H3 In this section we prove an inequality which can be considered as a transformation of Condition H3. Take any number ε > 0 and denote Xε = {x : ||x − x∗ || ≥ ε}. Consider the sets D ∩ Xε and M∗ ∩ Xε . Let x ∈ D ∩ Xε , y ∈ M∗ ∩ Xε and px = py, with c(x) < 0 and c(y) > 0. From Condition H3 it follows that ϕ(x, y) < 0.

(5.4)

We show that for every ε > 0 there exists δε > 0 such that ϕ(x, y) < −δε

(5.5)

for all x ∈ D (x = x∗ ), y ∈ M∗ ∩ Xε , for which px = py, c(x) < 0 and c(y) > 0.

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First we consider the case where x ∈ D ∩ Xε . In this case if (5.5) is not true then there exist sequences (xn ) and (yn ), for which pxn = pyn , c(xn ) < 0, c(yn ) > 0, xn ∈ D ∩ Xε , yn ∈ M∗ ∩ Xε and ¯, yn → y¯, ϕ(xn , yn ) → 0. xn → x From Lemma 17.4.1 it follows that the sequence {(u(xn ) − u∗ )/|c(xn )|} is bounded. Since ϕ(xn , yn ) → 0, the sequence {(u(yn ) − u∗ )/c(yn )} is also bounded and therefore from Lemma 17.3.1 we obtain c(¯ y ) > 0. We also obtain c(¯ x) < 0 from the inclusion x ¯ ∈ D ∩ Xε . Thus the function ϕ(x, y) is continuous at the point (¯ x, y¯). Then from ϕ(xn , yn ) → 0 it follows that ϕ(¯ x, y¯) = 0, which contradicts (5.4). We now consider the case where x ∈ D, x = x∗ . Assume that (5.5) does not hold. Then there exist sequences (xn ) and (yn ), for which pxn = ¯, yn → y¯, pyn , c(xn ) < 0, c(yn ) > 0, xn ∈ D, yn ∈ M∗ ∩ Xε and xn → x ¯ = x∗ , we have a contradiction similar to the first case. If ϕ(xn , yn ) → 0. If x x ¯ = x∗ , taking the inequality y¯ = x∗ into account we obtain a contradiction to the second part of Condition H3. Thus we have shown that (5.5) is true. Define the function ϕ(x, y)δ1 ,δ2 =

u(x) − u∗ u(y) − u∗ + , |c(x)| + δ1 c(y) + δ2

(δ1 ≥ 0, δ2 ≥ 0).

Since the support function c(·) is continuous, from Conditions H2 and H3 it follows that there exists a number b ∈ (0, +∞) such that u(x) − u∗ ≤b |c(x)|

for all x ∈ D, x = x∗ .

(5.6)

For a given ε > 0 we choose a number γ(ε) > 0 such that b−

νε ≤ −δε ; γε

(5.7)

here the number νε is defined by Lemma (17.3.1). Clearly u(y) − u∗ ≤ −νε

for all y ∈ M∗ ∩ Xε .

(5.8)

By using γ(ε) we divide the set M∗ ∩ Xε ∩ {y : c(y) > 0} into two parts: 1 γ(ε)}, 2 1 Y2 = {y ∈ M∗ ∩ Xε ∩ {y : c(y) > 0} : c(y) < γ(ε)}. 2 Y1 = {y ∈ M∗ ∩ Xε ∩ {y : c(y) > 0} : c(y) ≥

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Asymptotical stability of optimal paths in nonconvex problems

Consider the set Y2 . Denote δ¯2 = Then u(x) − u∗ ≤b |c(x)| + δ1

1 2

103

γ(ε) and take any number δ¯1 > 0.

for all x ∈ D, x = x∗ , δ1 ≤ δ¯1

(5.9)

and

1 1 c(y) + δ2 ≤ c(y) + δ¯2 < γ(ε) + γ(ε) = γ(ε) 2 2 for all 0 ≤ δ2 ≤ δ¯2 , y ∈ Y2 . Using (5.8) we obtain u(y) − u∗ νε νε ≤− ≤− c(y) + δ2 c(y) + δ2 γ(ε)

for all y ∈ Y2 .

(5.10)

Thus from (5.9) and (5.10) we have ϕ(x, y)δ1 ,δ2 ≤ b −

νε ≤ − δε γ(ε)

(5.11)

for all (x, y) and (δ1 , δ2 ) satisfying x ∈ D, x = x∗ , y ∈ Y2 , c(x) < 0, c(y) > 0, 1 0 ≤ δ1 ≤ δ¯1 , 0 ≤ δ2 ≤ δ¯2 = γ(ε). 2 Now consider the set Y1 . Since Y1 is a bounded closed set and c(y) ≥ γ(ε) > 0 for all y ∈ Y1 , then the function (u(y) − u∗ )/(c(y) + δ2 ) is uniformly continuous with respect to (y, δ2 ) on the closed set Y1 × [0, L], where L > 0. That is why for a given number δε there exists δˆ¯21 (ε) such that 1 2

1 u(y) − u∗ u(y) − u∗ ≤ δε − c(y) + δ2 c(y) 2

for all y ∈ Y1 , 0 ≤ δ2 ≤ δˆ¯21 (ε).

On the other hand u(x) − u∗ u(x) − u∗ ≤ |c(x)| + δ1 |c(x)|

for all x ∈ D, x = x∗ , u(x) ≥ u∗ , 0 ≤ δ1 ≤ δ¯1 .

Then if px = py we have ϕ(x, y)δ1 ,δ2 ≤ ϕ(x, y) +

1 1 1 δε ≤ − δε + δε = − δε , 2 2 2

for all x ∈ D, x = x∗ , u(x) ≥ u∗ , y ∈ Y1 , px = py, 0 ≤ δ1 ≤ δ¯1 and 0 ≤ δ2 ≤ δˆ¯21 (ε).

(5.12)

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Now consider the case when x ∈ D and u(x) < u∗ . Since the function c(y) is bounded on the set Y1 , then for ε > 0 there exist δˆ¯22 (ε) > 0 and δˆε > 0 such that 1 u(y) − u∗ ≤ − δˆε c(y) + δ2 2

for all y ∈ Y1 , 0 ≤ δ2 ≤ δˆ¯22 (ε).

Then ϕ(x, y)δ1 ,δ2 ≤

u(y) − u∗ 1 ≤ − δˆε , c(y) + δ2 2

(5.13)

for all x ∈ D, u(x) < u∗ , y ∈ Y1 , px = py, c(x) < 0, 0 ≤ δ1 ≤ δ¯1 and 0 ≤ δ2 ≤ δˆ¯22 (ε). Therefore from (5.11) to (5.13) we have ϕ(x, y)δ1 ,δ2 ≤ − δ¯ε ,

(5.14)

for all x ∈ D, x = x∗ , y ∈ M∗ ∩ Xε , px = py, c(y) > 0, 0 ≤ δ1 ≤ δ¯1 and 0 ≤ δ2 ≤ δ2 (ε). Here δ¯ε = min{ 12 δε ,

1 2

δˆε } and δ2 (ε) = min{ 12 γ(ε), δ¯ˆ21 (ε), δ¯ˆ22 (ε)}.

Consider the function (u(x) − u∗ )/(|c(x)| + δ1 ) with respect to (x, δ1 ). From Lemma 17.4.1 we obtain that this function is continuous on the set (D ∩ Xεˆ) × [0, δ¯1 ] for all εˆ > 0. Then for a given number 12 δ¯ε > 0 there exists η = η(ε, εˆ) > 0 such that u(x ) − u∗ 1 u(x) − u∗ − ≤ δ¯ε |c(x)| + δ1 |c(x )| + δ1 2 for all x ∈ cl (Vη (x )) = {x : ||x − x || ≤ η}, x ∈ D ∩ Xε and 0 ≤ δ1 ≤ δ¯1 . If px = py then we have 1¯ 1 δε ≤ − δ¯ε , 2 2   ∗  for all x ∈ cl (Vη (x )), x ∈ D ∩ Xε , y ∈ M ∩ Xε , px = py, c(y) > 0, 0 ≤ δ1 ≤ δ¯1 and 0 ≤ δ2 ≤ δ2 (ε). Denote δ  (ε) = min{ 12 δ¯ε , δ2 (ε)}. Obviously δ  (ε) > 0 if ε > 0 and also for every ε > 0 there exists δ  > 0 such that δ  (ε) ≥ δ  for all ε ≥ ε . Therefore there exists a continuous function δ  (ε), with respect to ε, such that ϕ(x, y)δ1 ,δ2 ≤ − ϕ(x , y)δ1 ,δ2 +

• δ  (ε) ≤ δ  (ε) for all ε > 0 and • for every ε > 0 there exists δ  > 0 such that δ  (ε) < δ  for all ε ≥ ε .

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Asymptotical stability of optimal paths in nonconvex problems

105

Let x ∈ D and y ∈ M∗ . Taking εˆ = ||x − x∗ || and e = ||y − x∗ ||, we define the functions δ(y) = δ  (ε) = δ  (||y − x∗ ||) and η(x , y) = η(ε, εˆ) = η(||y − x∗ ||, ||x − x∗ ||). Clearly in this case the function δ(y) is continuous with respect to y. Thus the following lemma is proved. Lemma 3. Assume that at the point x ∈ D, y ∈ M∗ we have px = py, c(x) < 0 and c(y) > 0. Then for every point x and numbers δ1 , δ2 satisfying

x ∈ cl Vη(x ,y) (x ) , c(x) < 0, 0 ≤ δ1 ≤ δ¯1 and 0 ≤ δ2 ≤ δ2 (y), the following inequality holds: u(x) u(y) + ≤ u∗ |c(x)| + δ1 c(y) + δ2

+

1 1 + |c(x)| + δ1 c(y) + δ2

, − δ(y).

Here the functions η(x , y) and δ(y) are such that δ(y) is continuous and ∧

∼

∧

for every ε > 0, ε> 0 there exist δ ε > 0 and η ε,∼ε > 0 such that ∧

∧

δ(y) ≥δ ε and η(x , y) ≥η ε,∼ε ∼

for all (x , y) for which ||x − x∗ || ≥ ε, ||y − x∗ || ≥ ε.

5.5 Sets of 1st and 2nd type: Some integral inequalities In this section, for a given trajectory x(t) ∈ XT we divide the interval [0, T ] into two types of intervals and prove some integral inequalities. We assume that x(t) is a given continuously differentiable trajectory.

5.5.1 Consider a set {t ∈ [0, T ] : x(t) ∈ int D}. This set is an open set and therefore it can be presented as a union of a countable (or finite) number of open intervals τk = (tk1 , tk2 ), k = 1, 2, 3, ..., where τk ∩ τl = ∅ if k = l. We denote the set of intervals τk by G = {τk : k = 1, 2, ...}. From the definition of the set D we have d · px(t) = p x (t) ≤ c(x(t)) < 0 for all t ∈ τk , k = 1, 2, .... dt Then px(tk1 ) > px(tk2 ) for all k. We introduce the notation piτk = px(tki ), i = 1, 2, ..., Pτk = [p2τk , p1τk ] and 0 Pτk = (p2τk , p1τk ).

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5.5.2 We divide the set G into the sets gm (G = ∪m gm ), such that for every set g = gm the following conditions hold: a) The set g consists of a countable (or finite) number of intervals τk , for which i - related intervals Pτ0k are disjoint; ii - if [t1τi , t2τi ] ≤ [t1τj , t2τj ] for some τi , τj ∈ g, then Pτi ≥ Pτj ; (here and henceforth the notation [t1τi , t2τi ] ≤ [t1τj , t2τj ] and Pτi ≥ Pτj means that t2τi ≤ t1τj and p2τi ≥ p1τj , respectively) iii - if [t1τ , t2τ ] ⊂ [t1g , t2g ] for τ ∈ G, then τ ∈ g; here t1g = inf τ ∈g t1τ and t2g = supτ ∈g t2τ . b) The set g is a maximal set satisfying Condition a); that is, the set g cannot be extended by taking other elements τ ∈ G such that a) holds. Sets gm (G = ∪m gm ) satisfying these conditions exist. For example we can construct a set g satisfying Conditions a) and b) in the following way. Take any interval τ1 = (t11 , t12 ). Denote by t2 a middle point of the interval [0, t11 ] : t2 = 12 t11 . If for all intervals τ ∈ G, for which τ ⊂ [t2 , t11 ], Conditions i and ii hold, then we take t3 = 12 t2 , otherwise we take t3 = 12 (t2 + t11 ). We repeat this process and obtain a convergent sequence tn . Let tn → t . In this case for all intervals τ ∈ G, for which τ ⊂ [t , t11 ], Conditions i and ii are satisfied. Similarly, in the interval [t12 , T ] we find the point t such that for all intervals τ ∈ G, for which τ ⊂ [t12 , t ], Conditions i and ii are satisfied. Therefore we obtain that for the set g which consists of all intervals τ ∈ G, for which τ ⊂ [t , t ], Condition a) holds. It is not difficult to see that for the set g Condition b) also holds. Thus we have constructed a set g1 = g satisfying Conditions a) and b). We can construct another set g2 taking G \ g1 in the same manner and so on. Therefore G = ∪m gm , where for all gm Conditions a) and b) hold. For every set g, we have intervals [t1g , t2g ] (see iii) and [p1g , p2g ], where p1g = supτ ∈g p1τ and p2g = inf τ ∈g p2τ . Definition 5. We say that g1 < g2 if: a) p2g1 < p1g2 and t2g1 < t1g2 ; b) there is no g ∈ G which belongs to the interval [t2g1 , t1g2 ]. Take some set g 1 ∈ G and consider all sets gm ∈ G, m = ±1, ±2, ±3, ... for which ... < g−2 < g−1 < g 1 < g1 < g2 < .... The number of sets gm may be at most countable. Denote G1 = {g 1 } ∪ {gm : m = ±1, ±2, ±3, ...}.

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Asymptotical stability of optimal paths in nonconvex problems

107

We take another set g 2 ∈ G \ G1 and construct a set G2 similar to G1 , and so on. We continue this procedure and obtain sets Gi . The number of these sets is either finite or countable. Clearly G = ∪i Gi . Denote t1Gi = inf g∈Gi t1g and t2Gi = supg∈Gi t2g . Clearly ∪i [t1Gi , t2Gi ] ⊂ [0, T ]

(5.15)

and (t1Gi , t2Gi ) ∩ (t1Gj , t2Gj ) = ∅ for all i = j. Proposition 1. Let gk ∈ Gi , k = 1, 2, .... Then a) if g1 < g2 < g3 < .... then x(t2Gi ) = x∗ ; b) if g1 > g2 > g3 > .... then x(t1Gi ) = x∗ . Proof. Consider case a). Take some gk . By Definition 5 it is clear that there exists an interval τk ∈ gk and a point tk ∈ τk such that x(tk ) ∈ D.

(5.16)

Consider the interval [t2gk , t1gk+1 ]. Since gk < gk+1 by Definition 5 we have px(t2gk ) < px(t1gk+1 ). Therefore there exists a point sk ∈ (t2gk , t1gk+1 ) such that ·

p x (sk ) ≥ 0, which implies x(sk ) ∈ M∗ .

(5.17)

On the other hand, |t2gk − t1gk | → 0 and |t1gk+1 − t2gk | → 0, as k → ∞. −

−

−

Then x(t) →x as t → t2Gi . In this case x(tk ) →x and x(sk ) →x as k → ∞. Therefore from the definition of the set D and from (5.16) and (5.17) we −

−

obtain x∈ D ∩ M∗ = {x∗ }; that is, x= x∗ . We can prove case b) in the same manner.



5.5.3 Take the set Gi . We denote by t1i an exact upper bound of the points t2Gm satisfying t2Gm ≤ t1Gi and by t2i an exact lower bound of the points t1Gm satisfying t1Gm ≥ t2Gi . Proposition 2. There exist points ti ∈ [t1i , t1Gi ] and ti ∈ [t2Gi , t2i ] such that x(ti ) = x(ti ) = x∗ . Proof. First we consider the interval [t1i , t1Gi ]. Two cases should be studied. 1. Assume that the exact upper bound t1i is not reached. In this case there exists a sequence of intervals [t1Gm , t2Gm ] such that t2Gm → t1i and t1Gm → t1i . Since the intervals (t1Gm , t2Gm ) are disjoint we obtain that x(t1i ) = x∗ . Therefore ti = t1i .

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2. Assume that t1i = t2Gm for any Gm . If there exists a sequence gk ∈ Gm , k = 1, 2, ..., such that g1 < g2 < ..., then from Proposition 1 it follows that x(t2Gm ) = x∗ . So we can take ti = t2Gm = t1i . Therefore we can assume that the set Gm consists of gmk , where gm1 > gm2 > .... Now consider the set Gi . If in this set there exists a sequence gil such that gi1 > gi2 > · · · then from Proposition 6.1 it follows that x(t1Gi ) = x∗ , so we can take ti = t1Gi . That is why we consider the case when the set Gi consists of gil , where gi1 < gi2 < .... Consider the elements gm1 and gi1 . We denote gm = gm1 and gi = gi1 . The elements gm and gi belong to the different sets Gm and Gi , so they cannot be compared by Definition 5. Since the second condition of this definition holds, the first condition is not satisfied, that is, p2Gm ≥ p2Gi . This means that the interval τ in gm and gi forms only one element of type g. This is a contradiction. Therefore we can take either ti = t1i or ti = t1Gi .  Similarly we can prove the proposition for the interval [t2Gi , t2i ]. Note We take ti = 0 (or ti = T ) if for the chosen set Gi there does not exist Gm such that t2Gm ≤ t1Gi (or t1Gm ≥ t2Gi , respectively). Therefore the following lemma is proved. Lemma 4. The interval [0, T ] can be divided into an at most countable number of intervals [0, t1 ], [t1k , t2k ] and [t2 , T ], such that the interiors of these intervals are disjoint and a) [0, T ] = [0, t1 ] ∪ {∪k [t1k , t2k ]} ∪ [t2 , T ]; b) in each interval [0, t1 ], [t1k , t2k ] and [t2 , T ] there is only one set G0 , Gk and GT , respectively, and G = G0 ∪ {∪k Gk } ∪ GT ; c) x(t1 ) = x(t1k ) = x(t2k ) = x(t2 ) = x∗ ,

k = 1, 2, 3, ....

5.5.4 In this subsection we give two lemmas. Lemma 5. Assume that the function x(t) is continuously differentiable on the interval [t1 , t2 ] and p1 < p2 , where pi = px(ti ), i = 1, 2. Then there exists an at most countable number of intervals [tk1 , tk2 ] ⊂ [t1 , t2 ] such that · a) p x (t) > 0, t ∈ [tk1 , tk2 ], k = 1, 2, ...; b) [pk1 , pk2 ] ⊂ [p1 , p2 ] and pk1 < pk2 for all k = 1, 2, ..., where pki = px(tki ), i = 1, 2;

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109

c) the intervals (pk1 , pk2 ) are disjoint and  (pk2 − pk1 ). p2 − p1 = k

Proof: For the sake of definiteness, we assume that px(t) ∈ [p1 , p2 ], t ∈ [t1 , t2 ]. Otherwise we can consider an interval [t1 , t2 ] ⊂ [t1 , t2 ], for which px(t) ∈ [p1 , p2 ], t ∈ [t1 , t2 ] and pi = px(ti ), i = 1, 2. We set t(q) = min{t ∈ [t1 , t2 ] : px(t) = q} for all q ∈ [p1 , p2 ] and then define a set m = {t(q) : q ∈ [p1 , p2 ]}. Clearly m ⊂ [t1 , t2 ]. Consider a function a(t) = px(t) defined on the set m. It is not difficult to see that for every q ∈ [p1 , p2 ] there is only one number . t(q), for which a(t(q)) = q, and also a (t(q)) ≥ 0, ∀q ∈ [p1 , p2 ]. We divide the interval [p1 , p2 ] into two parts as follows: .

.

P1 = {q : a (t(q)) = 0} and P2 = {q : a (t(q)) > 0}. Define the sets m(P1 ) = {t(q) : q ∈ P1 } and m(P2 ) = {t(q) : q ∈ P2 }. We denote by mΛ a set of points t ∈ [t1 , t2 ] which cannot be seen from the left (see [2]). It is known that the set mΛ can be presented in the form mΛ = ∪n (αn , βn ). Then we can write [t1 , t2 ] = m(P1 ) ∪ m(P2 ) ∪ mΛ ∪ (∪n {βn }).

Let q ∈ P2 . Since the function x(t) is continuously differentiable, there exists a number ε > 0, such that Vε (q) ⊂ P2 , where Vε (q) stands for the open ε-neighborhood of the point q. Therefore the set m(P2 ) is an open set and that is why it can be presented as m(P2 ) = ∪k (tk1 , tk2 ). Thus we have

t2 p2 − p1 = px(t2 ) − px(t1 ) =

·

t1 tk 2

 k

tk 1



βn

·

p x (t)dt +

n α n

·

·

p x (t)dt =

p x (t)dt + m(P1 )

p x (t)dt +

·

p x (t)dt. ∪n {βn }

110

M.A. Mamedov ·

.

It is not difficult to observe that p x (t) = a (t) = 0, ∀t ∈ m(P1 ), meas(∪n {βn }) = 0 and px(αn ) = px(βn ), n = 1, 2, ... (see [2]). Then we obtain   (px(tk2 ) − px(tk1 )) = (pk2 − pk1 ). p2 − p 1 = k

k

Therefore for the intervals [tk1 , tk2 ] all assertions of the lemma hold. Lemma 6. Assume that on the intervals [t1 , t2 ] and [s2 , s1 ] the following conditions hold: 

1. px(ti ) = px(si ) = pi , i = 1, 2. 2. x(t) ∈ int D, ∀t ∈ (t1 , t2 ). In particular, from this condition it follows · that p x (t) < 0, ∀t ∈ (t1 , t2 ). · 3. p x (s) > 0, ∀s ∈ (s2 , s1 ). Then

t2

s1

u(x(s))ds ≤ u [(t2 − t1 ) + (s1 − s2 )] −

u(x(t))dt + t1

s1

∗

s2

δ 2 (x(s))ds s2

where the function δ(x) is as defined in Lemma 17.5.1. Proof: Consider two cases. 

I. Let p∗ = px∗ = pi , i = 1, 2. We recall that pi = px(ti ), i = 1, 2. In this case from Conditions 2) and 3) we have ∼

ε= ρ (x∗ , {x(t) : t ∈ [t1 , t2 ]}) > 0 and ε = ρ (x∗ , {x(s) : s ∈ [s2 , s1 ]}) > 0. ∧

∧

Now we use Lemma 17.5.1. We define δ =δ ε > 0 and η =η ε,∼ε for the chosen ∼

numbers ε and ε. We take any number N > 0 and divide the interval [p2 , p1 ] into N equal parts [pk2 , pk1 ]. From Conditions 2 and 3 it follows that in this case the intervals [t1 , t2 ] and [s2 , s1 ] are also divided into N parts, say [tk1 , tk2 ] and [sk2 , sk1 ], respectively. Here px(tki ) = px(ski ) = pki , i = 1, 2, k = 1, ..., N. Clearly pk1 − pk2 =

p1 − p2 → 0 as N → ∞. N ∼

Since x(t) ∈ D and ||x(t) − x∗ || ≥ ε > 0 for all t ∈ [t1 , t2 ] then from Lemma 17.4.1 it follows that ·

p x (t) ≤ c(x(t)) < −η∼ε < 0. That is why for every k we have tk2 − tk1 → 0 as N → ∞. Therefore for a given η > 0 there exists a number N such that

5

Asymptotical stability of optimal paths in nonconvex problems

max

k t,s∈[tk 1 ,t2 ]

111

||x(t) − x(s)|| < η for all k = 1, ..., N.

(5.18)

Now we show that for all k = 1, ..., N sk1 − sk2 → 0 as N → ∞.

(5.19)

Suppose that (5.19) is not true. In this case there exists a sequence of intervals [sk2N , sk1N ], such that ski N → si , i = 1, 2, and s2 < s1 . Since pk1N −pk2N → 0 as N → ∞ then px(s2 ) = px(s1 ) = p , and moreover px(s) = p for all s ∈ [s2 , s1 ]. This is a contradiction. So (5.19) is true. A. Now we take any number k and fix it. For the sake of simplicity we denote the intervals [tk1 , tk2 ] and [sk2 , sk1 ] by [t1 , t2 ] and [s2 , s1 ], respectively. Let pi = px(ti ) = px(si ), i = 1, 2. Take any s ∈ (s2 , s1 ) and denote by t the point in the interval (t1 , t2 ) for which px(t ) = px(s). From (6.8) it follows that x(t) ∈ Vη (x(t )) for all t ∈ [t1 , t2 ]. Therefore we can apply Lemma 17.5.1. We also note that the following conditions hold: ·

• |cx(t))| ≤ |p x (t)| for all t ∈ (t1 , t2 ); · • cx(s)) ≥ p x (s) for all s ∈ (s2 , s1 ); • u(x(s)) ≤ u∗ for all s ∈ [s2 , s1 ]. Then from Lemma 17.5.1 we obtain u(x(t)) ·

|p x (t)| + δ1

+



u(x(s)) ·

p x (s) + δ2

≤u

1

∗

·

|p x (t)| + δ1

+

1



·

p x (s) + δ2

− δ(x(s)),

(5.20)

for all t ∈ (t1 , t2 ), s ∈ (s2 , s1 ), δ1 ∈ [0, δ¯1 ] and δ2 ∈ [0, δ(x(s))]. Denote ξ = mins∈[s2 ,s1 ] δ(x(s)). Clearly ξ > 0. Since the function δ(x) is ∼ continuous there is a point s∈ [s2 , s1 ] such that ∼

ξ = δ(x( s)). We transform t → π and s → ω as follows: s2 − s1 (t1 − t), t ∈ [t1 , t2 ], t1 − t2 ω = px(s) + ξ(s − s1 ), s ∈ [s2 , s1 ]. π = px(t) + ξ

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Clearly ∼

·

·

dπ = [p x (t)− ξ ]dt and dω = [p x (s) + ξ]ds, ∼

where ξ = ξ(s2 − s1 )/(t1 − t2 ). ∼

·

·

Since p x (t)− ξ < 0 and p x (s) + ξ > 0 then there exist inverse functions 



t = t(π) and s = s(ω). We also note that π1 = px(t1 ) = px(s1 ) = ω1 and 



π2 = px(t2 ) + ξ(s2 − s1 ) = ω2 . Therefore we have A

t2



=

u(x(t))dt + t1 π

2

=

=

s1 u(x(s))ds s2

u(x(t(π)))

ω1

u(x(s(ω)))

dω · x (s(ω)) + ξ p ω2  u(x(s(ω))) dω. ∼ + · · p x (s(ω)) + ξ |p x (t(ω))|+ ξ ·

∼ dπ

+

π1 p x (t(π))− ξ

ω1  u(x(t(ω))) ω2

∼

Let δ¯1 > ξ . Since ξ ≤ δ(x(t)) = δ(x(t(ω))), t(ω) ∈ [t1 , t2 ], s(ω) ∈ [s2 , s1 ], then from (6.10) we obtain 

ω1 

ω1 1 1 ∗ dω − δ(x(s(ω)))dω A≤ u ∼ + · · x (s(ω)) + ξ p x (t(ω))|+ |p ξ ω2 ω2 ⎛t ⎞

2

s1

s1 · ∗⎝ ⎠ =u dt + ds − δ(x(s))[p x (s) + ξ]ds t1 ∗

s2

s2

s1

≤ u [(t2 − t1 ) + (s1 − s2 )] −

ξδ(x(s)ds. s2 ∼

On the other hand δ(x(s)) ≥ ξ = δ(x( s)). Thus ∼

A ≤ u∗ [(t2 − t1 ) + (s1 − s2 )] − (s1 − s2 )δ 2 (x( s)). B. Now we consider different numbers k. The last inequality shows that ∼ for every k = 1, ..., N there is a point sk ∈ [sk2 , sk1 ] such that

5

Asymptotical stability of optimal paths in nonconvex problems

t2

s1 u(x(t))dt +

t1

113 ∼

u(x(s))ds ≤ u∗ [(tk2 − tk1 ) + (sk1 − sk2 )] − (sk1 − sk2 )δ 2 (x(sk )).

s2

Summing over k we obtain

t2

s1 u(x(t))dt +

t1

u(x(s))ds ≤ u∗ [(t2 − t1 ) + (s1 − s2 )] −

N 

∼

(sk1 − sk2 )δ 2 (x(sk )).

k=1

s2

Therefore the lemma is proved taking into account (5.19) and passing to the limit as N → ∞. II. Now consider the case when p∗ = pi for some i = 1, 2. For the sake of definiteness we assume that p∗ = p1 . Take any number α > 0 and consider the interval [p2 , p1 − α]. Denote by [t1 − t(α), t2 ] and [s2 , s1 − s(α)] the intervals which correspond to the interval [p2 , p1 − α]. Clearly t(α) → 0 and s(α) → 0 as α → 0. We apply the result proved in the first part of the lemma for the interval [p2 , p1 − α]. Then we pass to the limit as α → 0. Thus the lemma is proved.

5.5.5 We define two types of sets. Definition 6. The set π ⊂ [0, T ] is called a set of 1st type on the interval [p2 , p1 ] if the following conditions hold: a) The set π consists of two sets π1 and π2 , that is, π = π1 ∪ π2 , such that / int D, ∀t ∈ π2 . x(t) ∈ int D, ∀t ∈ π1 and x(t) ∈ b) The set π1 consists of an at most countable number of intervals dk , with end-points tk1 < tk2 and the intervals (px(tk2 ), px(tk1 )), k = 1, 2, ..., are disjoint. Clearly in this case the intervals d0k = (tk1 , tk2 ) are also disjoint. c) Both the inequalities p1 ≥ supk px(tk1 ) and p2 ≤ inf k px(tk2 ) hold. Definition 7. The set ω ⊂ [0, T ] is called a set of 2nd type on the interval [p2 , p1 ] if the following conditions hold: a) x(t) ∈ / int D, ∀t ∈ ω. b) The set ω consists of an at most countable number of intervals [sk2 , sk1 ], such that the intervals (px(sk2 ), px(sk1 )), k = 1, 2, ..., are nonempty and disjoint, and  [px(sk1 ) − px(sk2 )]. p1 − p 2 = k

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Lemma 7. Assume that π and ω are sets of 1st and 2nd type on the interval [p2 , p1 ], respectively. Then

u(x(t))dt ≤ u∗ meas(π ∪ ω) − [u∗ − u(x(t))]dt − δ 2 (x(t))dt, π∪ω

Q

E

where a) Q ∪ E = ω ∪ π2 = {t ∈ π ∪ ω : x(t) ∈ / int D}; b) for every ε > 0 there exists a number δε > 0 such that δ 2 (x) ≥ δε for all x for which ||x − x∗ || ≥ ε; c) for every δ > 0 there exists a number K(δ) < ∞ such that meas[(π ∪ ω) ∩ Zδ ] ≤ K(δ)meas[(Q ∪ E) ∩ Zδ ], where Zδ = {t ∈ [0, T ] : |px(t) − p∗ | ≥ δ}. Proof. Let π = π1 ∪ π2 , π1 = ∪k dk , ∪n νn ⊂ ω and νn = [sn2 , sn1 ] (see Definitions 6 and 7). We denote π10 = ∪k d0k and d0k = (tk1 , tk2 ). Clearly meas π1 = meas π10 and that is why below we deal with d0k . Denote pni = px(sni ), i = 1, 2. Clearly pn2 < pn1 . Since the function x(t) is absolutely continuous, from Lemma 5 it follows that there exists an at most nm n n countable number of intervals [snm 2 , s1 ] ⊂ [s2 , s1 ], m = 1, 2, ..., such that · nm i - p x (s) > 0, for all s ∈ [snm 2 , s1 ], n, m = 1, 2, ...; nm nm n n nm for all n, m, ii - [p2 , p1 ] ⊂ [p2 , p1 ] and p2 < pnm 1 = px(snm here pnm i i ), i = 1, 2; nm iii - the intervals (pnm 2 , p1 ), n, m = 1, 2, ..., are disjoint and pn1 − pn2 =



(pnm − pnm 1 2 ).

m

Therefore the set ω contains an at most countable number of intervals νm = m (sm 2 , s1 ), such that: m m m 1. the intervals (pm 2 , p1 ), m = 1, 2, . . . , are disjoint (here pi = px(si ), i = 1, 2, m = 1, 2, . . .) ; 2. ·

m p x (t) > 0, for all t ∈ ∪m (sm 2 , s1 ).

(5.21)

Now we take some interval d0k = (tk1 , tk2 ) and let pki = px(tki ), i = 1, 2. Denote km k k m m [pkm 2 , p1 ] = [p2 , p1 ] ∩ [p2 , p1 ].

(5.22)

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Asymptotical stability of optimal paths in nonconvex problems

115

·

Since p x (t) < 0, for all t ∈ d0k , from (5.21) it follows that there are two km km km intervals [tkm 1 , t2 ] and [s2 , s1 ] corresponding to the nonempty interval km , p ], and [pkm 2 1  km k k (tkm 2 − t 1 ) = t2 − t 1 . m

Applying Lemma 6 we obtain km

km

t2

s1

u(x(t))dt+ tkm 1

km

km km km u(x(s))ds ≤ u∗ [(tkm 2 −t1 )+(s1 −s2 )]−

skm 2

s1

δ 2 (x(s))ds.

skm 2

Summing over m and then over k we have tk 2

 k

km

u(x(t))dt +

s1 

u(x(s))ds

k,m km s2

tk 1

⎡ ⎤ skm 1    km ⎦ (skm − s ) − δ 2 (x(s))ds. ≤ u∗ ⎣ (tk2 − tk1 ) + 1 2 k

k,m

k,m km s2

km  Denote ω  = ∪k,m [skm 2 , s1 ]. Clearly ω ⊂ ω. Therefore

u(x(t))dt π∪ω

=

u(x(t))dt +

u(x(t))dt +

ω

π1

≤ u∗ (meas π1 + meas ω  ) −

−



u(x(t))dt +

ω\ω 

u(x(t))dt π2

δ 2 (x(s))ds ω

[u∗ − u(x(t))]dt + u∗ [meas π2 + meas (ω \ ω  )]

π2 ∪(ω\ω  )

= u∗ meas (π ∪ ω) −

Q

[u∗ − u(x(t))]dt −

δ 2 (x(t))dt, E

where Q = π2 ∪ (ω \ ω  ) and E = ω  . Now we check Conditions a), b) and c) of the lemma. Condition a) holds, because Q ∪ E = π2 ∪ (ω \ ω  ) ∪ ω  = π2 ∪ ω. Condition b) follows from Lemma 17.5.1. We now check Condition c). Take any number δ > 0 and denote Pδ = {l : |l − p∗ | ≥ δ}.

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M.A. Mamedov

km k k m m Consider the intervals [pkm 2 , p1 ], [p2 , p1 ] and [p2 , p1 ] (see (6.11)) cor0 responding to the interval dk , where

pk1 − pk2 =



km (pkm 1 − p2 ).

(5.23)

m

From Lemma 17.4.1 we have ·

p x (t) ≤ c(x(t)) < −ηδ for all t ∈ π1 ∩ Zδ .

(5.24)

On the other hand there exists a number K < +∞, for which ·

p x (t) ≤ K for all t ∈ [0, T ]. Therefore meas



[pk2 , pk1 ]



·

[−p x (t)]dt ≥ ηδ meas (dk ∩ Zδ ).

∩ Pδ = dk ∩Zδ

Summing over k we have 

I= meas [pk2 , pk1 ] ∩ Pδ ≤ ηδ meas (π1 ∩ Zδ ).

(5.25)

k

On the other hand, from (5.23) it follows that I=



 km meas [pkm 2 , p1 ] ∩ Pδ =

k,m

≤ K







·

p x (t)dt

k,m km km [s2 ,s1 ]∩Zδ

km  meas [skm 2 , s1 ] ∩ Zδ ≤ K meas (ω ∩ Zδ ) .

(5.26)

k,m

Thus from from (5.25) and (5.26) we obtain meas (π1 ∩ Zδ ) ≤

K K meas (E ∩ Zδ ) ≤ meas [(Q ∪ E) ∩ Zδ ]. ηδ ηδ

But Q ∪ E = π2 ∪ ω and therefore meas [(π ∪ ω) ∩ Zδ ] = meas (π1 ∩ Zδ ) + meas [(Q ∪ E) ∩ Zδ ] K ≤ meas [(Q ∪ E) ∩ Zδ ] + meas [(Q ∪ E) ∩ Zδ ]. ηδ Then Condition c) holds if we take Kδ = proved.

K ηδ

+ 1 and thus the lemma is 

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Asymptotical stability of optimal paths in nonconvex problems

117

5.6 Transformation of the functional (5.2) In this section we divide the sets G0 , Gk and GT (see Lemma 4) into sets of 1st and 2nd type such that Lemma 7 can be applied. Note that x(t) is a continuously differentiable trajectory.

5.6.1 Lemma 8. Assume that the set Gi consists of a finite number of elements gk , g1 < g2 < ... < gN . Then 2

tg N

u(x(t))dt =



u(x(t))dt +

k π ∪ω k k

t1g 1

u(x(t))dt. F

Here πk and ωk are the sets of 1st and 2nd type in the interval [pk2 , pk1 ] and the set F is either a set of 1st type in the interval [p2gN , p1g1 ], if p2gN ≤ p1g1 , or is a set of 2nd type in the interval [p1g1 , p2gN ], if p2gN > p1g1 . Proof. Take the set g1 and assume that [p2g1 , p1g1 ] and [t1g1 , t2g1 ] are the corresponding intervals. Note that we are using the notation introduced in Section 5.5. Take the set g2 . A. First we consider the case p1g2 < p1g1 . In this case there is a point t ∈ [t1g1 , t2g1 ] such that px(t1 )) = p1g2 . Denote π1 = [t1 , t2g1 ] and ω1 = [t2g1 , t1g2 ]. Note that by Definition 6 we have p2g1 < p1g2 . It is clear that π1 and ω1 are sets of 1st and 2nd type on the interval [p2g1 , p1g2 ], respectively. Therefore 1

2

tg2

u(x(t))dt =

t1g1

u(x(t))dt +

π1 ∪ω1

u(x(t))dt, π11

where π11 = [t1g1 , t1 ] ∪ [t1g2 , t2g2 ] is a set of 1st type on the interval [p2g2 , p1g1 ]. B. Now we assume that p1g2 ≥ p1g1 . In this case there is a point t1 ∈ [t2g1 , t1g2 ] such that px(t1 ) = p1g1 . Denote π1 = [t1g1 , t2g1 ] and ω1 = [t2g1 , t1 ]. Consider two cases. 1. Let p2g2 ≥ p1g1 . Then there is a point t2 ∈ [t1 , t1g2 ] such that px(t2 )) = p2g2 . In this case we denote

118

M.A. Mamedov

π2 = [t1g2 , t2g2 ], ω2 = [t2 , t1g2 ] ; and]; ω1 = [t1 , t2 ]. Therefore 2

tg2 u(x(t))dt =



u(x(t))dt +

i=1,2π ∪ω i i

t1g1

u(x(t))dt, ω1

where ω1 is a set of 2nd type on the interval [p1g1 , p2g2 ]. 2. Let p2g2 < p1g1 . Then there is a point t2 ∈ [t1 , t1g2 ] such that px(t2 )) = p1g1 . In this case we denote π1 = [t1g2 , t2 ], ω2 = [t1 , t1g2 ] and π1 = [t2 , t2g2 ]. Therefore 2

tg2 u(x(t))dt =



u(x(t))dt +

i=1,2π ∪ω i i

t1g1

u(x(t))dt, π1

where π1 is a set of 1st type on the interval [p2g2 , p1g1 ]. We repeat this procedure taking g3 , g4 , ...gN and thus the lemma is proved.  Lemma 9. Assume that gn ∈ Gi , n = 1, 2, ..., g1 < g2 < ..., and t2 = limn→∞ t2gn . Then

t

2

u(x(t))dt =



u(x(t))dt +

n π ∪ω n n

t1g1

u(x(t))dt. F

Here πn and ωn are sets of 1st and 2nd type in the interval [p2n , p1n ] and the set F is either a set of 1st type in the interval [p∗ , p1g1 ], if p∗ ≤ p1g1 , or is a set of 2nd type in the interval [p1g1 , p∗ ], if p∗ > p1g1 . Proof. We apply Lemma 8 for every n. From Proposition 1 we obtain that x(t) → x∗ as t → t2 , and therefore p2gn → p∗ as n → ∞. This completes the proof.  We can prove the following lemmas in a similar manner to that used for proving Lemmas 8 and 9. Lemma 10. Assume that the set Gi consists of a finite number of elements gk , g1 > g2 > ... > gN . Then 2

tg1 u(x(t))dt = t1g

N

 k π ∪ω k k

u(x(t))dt +

u(x(t))dt. F

5

Asymptotical stability of optimal paths in nonconvex problems

119

Here πk and ωk are sets of 1st and 2nd type in the interval [p2k , p1k ] and the set F is either a set of 1st type in the interval [p2g1 , p1gN ], if p1gN ≥ p2g1 , or is a set of 2nd type in the interval [p1gN , p2g1 ], if p1gN < p2g1 . Lemma 11. Assume that gn ∈ Gi , n = 1, 2, ..., g1 > g2 > ..., and t1 = limn→∞ t1gn . Then 2

tg1



u(x(t))dt =

u(x(t))dt +

n π ∪ω n n

t1

u(x(t))dt. F

Here πn and ωn are sets of 1st and 2nd type in the interval [p2n , p1n ] and the set F is either a set of 1st type in the interval [p2g1 , p∗ ], if p∗ ≥ p2g1 , or is a set of 2nd type in the interval [p∗ , p2g1 ], if p∗ < p2g1 . In the next lemma we combine the results obtained by Lemmas 9 and 11. Lemma 12. Assume that the set Gi consists of elements gn , n = ±1, ±2, . . ., where · · · < g−2 < g−1 < g1 < g2 < · · · , and where t1 = limn→−∞ t1gn and t2 = limn→∞ t2gn . Then

t

2

u(x(t))dt =



u(x(t))dt.

n π ∪ω n n

t1

Here πn and ωn are sets of 1st and 2nd type in the interval [p2n , p1n ]. Proof. We apply Lemmas 9 and 11 and obtain

t

2

u(x(t))dt =

 n

t1g1 t2g

−1 u(x(t))dt =

t1

u(x(t))dt +

 ∪ω  πn n

 n

 ∪ω  πn n

u(x(t))dt,

(5.27)

F

u(x(t))dt +

u(x(t))dt.

(5.28)

F 

We define π0 = F  ∪ F  and ω0 = [t2g−1 , t1g1 ]. Clearly they are sets of 1st and 2nd type in the interval [p2g−1 , p1g1 ] (note that p2g−1 < p1g1 by Definition 5). Therefore the lemma is proved if we sum (5.27) and (5.28). 

5.6.2 Now we use Lemma 4. We take any interval [t1k , t2k ] and let

120

M.A. Mamedov

[t1k , t2k ] = [t1k , t1Gk ] ∪ [t1Gk , t2Gk ] ∪ [t2Gk , t2k ]. We show that 2

tk u(x(t))dt =

 n

t1k

u(x(t))dt +

k ∪ω k πn n

u(x(t))dt,

(5.29)

Ek

where πnk and ωnk are sets of 1st and 2nd type in the interval [p2nk , p1nk ] and x(t) ∈ int D, ∀t ∈ E k . If the conditions of Lemma 12 hold then (5.29) is true if we take E k = [t1k , t1Gk ] ∪ [t2Gk , t2k ]. Otherwise we apply Lemmas 8–11 and obtain 2

tGk u(x(t))dt =

 n

t1G k

u(x(t))dt +

k ∪ω k πn n

u(x(t))dt.

Fk

If F k is a set of 2nd type then (5.29) is true if we take E k = F k . Assume that F k is a set of 1st type on some interval [p2 , p1 ]. In this case we set π0k = F k and ω0k = [t1k , t1Gk ] ∪ [t2Gk , t2k ]. We have x(t1k ) = x(t2k ) = x∗ (see Lemma 4) and therefore π0k and ω0k are sets of 1st and 2nd type in the interval [p2 , p1 ]. Thus (5.29) is true. Now we apply Lemmas 8–12 to the intervals [0, t1 ] and [t2 , T ]. We have

t

1

u(x(t))dt =

 n

0

T u(x(t))dt = t2

0 ∪ω 0 πn n

 n

u(x(t))dt + F0

T ∪ω T πn n

u(x(t))dt +

u(x(t))dt + FT

u(x(t))dt,

(5.30)

u(x(t))dt.

(5.31)

E0

u(x(t))dt + ET

Here • F 0 and F T are sets of 1st type (they may be empty); • [0, t1G0 ] ∪ [t2G0 , t1 ] ⊂ E 0 and [t2 , t1GT ] ∪ [t2GT , T ] ⊂ E T ; • x(t) ∈ / int D for all t ∈ E 0 ∪ E T . Thus, applying Lemma 4 and taking into account (5.29)–(5.31), we can prove the following lemma. Lemma 13. The interval [0, T ] can be divided into subintervals such that

5

Asymptotical stability of optimal paths in nonconvex problems

121

[0, T ] = ∪n (πn ∪ ωn ) ∪ F1 ∪ F2 ∪ E,

T u(x(t)) dt = 0

(5.32)



u(x(t)) dt +

n π ∪ω n n

u(x(t)) dt +

F1 ∪F2

u(x(t)) dt. E

(5.33) Here 1. The sets πn and ωn are sets of 1st and 2nd type, respectively, in the intervals [p2n , p1n ], n = 1, 2, .... 2. The sets F1 and F2 are sets of 1st type in the intervals [p21 , p11 ] and [p22 , p12 ], respectively, and x(t) ∈ int D, f or all t ∈ F1 ∪ F2 , p1i

−

p2i

≤ C < +∞, i = 1, 2.

(5.34) (5.35)

3. Also x(t) ∈ / int D, f or all t ∈ E.

(5.36)

4. For every δ > 0 there is a number C(δ) such that meas [(F1 ∪ F2 ) ∩ Zδ ] ≤ C(δ),

(5.37)

where the number C(δ) < ∞ does not depend on the trajectory x(t), on T or on the intervals in (5.32). Proof: We define F1 = {t ∈ F 0 : x(t) ∈ int D}, F2 = {t ∈ F T : x(t) ∈ int D} and E = ∪k E k ∪ E 0 ∪ E T . Then we replace π10 to π10 ∪ (F 0 \ F1 ) and π1T to π1T ∪ (F T \ F2 ) in (5.30) and (5.31) (note that after these replacements the conditions of Definition 6 still hold). We obtain (5.33) summing (5.29)– (5.31). It is not difficult to see that all assertions of the lemma hold. Note that (5.35) follows from the fact that the trajectory x(t) is uniformly bounded (see (5.3)). The inequality (5.37) follows from Lemma 17.4.1, taking into account Definition 6, and thus the lemma is proved. Lemma 14. There is a number L < +∞ such that

[u(x(t)) − u∗ ] dt < L,

(5.38)

F1 ∪F2

where L does not depend on the trajectory x(t), on T or on the intervals in (5.32). Proof: From Condition H3 it follows that there exist a number ε > 0 and a ∼ ∼ trajectory x (·) to the system (6.1), defined on [ 0, Tε ], such that p x (0) = ∼ p∗ − ε, p x (Tε ) = p∗ + ε and

122

M.A. Mamedov . ∼

p x (t) > 0 for almost all t ∈ [0, Tε ].

Define

(5.39)

∼

u(x (t))dt.

Rε = [0,Tε ]

Consider the set F1 and corresponding interval [p21 , p11 ]. Define a set F1ε = {t ∈ F 1 : |px(t) − p∗ | < ε}. We consider the most common case, when [p∗ − ε, p∗ + ε] ⊂ [p21 , p11 ]. In this case the sets F1ε and [0, Tε ] are sets of 1st and 2nd type in the interval ∼ [p∗ − ε, p∗ + ε] for the trajectories x(·) and x (·), respectively. We have



u(x(t))dt + Rε = u(x(t)) dt + u(x(t)) dt + u(x(t))dt. F1ε

F1

[0,Tε ]

F1 \F1ε

(5.40) ∼

We use Lemma 7. Note that this lemma can be applied for the trajectory x (·) (which may not be continuously differentiable) due to the inequality (5.39). Taking into account δ(x) ≥ 0 and u(x(t)) ≤ u∗ , for t ∈ Q, from Lemma 7 we obtain

u(x(t)) dt + u(x(t)) dt ≤ u∗ (meas F1ε + Tε ). (5.41) F1ε

[0,Tε ]

From (5.37) it follows that meas (F1 \ F1ε ) = meas (F1 ∩ Zε ) ≤ C(ε). Thus

u(x(t)) dt ≤ Cε ,

(5.42)

F1 \F1ε

where the number Cε < +∞ does not depend on T or on the trajectory x(t). Denote C  = Tε u∗ + Cε − Rε . Then from (5.40)–(5.42) we obtain

u(x(t)) dt ≤ u∗ meas F1ε + C  ≤ u∗ meas F1 + C  F1

and therefore

F1

[u(x(t)) − u∗ ] dt ≤ C  .

5

Asymptotical stability of optimal paths in nonconvex problems

123

By analogy we can prove that

[u(x(t)) − u∗ ] dt ≤ C  . F2

Thus the lemma is proved if we take L = C  + C  .

5.7 The proof of Theorem 13.6 From Condition M it follows that for every T > 0 there exists a trajectory xT (·) ∈ XT , for which

u(xT (t)) dt ≥ u∗ T − b.

(5.43)

[0,T ]

5.7.1 First we consider the case when x(t) is a continuously differentiable function. In this case we can apply Lemma 7. From Lemmas 4 and 14 we have

 u(x(t)) dt ≤ u(x(t)) dt+ u(x(t)) dt+L+u∗ +meas (F1 ∪F2 ). n π ∪ω n n

[0,T ]

E

Then applying Lemma 7 we obtain

u(x(t)) dt [0,T ]

≤



⎛ ⎝u∗ meas (πn ∪ ωn ) −

n

+ E

= u∗

−

u(x(t)) dt −

Qn

⎞ δ 2 (x(t)) dt⎠

En

u(x(t)) dt + L + u∗ + meas (F1 ∪ F2 )  

 meas (πn ∪ ωn ) + meas (F1 ∪ F2 ) + meas E

n

[u∗ − u(x(t))] dt −

Q ∗



= u meas [0, T ] −

δ 2 (x(t)) dt + L A

∗

[u − u(x(t))] dt − Q

δ 2 (x(t)) dt + L. A

124

M.A. Mamedov

Here Q = (∪n Qn ) ∪ E and A = ∪n En . Taking (5.43) into account we have

u(x(t)) dt − u(xT (t)) dt ≤ − [u∗ − u(x(t))] dt [0,T ]

Q

[0,T ]

−

δ 2 (x(t)) dt + L + b, A

that is,

JT (x(·)) − JT (xT (·)) ≤ −

[u∗ − u(x(t))] dt −

Q

δ 2 (x(t)) dt + L + b. A

(5.44) Here Q = (∪n Qn ) ∪ E and A = ∪n En

(5.45)

and the following conditions hold: a) Q ∪ A = { t ∈ [0, T ] : x(t) ∈ / int D};

(5.46)

[0, T ] = ∪n (πn ∪ ωn ) ∪ (F1 ∪ F2 ) ∪ E;

(5.47)

b)

c) for every δ > 0 there exist K(δ) < +∞ and C(δ) < +∞ such that meas [(πn ∪ ωn ) ∩ Zδ ] ≤ K(δ) meas [(Qn ∪ En ) ∩ Zδ ] and meas [(F1 ∪ F2 ) ∩ Zδ ] ≤ C(δ);

(5.48) (5.49)

(recalling that Zδ = {t ∈ [0, T ] : |px(t) − p∗ | ≥ δ}) d) for every ε > 0 there exists δε > 0 such that δ 2 (x) ≥ δε for all x, ||x − x∗ || ≥ ε.

(5.50)

The first assertion of the theorem follows from (5.44), (5.46) and (5.50) for the case under consideration (that is, x(t) continuously differentiable). We now prove the second assertion. Let ε > 0 and δ > 0 be given numbers and x(·) a continuously differentiable ξ-optimal trajectory. We denote Xε = {t ∈ [0, T ] : ||x(t) − x∗ || ≥ ε}.

5

Asymptotical stability of optimal paths in nonconvex problems

125

∼

First we show that there exists a number K ε,ξ < +∞ which does not depend on T > 0 and meas [(Q ∪ A) ∩ Xε ] ≤

∼

K ε,ξ .

(5.51)

Assume that (5.51) is not true. In this case there exist sequences Tk → ∞, k → ∞ and sequences of trajectories {xk (·)} (every xk (·) is a ξ-optimal Kε,ξ trajectory in the interval [0, Tk ]) and {xTk (·)} (satisfying (5.43) for every T = Tk ) such that k meas [(Qk ∪ Ak ) ∩ Xεk ] ≥ Kε,ξ

as

k → ∞.

(5.52)

From Lemma 17.3.1 and (5.50) we have u∗ − u(xk (t)) ≥ νε δ 2 (xk (t)) ≥ δε2

if if

t ∈ Qk ∪ Xεk and t ∈ Ak ∩ Xεk .

Denote ν = min {νε , δε2 } > 0. From (5.44) it follows that JTk (xk (·)) − JTk (xTk (·)) ≤ L + b − ν meas [(Qk ∪ Ak ) ∩ Xεk ]. Therefore, for sufficiently large numbers k, we have JTk (xk (·)) ≤ JTk (xTk (·)) − 2 ξ ≤ JT∗k − 2 ξ, which means that xk (t) is not a ξ-optimal trajectory. This is a contradiction. Thus (5.51) is true. 1 < +∞ such that Now we show that for every δ > 0 there is a number Kδ,ξ 1 meas Zδ ≤ Kδ,ξ .

(5.53)

From (5.47)–(5.49) we have  meas [(πn ∪ ωn ) ∩ Zδ ] + meas [(F1 ∪ F2 ) ∩ Zδ ] meas Zδ = n

+ meas (E ∩ Zδ )  K(δ) meas [(Qn ∪ En ) ∩ Zδ ] + C(δ) + meas (E ∩ Zδ ) ≤ n ∼

≤K (δ) meas [([∪n (Qn ∪ En )] ∩ Zδ ) ∪ (E ∩ Zδ )] + C(δ) ∼

=K (δ) meas [(Q ∪ A) ∩ Zδ ] + C(δ), ∼

where K (δ) = max{1, K(δ)}. Since Zδ ⊂ Xδ , taking (5.51) into account we obtain (5.53), where

126

M.A. Mamedov ∼

∼

1 Kδ,ξ =K (δ) K δ,ξ +C(δ). 0 0 We denote Xε/2 = {t ∈ [0, T ] : ||x(t) − x∗ || > ε/2.}. Clearly Xε/2 is an open set and therefore can be presented as a union of an at most countable ∼ 0 = ∪k τ k . Out of these intervals we number of open intervals, say Xε/2 choose further intervals, which have a nonempty intersection with Xε , say these are τk , k = 1, 2, .... Then we have 0 Xε ⊂ ∪k τk ⊂ Xε/2 .

(5.54)

Since a derivative of the function x(t) is bounded, it is not difficult to see that there exists a number σε > 0 such that meas τk ≥ σε

for all k.

(5.55)

But the interval [0, T ] is bounded and therefore the number of intervals τk is finite too. Let k = 1, 2, 3, ..., NT (ε). We divide every interval τk into two parts: τk1 = {t ∈ τk : x(t) ∈ int D} and τk2 = {t ∈ τk : x(t) ∈ / int D}. From (5.46) and (5.54) we obtain 0 ∪k τk2 ⊂ (Q ∪ A) ∩ Xε/2

and therefore from (5.51) it follows that ∼

meas (∪k τk2 ) ≤ K ε/2,ξ .

(5.56)

Now we apply Lemma 17.4.1. We have ·

p x (t) ≤ − ηε/2 ,

t ∈ ∪k τk1 .

(5.57)

Define p1k = supt∈τk px(t) and p2k = inf t∈τk px(t). It is clear that ∼

p1k − p2k ≤ C ,

k = 1, 2, 3, ..., NT (ε),

(5.58)

and ·

|p x (t)| ≤ K,

for all t.

(5.59)

∼

Here the numbers C and K do not depend on T > 0, x(·), ε or ξ. We divide the interval τk into three parts: ·

τk− = {t ∈ τk : p x (t) < 0}, ·

τk+ = {t ∈ τk : p x (t) > 0}.

·

τk0 = {t ∈ τk : p x (t) = 0} and

5

Asymptotical stability of optimal paths in nonconvex problems

Then we have p1k − p2k

* * * * * * * *

* * * * · · · * * ≥ ** p x (t)dt ** = * p x (t)dt + p x (t)dt * . * * * * * τ− * τk τ+ k

We define

127

α=−



·

p x (t)dt and β =

τk−



k

·

p x (t)dt.

Clearly α > 0, β > 0

τk+

and

) p1k − p2k ≥

−α + β, α − β,

if α < β, if α ≥ β.

(5.60)

From (5.59) we obtain 0 < β ≤ K meas τk+ .

(5.61)

On the other hand, τk1 ⊂ τk− and therefore from (5.57) we have α ≥ ηε/2 meas τk− ≥ ηε/2 meas τk1 .

(5.62)

Consider two cases. a) α ≥ β. Then from (5.60)–(5.62) we obtain ∼

+ 1 2 1 C ≥ pk − pk ≥ α − β ≥ ηε/2 meas τk − K meas τk .

Since τk+ ⊂ τk2 , then from (5.56) it follows that Therefore from (5.63) we have  , meas τk1 ≤ Cε,ξ

meas τk+ ≤ ∼

 where Cε,ξ = (C + K· K ε/2,ξ )/ηε/2 .

(5.63) ∼

K ε/2,ξ .

(5.64)

b) α < β. Then from (5.61) and (5.62) we obtain ∼

ηε/2 meas τk1 < K meas τk+ ≤ K· K ε/2,ξ or  meas τk1 < Cε,ξ ,

∼

 where Cε,ξ = K· K ε/2,ξ /ηε/2 .

(5.65)

Thus from (5.64) and (5.65) we obtain   meas τk1 ≤ Cε,ξ = max{Cε,ξ , Cε,ξ },

k = 1, 2, ..., NT (ε),

and then meas (∪k τk1 ) ≤ NT (ε) Cε,ξ .

(5.66)

128

M.A. Mamedov

Now we show that for every ε > 0 and ξ > 0 there exists a number  < +∞ such that Kε,ξ  . meas (∪k τk1 ) ≤ Kε,ξ

(5.67)

Assume that (5.67) is not true. Then from (5.66) it follows that NT (ε) → ∞ as T → ∞. Consider the intervals τk for which the following conditions hold: meas τk1 ≥

1 σε and meas τk2 ≤ λ meas τk1 , 2

(5.68)

where λ is any fixed number. Since NT (ε) → ∞, then from (5.55) and (5.56) it follows that the number of intervals τk satisfying (5.68) increases infinitely as T → ∞. On the other hand, the number of intervals τk , for which the conditions α < β, meas τk2 > λ meas τk1 and λ = ηε/2 /K hold, is finite. Therefore the number of of intervals τk for which the conditions α ≤ β and (5.68) hold infinitely increases as T → ∞. We denote the number of such intervals by NT and for the sake of definiteness assume that these are intervals τk , k = 1, 2, ..., NT . We set λ = ηε/2 /2K for every τk . Then from (5.63) and (5.68) we have p1k − p2k ≥ ηε/2 meas τk1 − K·

ηε/2 1 meas τk1 = ηε/2 meas τk1 . 2K 2

Taking (5.55) into account we obtain p1k − p2k ≥ eε ,

k = 1, 2, ..., NT ,

(5.69)

where eε =

1 ηε/2 σε > 0 2

1 8 eε . From (5.69) 1 2 [sk , sk ] ⊂ τk such that

Let δ = d

dk =

and NT → ∞ as T → ∞.

it follows that for every τk there exists an interval

|p x(t) − p∗ | ≥ δ, t ∈ dk ,

p x(s1k ) = sup p x(t), t∈dk

p x(s2k ) = inf p x(t) t∈dk

and p x(s1k ) − p x(s2k ) = δ.

From (5.59) we have * * * *

* * · · · * * δ = * p x (t) dt* ≤ |p x (t)| dt ≤ |p x (t)| dt ≤ K·meas dk . * * * [s1 ,s2 ] * [s1 ,s2 ] dk k

k

k

k

5

Asymptotical stability of optimal paths in nonconvex problems

129

Then meas dk ≥ δ/K > 0. Clearly dk ⊂ Zδ and therefore

T meas Zδ ≥ meas ∪N k=1 dk =

NT 

meas dk ≥ NT

k=1

δ . K

This means that meas Zδ → ∞ as T → ∞, which contradicts (5.53). Thus (5.67) is true. Then taking (5.56) into account we obtain meas ∪k τk =



∼

 (meas τk1 + meas τk2 ) ≤ K ε/2,ξ +Kε,ξ .

k

Therefore from (5.54) it follows that meas Xε = meas ∪k τk ≤ Kε,ξ , ∼

 where Kε,ξ =K ε/2,ξ +Kε,ξ . Thus we have proved that the second assertion of the theorem is true for the case when x(t) is a continuously differentiable function.

5.7.2 We now take any trajectory x(·) to the system (6.1). It is known (see, for example, [3]) that for a given number δ > 0 (we take δ < ε/2) there exists a ∼ continuously differentiable trajectory x (·) to the system (6.1) such that ∼

|| x(t)− x (t)|| ≤ δ for all t ∈ [0, T ]. Since the function u is continuous then there exists η(δ) > 0 such that ∼

u(x (t)) ≥ u(x(t)) − η(δ) for all t ∈ [0, T ]. Therefore

∼

u(x (t)) dt ≥ [0,T ]

u(x(t)) dt − T η(δ). [0,T ]

Let ξ > 0 be a given number. For every T > 0 we choose a number δ such that T η(δ) ≤ ξ. Then

∼ ∼ u(x(t)) dt ≤ u(x (t)) dt + T η(δ) ≤ u(x (t)) dt + ξ, (5.70) [0,T ]

[0,T ]

[0,T ]

130

M.A. Mamedov

that is,



∗

[ u(x(t)) − u ] dt ≤ [0,T ]

∼

[ u(x (t)) − u∗ ] dt + ξ.

[0,T ] ∼

Since the function x (·) is continuously differentiable then the second integral in this inequality is bounded (see the first part of the proof), and therefore the first assertion of the theorem is proved. Now we prove the second assertion of Theorem 13.6. We will use (5.70). Take a number ε > 0 and assume that x(·) is a ξ-optimal trajectory, that is, JT (x(·)) ≥ JT∗ − ξ. From (5.70) we have ∼

JT (x (·)) ≥ JT (x(·)) − ξ ≥ JT∗ − 2ξ. ∼

Thus x (·) is a continuously differentiable 2ξ-optimal trajectory. That is why (see the first part of the proof) for the numbers ε/2 > 0 and 2ξ > 0 there exists Kε,ξ < +∞ such that ∼

meas { t ∈ [0, T ] : || x (t) − x∗ || ≥ ε/2} ≤ Kε,ξ . If || x(t ) − x∗ || ≥ ε for any t then ∼

∼

|| x (t ) − x∗ || ≥ || x(t ) − x∗ || − || x(t )− x (t ) || ≥ ε − δ ≥

ε . 2

Therefore ∼

{ t ∈ [0, T ] : || x(t) − x∗ || ≥ ε} ⊂ { t ∈ [0, T ] : || x (t) − x∗ || ≥ ε/2}, which implies that the proof of the second assertion of the theorem is completed, that is, meas { t ∈ [0, T ] : || x(t) − x∗ || ≥ ε} ≤ Kε,ξ . Now we prove the third assertion of the theorem. Let x(·) be an optimal trajectory and x(t1 ) = x(t2 ) = x∗ . Consider a trajectory x∗ (·) defined by the formula ) x(t) if t ∈ [0, t1 ] ∪ [t2 , T ], x∗ (t) = t ∈ [t1 , t2 ]. x∗ if

5

Asymptotical stability of optimal paths in nonconvex problems

131

Assume that the third assertion of the theorem is not true, that is, there is a point t ∈ (t1 , t2 ) such that ||x(t ) − x∗ || = c > 0. Consider the function x(·). In [3] it is proved that there is a sequence of continuously differentiable trajectories xn (·), t ∈ [t1 , T ], which is uniformly convergent to x(·) on [t1 , T ] and for which xn (t1 ) = x(t1 ) = x∗ . That is, for every δ > 0 there exists a number Nδ such that max || xn (t) − x(t) || ≤ δ for all n ≥ Nδ .

t∈[t1 ,T ]

On the other hand, for every δ > 0 there exists a number η(δ) > 0 such that η(δ) → 0 as δ → 0 and | u(x(t)) − u(xn (t)) | ≤ η(δ) for all t ∈ [t1 , T ].

(5.71)

Then we have

u(x(t)) dt ≤

[t1 ,T ]

u(xn (t)) dt + T η(δ).

(5.72)

[t1 ,T ]

Take a sequence of points tn ∈ (t , t2 ) such that tn → t2 as n → ∞. Clearly in this case xn (tn ) → x∗ . We apply Lemma 13 for the interval [t1 , tn ] and obtain (see also (5.31))

u(xn (t)) dt [t1

,tn ]

=



u(xn (t)) dt +

k π n ∪ω n k k

Fn

u(xn (t)) dt +

u(xn (t)) dt.

(5.73)

En

Here x(t) ∈ int D ∀t ∈ F n and F n is a set of 1st type on the interval [pxn (tn ), p∗ ] if pxn (tn ) < p∗ . Since xn (tn ) → x∗ , pxn (tn ) → p∗ and thus for every t ∈ F n we have u(xn (t)) → u∗ as n → ∞. Therefore

αn =

[u(xn (t)) − u∗ ] dt → 0 as n → ∞.

Fn

We also note that from xn (t) ∈ / int D, t ∈ E n , it follows that

En

u(xn (t)) dt ≤ u∗ meas E n .

132

M.A. Mamedov

Now we use Lemma 7 and obtain  u(xn (t)) dt k π n ∪ω n k k

=u

∗

meas [∪k (πkn

∪

ωkn )]

−

∗

[u − u(xn (t))] dt −

∪k Qn k

δ 2 (xn (t)) dt.

∪k Ekn ∼

We take a number δ < c/2. Then there exists a number β > 0 such that ∼

meas [∪k (Qnk ∪ Ekn )] ≥ β . Then there exists a number β > 0 for which 

u(xn (t)) dt ≤ u∗ meas [∪k (πkn ∪ ωkn )] − β.

k π n ∪ω n k k

Therefore from (5.73) we have

u(xn (t)) dt ≤ u∗ {meas [∪k (πkn ∪ ωkn )] + meas F n + meas E n }

[t1 ,tn ]

+ αn − β or

u(xn (t)) dt ≤ u∗ (tn − t1 ) + αn − β.

(5.74)

[t1 ,tn ]

From (5.71) we obtain

u(xn (t)) dt [t2 ,T ]

≤

u(x(t)) dt + T η(δ) =

[t2 ,T ]

Thus from (5.72)–(5.75) we have

[t2 ,T ]

u(x∗ (t)) dt + T η(δ).

(5.75)

5

Asymptotical stability of optimal paths in nonconvex problems

133

u(x(t)) dt [t1 ,T ]

≤

u(xn (t)) dt + T η(δ) [t1 ,T ]

=

u(xn (t)) dt +

[t1 ,tn ]

u(xn (t)) dt +

[tn ,t2 ]

≤ u∗ (tn − t1 ) + u∗ (t2 − tn ) +

u(xn (t)) dt + T η(δ)

[t2 ,T ]

u(x∗ (t)) dt

[t2 ,T ]

+ αn − β + λn + 2T η(δ)

= u(x∗ (t)) dt + αn − β + λn + 2T η(δ). [t1 ,T ]

Here λn =

[ u(xn (t)) − u∗ ] dt → 0 as n → ∞,

[tn ,t2 ]

because tn → t2 . We choose the numbers δ > 0 and n such that the following inequality holds: αn + λn + 2T η(δ) < β. In this case we have



u(x∗ (t)) dt

u(x(t)) dt < [t1 ,T ]

and therefore

[t1 ,T ]



u(x(t)) dt <

[0,T ]

u(x∗ (t)) dt,

[0,T ]

which means that x(t) is not optimal. This is a contradiction. Thus the theorem is proved.

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