Variational Methods in Partially Ordered Spaces (CMS Books in Mathematics)

Canadian Mathematical Society Societe mathematique du Canada Editors-in - Chief Re´dacteurs-en-chef Jonathan Borwein Peter Borwein

Springer

New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo

Alfred Go¨pfert Hassan Riahi

Christiane Tammer Constantin Za˘linescu

Variational Methods in Partially Ordered Spaces With 13 Illustrations

Alfred Go¨pfert Department of Mathematics and Computer Science Martin-Luther-University Halle Wittenberg Halle D-06099 Germany [email protected]

Christiane Tammer Department of Mathematics and Computer Science Martin-Luther-University Halle-Wittenberg Halle D-06099 Germany [email protected]

Hassan Riahi Faculte´ des Sciences Semlalia, Mathematiques Universite´ Cadi Ayyad B.P. S 15 Marrakech 40000 Morocco [email protected]

Constantin Za˘linescu Faculty of Mathematics University of “Al. I. Cuza” Iasi Bd. Copou, Nr. 11 700506 Ia\c si Romania [email protected]

Editors-in-Chief Re´dacteurs-en-chef Jonathan Borwein Peter Borwein Centre for Experimental and Constructive Mathematics Department of Mathematics and Statistics Simon Fraser University Burnaby, British Columbia V5A 1S6 Canada [email protected] Mathematics Subject Classification (2000): 00A71, 93A30, 65K10 Library of Congress Cataloging-in-Publication Data Variational methods in partially ordered spaces / Alfred Go¨pfert ... [et al.]. p. cm. — (CMS books in mathematics ; 17) Includes bibliographical references and index. ISBN 0-387-00452-1 (alk. paper) 1. Partially ordered spaces. 2. Vector spaces. 3. Mathematical optimization. I. Go¨pfert, A. (Alfred) II. Series. QA611.3 V37 2003 512′.52—dc21 2003044935 ISBN 0-387-00452-1

Printed on acid-free paper.

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Preface

In mathematical modeling of processes occurring in industrial systems, logistics, management science, operations research, networks, and control theory one often encounters optimization problems involving more than one objective function, so that multiobjective optimization (or vector optimization, initiated by W. Pareto) has received new impetus. The growing interest in multiobjective problems, both from the theoretical point of view and as it concerns applications to real problems, asks for a general scheme that embraces several existing developments and stimulates new ones. With this book we intend to give direct access to new results and new applications of this quickly growing field.

Mathematical Background In particular, we discuss basic tools of partially ordered spaces and apply them to variational methods in nonlinear analysis and for optimization problems; i.e., we present the relevant functional analysis for our presentations, especially separation theorems for not necessarily convex sets, which are important for the characterization of solutions, for the proof of existence results and optimality conditions in multicriteria optimization. We use the optimality conditions in order to derive numerical algorithms for special classes of vector optimization problems.

Purpose of This Book We believe that our book will be of interest to graduate students in mathematics, economics, and engineering as well as researchers in pure and applied mathematics, economics, engineering, geography, and town planning. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book.

vi

Preface

On the one hand, the book has the character of a monograph, because the authors use many of their own results and applications; on the other hand, it is a textbook, because we would like to present in a sense a state of the art of the field in an understandable, useful, and teachable way.

Organization Firstly, we shall give some simple examples to show which kinds of problems can be handled with the methods of the book. Then the three main chapters follow. In the first of them we deal with connections between order structures and topological structures of sets, give a new nonconvex separation theorem, which is very useful for scalarization, and study different properties of multifunctions. The second of them contains our results concerning the theory of multicriteria optimization and equilibrium problems directly. Approximate efficiency, scalarization, new existence results with respect to different order relations, well-posedness, sensitivity, duality, and optimality conditions with respect to general vector optimization problems and a big section on minimal point theorems belong to this chapter as well as new results of vector equilibrium problems and their applications to vector-valued variational inequalities. Those new theoretical results are applied in the last chapter of the book in order to construct numerical algorithms, especially proximal-point algorithms and geometrical algorithms based on duality assertions. It is possible to use the special structure of several classes of multicriteria optimization problems (location problems, approximation problems, fractional programming problems, multicriteria control problems) for deriving optimality conditions and corresponding algorithms. We discuss concrete applications (approximation problems, location problems in town planning, multicriteria equilibrium problems, fractional programming) with solution procedures and in some cases with corresponding software. Here and in the whole book there are examples to illustrate the results or to check stated conditions. The chapters are followed by a list of references, a list of symbols and a big enough index. The book was written by four authors, we wrote it together, and it was at every time stimulating and profitable to consider the problems from different sides. A. G¨ opfert, Chr. Tammer, and C. Z˘ alinescu contributed to Sections 1.1, 2.1–2.3, 3.1, and 3.10; C. Z˘ alinescu wrote Sections 2.4–2.7 and 3.2–3.6; H. Riahi wrote Sections 3.8, 3.9, 4.2.4, and 4.2.5; and A. G¨opfert and Chr. Tammer wrote Sections 1.2–1.6, 3.7, 3.11, 4.1, 4.2.1–4.2.3, and 4.3–4.6.

Preface

vii

Acknowledgments We are grateful to Manfred Goebel, Sylke Sauter, Kristin Winkler, and Eugen Z˘ alinescu for numerous suggestions for technically improving the manuscript. Each author is grateful to her/ his coauthors for their wonderful typing of the manuscript. We would like to thank an anonymous referee for his/ her remarks. We are happy to publish this monograph in the series Canadian Mathematical Society Series of Monographs and Advanced Texts by Springer.

Halle, Germany Halle, Germany Marrakech, Morocco Ia¸si, Romania January 2003

Alfred G¨opfert Christiane Tammer Hassan Riahi Constantin Z˘alinescu

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Cones in Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Equilibrium Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Location Problems in Town Planning . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Multicriteria Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Multicriteria Fractional Programming Problems . . . . . . . . . . . . . 10 1.6 Stochastic Efficiency in a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2

Functional Analysis over Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Order Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Functional Analysis and Convexity . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Separation Theorems for Not Necessarily Convex Sets . . . . . . . . 2.4 Convexity Notions for Sets and Multifunctions . . . . . . . . . . . . . . 2.5 Continuity Notions for Multifunctions . . . . . . . . . . . . . . . . . . . . . . 2.6 Continuity Properties of Multifunctions Under Convexity Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Tangent Cones and Differentiability of Multifunctions . . . . . . . .

3

Optimization in Partially Ordered Spaces . . . . . . . . . . . . . . . . . . 3.1 Solution Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Approximate Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 A General Scalarization Method . . . . . . . . . . . . . . . . . . . . . 3.2 Existence Results for Efficient Points . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Preliminary Notions and Results Concerning Transitive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Existence of Maximal Elements with Respect to Transitive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 26 39 45 51 70 73 81 81 81 84 86 87 89

x

Contents

3.3 3.4 3.5

3.6 3.7

3.8

3.9

3.10

3.11

3.2.3 Existence of Efficient Points with Respect to Cones . . . . 94 3.2.4 Types of Convex Cones and Compactness with Respect to Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.2.5 Classification of Existence Results for Efficient Points . . 104 3.2.6 Some Density and Connectedness Results . . . . . . . . . . . . . 110 Continuity Properties with Respect to a Scalarization Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Well-Posedness of Vector Optimization Problems . . . . . . . . . . . . 119 Continuity Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.5.1 Continuity Properties of Optimal-Value Multifunctions . 122 3.5.2 Continuity Properties for the Optimal Multifunction in the Case of Moving Cones . . . . . . . . . . . . . . . . . . . . . . . . 135 3.5.3 Continuity Properties for the Solution Multifunction . . . 138 Sensitivity of Vector Optimization Problems . . . . . . . . . . . . . . . . 141 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.7.1 Duality Without Scalarization . . . . . . . . . . . . . . . . . . . . . . 155 3.7.2 Duality by Scalarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 3.7.3 Duality for Approximation Problems . . . . . . . . . . . . . . . . . 162 Vector Equilibrium Problems and Related Topics . . . . . . . . . . . . 168 3.8.1 Vector Equilibrium Problems . . . . . . . . . . . . . . . . . . . . . . . 169 3.8.2 General Vector Monotonicity . . . . . . . . . . . . . . . . . . . . . . . 171 3.8.3 Existence of Vector Equilibria by Use of the Generalized KKM Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.8.4 Existence by Scalarization of Vector Equilibrium Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 3.8.5 Some Knowledge About the Assumptions . . . . . . . . . . . . 177 3.8.6 Some Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3.8.7 Mixed Vector Equilibrium Problems . . . . . . . . . . . . . . . . . 183 Applications to Vector Variational Inequalities . . . . . . . . . . . . . . 186 3.9.1 Vector Variational-Like Inequalities . . . . . . . . . . . . . . . . . . 186 3.9.2 Perturbed Vector Variational Inequalities . . . . . . . . . . . . . 188 3.9.3 Hemivariational Inequality Systems . . . . . . . . . . . . . . . . . . 189 3.9.4 Vector Complementarity Problems . . . . . . . . . . . . . . . . . . . 192 3.9.5 Application to Vector Optimization Problems . . . . . . . . . 194 3.9.6 Minimax Theorem for Vector-Valued Mappings . . . . . . . 196 Minimal-Point Theorems in Product Spaces and Corresponding Variational Principles . . . . . . . . . . . . . . . . . . . . . . . 197 3.10.1 Not Authentic Minimal-Point Theorems . . . . . . . . . . . . . . 199 3.10.2 Authentic Minimal-Point Theorems . . . . . . . . . . . . . . . . . . 202 3.10.3 Minimal-Point Theorems and Gauge Techniques . . . . . . . 205 3.10.4 Minimal-Point Theorems and Cone-Valued Metrics . . . . 209 3.10.5 Fixed Point Theorems of Kirk–Caristi Type . . . . . . . . . . 212 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3.11.1 Lagrange Multipliers and Saddle Point Assertions . . . . . 213 3.11.2 ε-Saddle Point Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

Contents

4

xi

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.1 Approximation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.1.1 General Approximation Problems . . . . . . . . . . . . . . . . . . . 229 4.1.2 Finite-dimensional Approximation Problems . . . . . . . . . . 235 4.1.3 Lp -Approximation Problems . . . . . . . . . . . . . . . . . . . . . . . . 241 4.1.4 Example: The Inverse Stefan Problem . . . . . . . . . . . . . . . . 242 4.2 Solution Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.2.1 A Proximal-Point Algorithm for Real-Valued Control Approximation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.2.2 Computer Programs for the Application of the Proximal-Point Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 259 4.2.3 An Interactive Algorithm for the Vector Control Approximation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.2.4 Proximal Algorithms for Vector Equilibrium Problems . 266 4.2.5 Relaxation and Penalization . . . . . . . . . . . . . . . . . . . . . . . . 277 4.3 Location Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 4.3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 282 4.3.2 An Algorithm for the Multicriteria Location Problem . . 285 4.3.3 A Mathematica Program for Solving the Multicriteria Location Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 4.3.4 Comparison of Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . 287 4.3.5 Application to a Problem of Town Planning . . . . . . . . . . 289 4.4 Multicriteria Fractional Programming . . . . . . . . . . . . . . . . . . . . . . 294 4.4.1 Solution Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 4.4.2 Generalized Dinkelbach Transformation . . . . . . . . . . . . . . 297 4.4.3 Possibilities for a Solution Approach . . . . . . . . . . . . . . . . . 300 4.5 Multicriteria Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 4.5.1 The Formulation of the Problem . . . . . . . . . . . . . . . . . . . . 303 4.5.2 An ε-Minimum Principle for Multicriteria Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 4.5.3 A Multicriteria Stochastic Control Problem . . . . . . . . . . . 309 4.6 Stochastic Efficiency in a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

List of Figures

1.3.1 The set of efficient elements of the multicriteria location problem (P) with the maximum norm . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The set of efficient points of the multicriteria location problem (P) with the Lebesgue norm instead of the maximum norm . . . .

8 8

2.3.1 Level sets of the functional ϕ in (2.23). . . . . . . . . . . . . . . . . . . . . . . 42 3.1.1 The set of approximately efficient elements (where the distance between it and the set of efficient elements is unbounded). . . . . . 82 3.1.2 The set of approximately efficient elements with a bigger cone C. 83 4.2.1 Application of Algorithm II in a C++ computer program. . . . . . . 260 4.2.2 Solutions x10 , x20 , and x30 of the location problem generated by the proximal-point algorithm choosing different weights αi (i = 1, . . . , n). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 4.3.3 The set of efficient elements of the multicriteria location problem (P) with the maximum norm. . . . . . . . . . . . . . . . . . . . . . . 287 4.3.4 The set of efficient points of the multicriteria location problem (P) with the Lebesgue norm instead of the maximum norm. . . . 288 4.3.5 Comparison of the alternatives x1 , x2 , x3 with respect to the criteria C1 , C2 , C3 , C4 , C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 4.3.6 The set of efficient points of the multicriteria location problem (PL ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Leb 4.3.7 The set of efficient points XEff of (PL ) with the Lebesgue norm instead of the maximum norm. . . . . . . . . . . . . . . . . . . . . . . . . 292 4.3.8 Radar chart for the alternatives x1 , x2 , and x3 . . . . . . . . . . . . . . . 293

1 Examples

1.1 Cones in Vector Spaces Vector optimization in partially ordered spaces requires, among other things, that one studies properties of cones; as arguments recall: • • • • • •

Conic approximation of sets, e.g., in order to prove necessary optimality conditions; think of tangent cones, generated cones . . . . Cone-valued mappings; think of the subdifferential of indicator functions, which coincide with the normal cone, used in order to state optimality conditions for optimization problems with nonfunctional constraints. Order relations in linear spaces induced by cones; think of Pareto efficiency and the consideration of smaller or larger cones in order to vary the set of efficient points. Cones in order to represent inequalities; think of side restrictions or vector variational inequalities. Cones in connection with some theoretical procedures; think of Phelps’s cone related to Ekeland’s variational principle in normed spaces, minimal points with respect to cone orderings. Dual cones or polar cones; think of dual assertions and the bipolar theorem.

A cone in a vector space is an algebraic notion, but its use in theory and applications of optimization demands that one considers cones in topological vector spaces and studies, besides algebraic properties such as pointedness and convexity, also analytical ones such as those of being closed, normal, Daniell, nuclear, or having a base with special properties. A short overview of such qualities can be found at the end of this section; the corresponding proofs are given by hints to references or to later sections. The examples of cones given below show interesting results on cones in infinite-dimensional vector spaces, which are important in vector optimization and control theory. Example 1.1.1. Let X be a normed vector space with dim X = ∞, and let x
: X → R be a linear but not continuous function. Then the cone

2

1 Examples

C = {0} ∪ {x | x
(x) > 0} has a base B = {x ∈ C | x
(x) = 1}, but 0 ∈ cl B, since B is dense in C. For bases see Section 2.2. Example 1.1.2. Let X be a locally convex space and C a convex cone in X. Then C # = ∅ ⇒ cl C ∩ (−C) = {0} ⇒ C is pointed, where


 C # := x∗ ∈ X ∗ | x∗ (x) > 0 ∀ x ∈ C \ {0} .

The first implication becomes an equivalence if dim X < ∞.  The converse of the second implication is not true, even if X = R2 ; C = (0, ∞) × R) ∪ ({0} × [0, ∞) is a counterexample. To prove the first implication consider u∗ ∈ C # . Assume that there exists c ∈ cl C ∩ (−C), c = 0. Then there exists a sequence (cn )n∈N ⊂ C such that (cn ) → c. Because u∗ (cn ) ≥ 0, we have that u∗ (c) = lim u∗ (cn ) ≥ 0, which contradicts the fact that u∗ ∈ C # and c ∈ −C \ {0}. Assume dim X < ∞; let us prove the converse of the first implication. Let C = {0} (if C = {0} then C # = X ∗ ) and suppose that cl C ∩ (−C) = {0}. Then 0∈ / raint C + ⊂ C # . (1.1) Indeed, if 0 ∈ raint C + , then C + is a linear subspace, and consequently = cl C is a linear subspace. This implies that cl C ∩ (−C) = −C = {0}, C a contradiction. Now let x∗ ∈ raint C + . Assume that x∗ ∈ / C # ; then there ∗ ∗ + exists x ∈ C \ {0} such that x (x) = 0. Let x ∈ C . Then there is λ > 0 such that (1 + λ)x∗ − λx∗ ∈ C + . So ((1 + λ)x∗ − λx∗ )(x) = −λx∗ (x) ≥ 0, whence x∗ (x) ≤ 0; it follows that −x ∈ C ++ = cl C. It follows that cl C ∩ (−C) = {0}, a contradiction. Taking into account dim X < ∞, (1.1) gives C # = ∅, since raint C + is nonempty (recall that every nonempty finite-dimensional convex set has a nonempty relative algebraic interior; see [168, p. 9]). ++

Example 1.1.3. The convex cone C in a Banach space X has the angle property if for some ε ∈ (0, 1] and x∗ ∈ X ∗ \ {0} we have C ⊂ {x ∈ X | x∗ (x) ≥ ε x∗ · x }. It follows that x∗ ∈ C # . Since for X = Rn = X ∗ the last inequality means cos(x∗ , x) ≥ ε, it is clear where the name “angle property” comes from. The class of convex cones with the angle property is very large (for all ε ∈ (0, 1) and x∗ ∈ X ∗ \ {0} the set {x ∈ X | x∗ (x) ≥ ε x∗ · x } is a closed convex cone with the angle property and nonempty interior). In fact, in normed spaces, a convex cone has the angle property iff it is well-based. So, the cone Rn+ in Rn has the angle property (with x∗ = (1, 1, . . . , 1) ∈ Rn and ε = 2−1/2 ), but the ordinary order cone + 2 ⊂ 2 does not have it. ∗ Indeed, if for some x∗ ∈ 2 \ {0} and some ε > 0, + 2 ⊂ {x ∈ 2 | x (x) ≥ + ∗ ∗ ∗ ε x · x }, then xn ≥ ε x (because en = (0, . . . , 1, . . .) ∈ 2 ), whence the contradiction ε x∗ ≤ 0 (since (x∗n ) → 0 for n → ∞).

1.2 Equilibrium problems

3

Overview of Several Properties of Cones Let X be a Banach space, C and K proper (i.e., {0} = C, K = X) convex cones in X, K + the continuous dual cone of K, and K # := {x∗ ∈ K + | x∗ (x) > 0 ∀ x ∈ K \ {0}} the quasi-interior of K + . Then the relations below hold. For more relationships among different kinds of cones and spaces look at Section 3.2; for cones with base see Section 2.2. K has compact base =⇒


K=K 
X = Rn K has angle property

⇐⇒

∃ C : K \ {0} ⊂ int C

⇐⇒

K is well-based





K well-based

 K is normal

K=K, X=Rn

⇐⇒

K is based ⇐⇒ 
K=K
X separable

K is Daniell

int K + = ∅

K # = ∅

⇐=

K pointed 



⇐= K + − K + = X ∗ 


⇐⇒

K is normal

=⇒ K is w-normal.

1.2 Equilibrium Problems Let us consider a common scalar optimization problem ϕ(x) → min s.t. x ∈ B,

(1.2)

where B is a given nonempty set in a space X and ϕ : B → R a given function. Let x ∈ B be a solution of (1.2); that is, ϕ(x) ≤ ϕ(y) ∀ y ∈ B.

(1.3)

Then, setting f (x, y) := ϕ(x) − ϕ(y) for x, y ∈ B, x solves also the problem find x ∈ B such that f (x, y) ≤ 0 ∀ y ∈ B.

(1.4)

For given B and f : B × B → R, a problem of the kind (1.4) is called an equilibrium problem and B its feasible set. A large number of quite different problems can be subsumed under the class of equilibrium problems as, e.g.,

4

1 Examples

saddle point problems, Nash equilibria in noncooperative games, complementarity problems, variational inequalities, and fixed point problems. To sketch the last one, consider X a Hilbert space and T : B → B a given mapping. With f (x, y) := (x − T x | x − y) we have that x ∈ B is a fixed point of T (i.e., T x = x) if and only if x satisfies f (x, y) ≤ 0 ∀ y ∈ B. Indeed, if x is an equilibrium point, then taking y := T x, we have 0 ≥ (x − T x | x − T x) = x − T x 2 , and so x = T x as claimed. The other direction of the assertion is obvious. There are powerful results that ensure the existence of a solution of the equilibrium problem (1.4); one of the most famous is Fan’s theorem; see [111]: Theorem 1.2.1. Let B be a compact convex subset of the Hausdorff locally convex space (H.l.c.s.) X and let f : B × B → R be a function satisfying ∀ y ∈ B : x → f (x, y) is lower semicontinuous, ∀ x ∈ B : y → f (x, y) is concave, ∀ y ∈ B : f (y, y) ≤ 0. Then there is x ∈ B with f (x, y) ≤ 0 for every y ∈ B. Since we mainly deal with multicriteria problems, we now take B ⊂ X as above and ϕ : B → Y , where Y is the Euclidean space Rn (n > 1) or any locally convex space (l.c.s.). In Y we consider a convex pointed cone C, for instance C = Rn+ if Y = Rn . Then, as usual and in accordance with (1.3), x ∈ B solves the multicriteria problem ϕ(x) → min, x ∈ B, if ∀ y ∈ B : ϕ(y) ∈ / ϕ(x) − (C \ {0}).

(1.5)

Sometimes, if the cone C has a nonempty interior (e.g., C = Rn+ when X = Rn ), one looks for so-called weak solutions of multicriteria problems replacing (1.5) by ∀ y ∈ B : ϕ(y) ∈ / ϕ(x) − int C. (1.6) As above, we introduce f (x, y) := ϕ(x) − ϕ(y) for x, y ∈ B, and so in accordance with (1.4) we get the following vector equilibrium problem: find x ∈ B (sometimes called vector equilibrium) with the property ∀ y ∈ B : f (x, y) ∈ / C \ {0},

(1.7)

or, considering weak solutions of multicriteria problems, find x ∈ B (sometimes called weak vector equilibrium) with the property ∀ y ∈ B : f (x, y) ∈ / int C.

(1.8)

Looking for existence theorems for solutions of (1.7) we mention Sections 3.8, 3.9, 4.2.4, and 4.2.5, which are devoted to vector-valued equilibrium problems.

1.2 Equilibrium problems

5

Vector equilibria play an important role in mathematical economics, e.g., if one deals with traffic control. As an example we describe an extension of Wardrop’s principle for weak traffic equilibria to the case in which the route flows have to satisfy the travel demands between origin–destination pairs and route capacity restrictions, and the travel cost function is a mapping. Let us consider a traffic (or transportation) network (N, L, P ), where N denotes a finite set of nodes, L a finite set of links, P ⊆ N × N a set of origin–destination pairs, and |P | the cardinality of P . We denote by • d ∈ R|P | the travel demand vector with dp > 0 for all p ∈ P ; • R a finite set of routes (paths) between the origin–destination pairs such that for each p ∈ P , the set R(p) of routes in R connecting p is nonempty; • Q the pair–route incidence matrix (Qpr = 1 if r ∈ R(p), Qpr = 0 otherwise); i.e., Q is a matrix of type (|P |, |R|). We suppose that the demand vector is given on R|P | . Then we introduce the set K of traffic flow vectors (path flow vectors), where K := {v ∈ R|R| | 0 ≤ vr ∀ r ∈ R, Qv = d}. Note that,  in more detailed form, the condition Qv = d states that for all p ∈ P , r∈R(p) vr = dp . The set K is the feasible set for the desired vector equilibrium problem. In order to formulate the vector equilibrium problem, a valuation of the traffic flow vectors is necessary. Therefore, let Y be a topological vector space partially ordered by a suitable convex, pointed cone C with int C = ∅ and F a travel cost function, which assigns to each route flow vector v ∈ R|R| a vector of marginal travel costs F (v) ∈ L(R|R| , Y ), where L(R|R| , Y ) is the set of all linear operators from R|R| into Y . The weak traffic equilibrium problem we describe consists in finding u ∈ K such that (in accordance to (1.4)) φ(u, v) := −F (u)(v − u) ∈ / int C ∀ v ∈ K,

(1.9)

recalling that in the scalar case Y = R1 , the product F (u)v is a price for v. Any route flow u ∈ K that satisfies (1.9) is called a weak equilibrium flow. It has been shown that u is a weak equilibrium flow if and only if u satisfies a generalized vector form of Wardrop’s principle, namely, for all origin–destination p ∈ P and all routes s, t ∈ R connecting p, i.e., s, t ∈ R(p), F (u)s − F (u)t ∈ − int C ⇒ us = 0.

(1.10)

To explain F (u)s consider u ∈ R|R| ; then u = (u1 , . . . , us , . . . , u|R| ), and for v = (0, . . . , 0, us , 0, . . . 0) it is F (u)s = F (u)v. Condition (1.10), due to its decomposed (user-oriented) form, is often more practical than the original definition (1.9), since it deals with pairs p and paths s, t ∈ R(p) directly: When the traffic flow is in vector equilibrium, users choose only Pareto-optimal or vector minimum paths to travel on. In the case

6

1 Examples

Y = Rm , C = Rm + it is easily seen that (1.10) follows from (1.9). Let u ∈ K satisfy (1.9); take p ∈ P and s, t ∈ R(p). Choose a flow v such that ⎧ ⎪ if λ = t, λ = s, ⎨uλ vλ = 0 if λ = s, (λ = 1, 2, . . . , |R|). ⎪ ⎩ ut + us if λ = t,   Then v λ ∈ K, since 1≤λ≤|R| v λ = 1≤λ≤|R| uλ = dp . Consequently, from (1.9), writing F (u) as a matrix (Fµλ ) ∈ Rm×|R| , we have ∀ µ = 1, . . . , m,

Fµλ (v λ − uλ ) = Fµt (us ) + Fµs (−us ) = (Fµt − Fµs )(us ) ≮ 0. 1≤λ≤|R|

But from the first condition in (1.10) it is Fµs − Fµt < 0 ∀ µ, so us = 0. For further references see [86, 88, 134, 226, 254, 370, 382]. For other applications of multicriteria decision-making to economic problems see [15].

1.3 Location Problems in Town Planning Urban development is connected with conflicting requirements of areas for dwelling, traffic, disposal of waste, recovery, trade, and others. Using methods of location theory may be one way of supporting urban planning to determine the best location for a special new layout or for arrangements. For references see e.g. [102, 101, 152, 156, 171, 173, 174, 170, 219, 305, 368, 369, 371]. In our investigations we consider the special situation in East German towns. One of the actual main problems of town planning is the traffic problem, due to the extremely high increase of motorized individual traffic in recent years. The lack of parking space is a part of the traffic problem. This is typical for many newly built residential areas in East Germany. Such a residential area is Halle-Silberh¨ ohe, which was built at the beginning of the 1980s. In this district 5, 6, and 11-story blocks dominate. In our example we consider two residential sections, which count about 9300 inhabitants. This area has a size of 800 m × 1000 m. There exist 1750 parking facilities, representing a shortage of 1950. The impact is that many inhabitants park their cars on green areas. One way to solve this problem of inadequate parking facilities is to build multistory parking garages. Now, the problem is to find the best location for such a multistory garages. It would be possible to formulate our problem as a real-valued location problem (Fermat–Weber problem), which is the problem to determine a location x of a new facility such that the weighted sum of distances between n given facilities ai (i = 1, . . . , n) and x is minimal. Using this approach it is very difficult to say how the weights λi (i = 1, . . . , n) are to be chosen. Another difficulty may arise if the solution of the corresponding optimal location is not

1.3 Location problems in town planning

7

practically useful. Then we need new weights, and again we don’t know how to choose the weights. So the following approach is of interest: We formulate the problem as a vector-valued (or synonymously vector or multicriteria) location problem ⎞ ⎛

x − a1 max ⎜ x − a2 max ⎟ ⎟ −→ v − min , ⎜ (P ) ⎠ ⎝ ··· x∈R2

x − an max where x, ai ∈ R2 (i = 1, . . . , n),

x max = max{| x1 |, | x2 |}, and “v − minx∈R2 ” means that we study the problem of determining the set of efficient points of an objective function f : X −→ Rn with respect to a cone C ⊂ Rn : Eff(f [X], C) := {f (x) | x ∈ X,

f [X] ∩ (f (x) − (C \ {0})) = ∅}.

Remark 1.3.1. For applications in town planning it is important that we can choose different norms in the formulation of (P ). The decision which of the norms will be used depends on the course of the roads in the city or in the district or on other influences coming from the practical background of the planning problem. In the following example we study the problem (P ) with C = Rn+ , where denotes the usual ordering cone in n-dimensional Euclidean space. In Section 4.3 we consider a location problem in town planning, formulate a multicriteria location problem, derive optimality conditions, and present several algorithms for solving multicriteria location problems and the corresponding computer programs. It is well known that the set of solutions in vector optimization (set of efficient elements) may be large, and so we will carry out a comparison of alternatives by using a graphical representation. For the problem in town planning mentioned above we fix the given points a1 = (−1.5, 3.5), a2 = (1, 3), a3 = (1, 0), a4 = (−3, −2), a5 = (3.5, −1.5), a6 = (2, 2), a7 = (−2, 2), a8 = (4, 1), a9 = (−3, 2). If the decision-maker prefers the maximum norm, we get the solution set of the multicriteria location problem (P ) as shown in Figure 1.3.1. Rn+

But if the decision-maker prefers the dual norm to the maximum norm (this norm is called the Lebesgue norm), the solution set has the form shown in Figure 1.3.2

8

1 Examples

x @

x @

@

x @ @

@ @ @

x @

@

@ @

@

@x @ @

x

@

@

@ @

@ @x @

@

@ @

@ @x

x Figure 1.3.1. The set of efficient elements of the multicriteria location problem (P) with the maximum norm

x x x

x

x x x

x x Figure 1.3.2. The set of efficient points of the multicriteria location problem (P) with the Lebesgue norm instead of the maximum norm

1.4 Multicriteria Control Problems In control theory often one has the problem to minimize more than one objective function, for instance, a cost functional as well as the distance between the final state and a given point. To realize this task usually one takes as objective function a weighted sum of the different objectives. However, the more natural way would be to study

1.4 Multicriteria control problems

9

the set of efficient points of a vector optimization problem with the given objective functions. It is well known that the weighted sum is only a special surrogate problem to find efficient points, which has the disadvantage that in the nonconvex case one cannot find all efficient elements in this way. In order to formulate a multicriteria control problem we introduce a system of differential equations that describe the motion of the controlled system. Let x(t) be an n-dimensional phase vector that characterizes the state of the controlled system at time t. Furthermore, let u(t) be an m-dimensional vector that characterizes the controlling action realized at time t. In particular, consider in a time interval 0 ≤ t ≤ T the system of ordinary differential equations with initial condition ⎫ dx (t) = ϕ(t, x(t), u(t)), ⎬ dt (1.11) ⎭ x(0) = x ∈ Rn . 0

It is assumed that the controlling actions u(t) satisfy the restrictions u(t)

∈ U ⊂ Rm .

The vector x is often called the state, and u is called the control of (1.11); a pair (x, u) satisfying (1.11) and the control restriction is called an admissible solution (or process) of (1.11). Under additional assumptions it is possible to ensure the existence of a solution (x, u) of (1.11) on the whole time interval [0, T ] or at least almost everywhere on [0, T ] (compare Section 4.5). Introducing the criteria or performance space, the objective function f : (X × U ) −→ (Y, CY ), where Y is a linear space, and CY ⊂ Y is a proper convex cone, we have as multicriteria optimal control problem (also called multiobjective or vector-valued or vector optimal control problem) (P): Find some control u ¯ such that the corresponding trajectory x ¯ satisfies f (x, u) ∈ / f (¯ x, u ¯) − (CY \ {0}) for all solution pairs (x, u) of (1.11). The pair (¯ x, u ¯) is then called an optimal process of (P ). If we study the problem to minimize the distance f1 of the final state x(T ) of the system (1.11) and a given point as well as a cost functional T Φ(t, x(t), u(t))dt by a control u, we have a special case of (P ) with 0   f1 (x(T )) f (x, u) =  T . Φ(t, x(t), u(t))dt 0 In the following we explain that cooperative differential games are special cases of the multicriteria control problem (P ). We consider a game with n ≥ 2 players and define

10

1 Examples

Y := Y1 × Y2 × · · · × Yn , the product of the criteria spaces of each of the n players, CY := CY1 × CY2 × · · · × CYn , the product of n proper convex cones, f (x, u) := (f1 (x, u), . . . , fn (x, u)), the vector of the loss functions of the n players, U := U1 × U2 × · · · × Un , the product of the control sets of the n players, and define u := (u1 , . . . , un ). The player j tries to minimize his cost function fj (the utility or profit function is −fj ) with respect to the partial order induced by the cone CYj influencing the system (1.11) by means of the function uj ∈ Uj . And “cooperative game” means that a state is considered optimal if no player can reduce his costs without increasing the costs of another player. Such a situation is possible because each cost function fj depends on the same control tuple u. The optimal process gives the Pareto minimum of the cost function f (x, u). It is well known that it is difficult to show the existence of optimal (or efficient) controls of (P ), whereas suboptimal controls exist under very weak conditions. So it is important to derive some assertions for suboptimal controls. This is done in Section 4.5. There an application of a variational principle for vector optimization problems yields an ε-minimum principle for (P), which is (for ε = 0) closely related to Pontryagin’s minimum principle (cf. Section 4.5).

1.5 Multicriteria Fractional Programming Problems Many problems in economics can be formulated as fractional programming problems. In the papers of Hirche ([166]), Schaible ([315]), Schaible and Ibaraki ([316]) the following real-valued fractional programming problem is considered: aT x ϕ(x) := aT1 x + T2 −→ max, x∈A b2 x where x ∈ Rn , a1 , a2 , b2 ∈ Rn+ , aT1 x, aT2 x, . . . are scalar products, bT2 x = 0, and A = {x ∈ Rn | Bx − b ∈ −K}. The cone K is a convex cone in Rm . The objective function aT1 x describes the production scope, and viability.

aT 2 x bT 2 x

describes the

1.6 Stochastic efficiency in a set

11

From the practical as well as from the mathematical point of view it is more useful to formulate the problem as a multicriteria fractional programming problem:   T a1 x aT2 x , T −→ v − max . f (x) = x∈A 1 b2 x Applying the Dinkelbach transformation, one gets a useful surrogate parametric optimization problem and the corresponding algorithms:   fλ (x) = aT1 x − λ1 , aT2 x − λ2 bT2 x −→ v − max, x∈A

where λ ∈ R2 . In Section 4.4 we discuss possibilities to handle this transformed vector optimization problem by means of parametric optimization. We then derive a three-level dialogue algorithm in order to solve the transformed problem.

1.6 Stochastic Efficiency in a Set Uncertainty is the key ingredient in many decision problems. Financial planning, cancer screening, and airline scheduling are just examples of areas in which ignoring uncertainty may lead to inferior or simply wrong decisions. There are many ways to model uncertainty; one that has proved particularly fruitful is to use probabilistic models. Two methods are frequently used for modeling choice among uncertainty prospects: stochastic dominance (Ogryczak and Ruszczynski [282], [283], Whitmore and Findlay ([377]); Levy ([236]) and mean-risk analysis (Markowitz ([251]). The stochastic dominance is based on an axiomatic model of risk-averse preferences: It leads to conclusions that are consistent with the axioms. Unfortunately, the stochastic dominance approach does not provide us with a simple computational recipe; it is, in fact, a multiple criteria model with a continuum of criteria. The mean-risk approach quantifies the problem in a lucid form of only two criteria: • •

The mean, representing the expected outcome; The risk, a scalar measure of the variability of outcomes.

The mean-risk model is appealing to decision-makers and allows a simple trade-off analysis, analytical or geometrical. On the other hand, mean-risk approaches are not capable of modeling the gamut of risk-averse preferences. Moreover, for typical dispersion statistics used as risk measures, the mean-risk approach may lead to inferior conclusions (compare Section 4.6).

12

1 Examples

The seminal portfolio optimization model of Markowitz ([249]) uses the variance as the risk measure in the mean-risk analysis, which results in a formulation of a quadratic programming model. Since then, many authors have pointed out that the mean-variance model is, in general, not consistent with stochastic dominance rules. The use of the semivariance rather than variance as the risk measure was already suggested by Markowitz ([250]) himself.

2 Functional Analysis over Cones

2.1 Order Structures As seen in the introduction, we are concerned with certain sets M with order structures. In the sequel we give the basic definitions. As usual, when M is a nonempty set, M × M is the set of ordered pairs of elements of M : M × M := {(x1 , x2 ) | x1 , x2 ∈ M }. Definition 2.1.1. Let M be a nonempty set and R a nonempty subset of M × M . Then R is called a binary relation or an order structure on M , and (M, R) is called a set M with order structure R. The fact that (x1 , x2 ) ∈ R will be denoted by x1 Rx2 . We say that R is (a) reflexive if ∀ x ∈ M : xRx, (b) transitive if ∀ x1 , x2 , x3 ∈ M : x1 Rx2 , x2 Rx3 ⇒ x1 Rx3 , (c) antisymmetric if ∀ x1 , x2 ∈ M : x1 Rx2 , x2 Rx1 ⇒ x1 = x2 . R is called a preorder on M if R satisfies (b) and a partial order on M if R satisfies (a), (b), and (c). In both cases the fact that (x1 , x2 ) ∈ R is denoted by x1 ≤R x2 , or simply x1 ≤ x2 if there is no risk of confusion. The binary relation R is called a linear or total order if R is a partial order and (d) for all x1 , x2 ∈ M either x1 ≤R x2 or x2 ≤R x1 , i.e., any two elements of M are comparable. Finally, if each nonempty subset M
of M has a first element x
(meaning that x
∈ M
and x
≤R x ∀ x ∈ M
), then M is called well-ordered.1 1

Recall Zermelo’s theorem: For every nonempty set M there exists a partial order R on M such that (M, R) is well-ordered.

14

2 Functional analysis over cones

An important example of a relation is ∆M := {(x, x) | x ∈ M }. It is obvious that ∆M is reflexive, transitive, and antisymmetric, but (d) is satisfied only when M is a singleton. Recall that the inverse of the relation R ⊂ M × M is the relation R−1 := {(x1 , x2 ) ∈ M × M | (x2 , x1 ) ∈ R}; if S is another relation on M , the composition of R and S is the relation S ◦ R := {(x1 , x3 ) | ∃ x2 ∈ M : (x1 , x2 ) ∈ R, (x2 , x3 ) ∈ S}. Using this notation, the conditions (a), (b), (c), and (d) are equivalent to ∆M ⊂ R, R ◦ R ⊂ R, R ∩ R−1 ⊂ ∆M and R ∪ R−1 = M × M , respectively. Definition 2.1.2. Let R be an order structure on the nonempty set M and let M0 ⊂ M be nonempty. The element x0 ∈ M0 is called a maximal (minimal) element of M0 relative to R if ∀ x ∈ M0 : x0 Rx ⇒ xRx0

(∀ x ∈ M0 : xRx0 ⇒ x0 Rx).

(2.1)

The class of maximal (minimal) elements of M0 with respect to (w.r.t. for short) R is denoted by Max(M0 , R) (Min(M0 , R)). Note that x0 is a maximal element of M0 w.r.t. R iff x0 is a minimal element of M0 w.r.t. R−1 , and so Max(M0 , R) = Min(M0 , R−1 ). Remark 2.1.3. If the order structure R in Definition 2.1.2 is antisymmetric, then x0 ∈ M0 is maximal (minimal) iff ∀ x ∈ M0 : x0 Rx ⇒ x = x0

(∀ x ∈ M0 : xRx0 ⇒ x0 = x).

(2.2)

Partial orders R play an important role for introducing a solution concept in multicriteria optimization (compare Chapter 3 and 4). In these chapters we denote, as usual, sets of maximal or minimal elements by Eff Max (M0 , R) and Eff Min (M0 , R), respectively. Furthermore, these sets are called sets of (maximal- or minimal-) efficient points of M0 with respect to R. For brevity we sometimes leave out maximal, minimal, Max or Min. Remark 2.1.4. When R is an order structure on M and ∅ = M0 ⊂ M , then R0 := R ∩ (M0 × M0 ) is an order structure on M0 ; in such a situation the set M0 will always be endowed with the order structure R0 if not stated explicitly otherwise. If R is a preorder (partial order, linear order) on M , then R0 is a preorder (partial order, linear order) on M0 . So x0 is a maximal (minimal) element of M0 relative to R iff x0 is a maximal (minimal) element of M0 relative to R0 .

2.1 Order structures

15

Example 2.1.5. (1) Let X be a nonempty set and let M := P(X) be the class of subsets of X. The binary relation R := {(A, B) ∈ M × M | A ⊂ B} is a partial order on M ; if X has at least two elements, then R is not a linear order. (2) Let N be the set of nonnegative integers and RN := {(n1 , n2 ) ∈ N × N | ∃ p ∈ N : n2 = n1 + p}. Then N is well-ordered by RN . Of course, RN defines the usual order relation on N, and n1 RN n2 will always be denoted by n1 ≤ n2 or, equivalently, n2 ≥ n1 . (3) Let R be the set of real numbers and let R+ := [0, ∞[ be the set of nonnegative real numbers. The usual order structure on R is defined by R1 := {(x1 , x2 ) ∈ R × R | ∃ y ∈ R+ : x2 = x1 + y}. Then R1 is a linear order on R, but R is not well-ordered by R1 . As usual, the fact x1 R1 x2 will always be denoted by x1 ≤ x2 or, equivalently, x2 ≥ x1 . (4) Let n ∈ N, n ≥ 2. Consider the binary relation Rn on Rn defined by Rn := {(x, y) ∈ Rn × Rn | ∀ i ∈ 1, n : xi ≤ yi }, where x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) and 1, n := {i ∈ N | 1 ≤ i ≤ n}. Then Rn is a partial order on Rn , but Rn is not a linear order; for example, the elements e1 and e2 are not comparable w.r.t. Rn , where ei := (0, . . . , 0, 1, 0, . . . , 0) ∈ Rn (all components of ei being 0 excepting the ith one, which is 1). Remark 2.1.6. Notice that every well-ordered subset W of R (endowed with its usual partial order recalled above) is at most countable. Indeed, every element y ∈ W , excepting the greatest element w of W (if it exists), has a successor s(y) ∈ W ; of course, if y, y
∈ W , y < y
, then s(y) ≤ y
. So, fixing qy ∈ Q such that y < qy < s(y) for y ∈ W \ {w}, we get an injective function from W \ {w} into Q, and so W is at most countable. Even when R is a partial order on M , a nonempty subset M0 of M may have zero, one, or several maximal elements, but if R is a linear order, then every subset has at most one maximal (minimal) element. Definition 2.1.7. Let ∅ = M0 ⊂ M and let R be an order structure on M . We say that M0 is lower (upper) bounded (w.r.t. R) if there exists a ∈ M such that aRx (xRa) for every x ∈ M0 ; a is called a lower (upper) bound of M0 (w.r.t. R). If, moreover, R is a partial order, we say that a ∈ M is the infimum (supremum) of M0 if a is a lower (upper) bound of M0 and for any lower (upper) bound a
of M0 we have that a
Ra (aRa
).

16

2 Functional analysis over cones

An important problem in vector optimization is the existence of maximal elements w.r.t. order structures. Of fundamental importance in this sense is Zorn’s lemma (or Zorn’s axiom). Axiom 2.1.8 (Zorn) Let (M, ≤) be a reflexively preordered set. If every nonempty totally ordered subset of M is upper bounded, then M has maximal elements. Very often, order structures are defined on subsets of linear spaces. We will consider only real linear spaces, so later on we shall omit the word real. Definition 2.1.9. Let X be a nonempty set. We say that X is a (real) linear space or a set equipped with linear structure if an addition (that is, a mapping + : X × X → X) and a multiplication by scalars (that is, a mapping · : R × X → X) are defined satisfying the following conditions: (a) ∀ x, y, z ∈ X : (x + y) + z = x + (y + z) (associativity), (b) ∀ x, y ∈ X : x + y = y + x (commutativity), (c) ∃ 0 ∈ X, ∀ x ∈ X : x + 0 = x (null element), (d) ∀ x ∈ X, ∃ x
∈ X : x + x
= 0; we write x
= −x, (e) ∀ x, y ∈ X, ∀ λ ∈ R : λ(x + y) = λx + λy, (f) ∀ x ∈ X, ∀ λ, µ ∈ R : (λ + µ)x = λx + µx, (g) ∀ x ∈ X, ∀ λ, µ ∈ R : λ(µx) = (λµ)x, (h) ∀ x ∈ X : 1x = x (unity element). A linear space is often called a vector space. Defining x + y := (x1 + y1 , . . . , xn + yn ) and λx := (λx1 , . . . , λxn ) for x, y ∈ Rn and λ ∈ R, we have that Rn is a linear space. In the sequel we shall use the following notation: Let A, B be nonempty subsets of a linear space X, ∅ = Γ ⊂ R, x ∈ X, and λ ∈ R; then A + B := {x + y | x ∈ A, y ∈ B}, A + ∅ := ∅ + A := ∅, x + A := {x} + A, and Γ · A := {γx | γ ∈ Γ, x ∈ A}, λA = {λ} · A, Γ · ∅ = ∅. In particular, A − B = A + (−1)B = {x − y | x ∈ A, y ∈ B}. If, moreover, I is a nonempty set and ∅ = Ai ⊂ X for every i ∈ I, we have         λ(A + B) = λA + λB, λ Ai = λAi , λ Ai = λAi . i∈I

i∈I

i∈I

i∈I

Let X be a linear space and ∅ = X0 ⊂ X; recall that X0 is a linear subspace of X if λx + µy ∈ X0 for all x, y ∈ X0 and λ, µ ∈ R. This shows that X0 is stable w.r.t. addition and multiplication by scalars, and X0 endowed with these operations is itself a linear space. Also recall that the nonempty set M ⊂ X is affine (or a linear manifold) if λx + (1 − λ)y ∈ M for all

2.1 Order structures

17

x, y ∈ M and λ ∈ R; it is known that M is affine iff there exist x0 ∈ X and X0 a linear subspace of X such that M = x0 + X0 (X0 being called the parallel linear subspace to M ). The nonempty set C of X is called convex if [x, y] := {λx + (1 − λ)y | λ ∈ [0, 1]} ⊂ C for all x, y ∈ C. By convention the empty set ∅ is considered to be affine and convex. It is obvious that a linear subspace is affine and an affine set is convex; moreover, any intersection of linear subspaces, affine sets, or convex sets is a linear subspace, an affine set, or a convex set, respectively. These properties give us the possibility to introduce the linear hull, the affine hull, and the convex hull of a nonempty set A ⊂ X as being, respectively,  lin A := {Y ⊂ X | A ⊂ Y, Y linear subspace} ,  aff A := {M ⊂ X | A ⊂ M, M linear manifold} ,  conv A := {C ⊂ X | A ⊂ C, C convex set} . It is obvious (and well known) that for X = Rn and R = Rn (from Example 2.1.5 (4)) one has ∀ x1 , x2 ∈ X, ∀ λ ∈ R : x1 Rx2 , 0 ≤ λ ⇒ λx1 Rλx2 , ∀ x1 , x2 , x ∈ X : x1 Rx2 ⇒ (x1 + x)R(x2 + x).

(2.3) (2.4)

It is simple to give examples of relations satisfying (2.4). In fact, a nonempty relation R on the linear space X satisfies (2.4) if and only if there exists ∅ = D ⊂ X such that R = RD , where RD := {(x1 , x2 ) ∈ X × X | x2 − x1 ∈ D}. Moreover, RD is reflexive if and only if 0 ∈ D, and RD is transitive if and only if D + D ⊂ D. Definition 2.1.10. Let R be an order structure on the linear space X; we say that R is compatible with the linear structure of X if (2.3) and (2.4) hold. Fortunately, there exists a large class of relations R on a linear space X that are compatible with the linear structure of X: This is the class of relations determined by cones of X. Definition 2.1.11. A nonempty set C ⊂ X is a cone if (a) ∀ x ∈ C, ∀ λ ∈ R+ : λx ∈ C. Of course, if C is a cone, then 0 ∈ C. The cone C is called: (b) convex if ∀ x1 , x2 ∈ C : x1 + x2 ∈ C, (c) nontrivial or proper if C = {0} and C = X, (d) reproducing if C − C = X, (e) pointed if C ∩ (−C) = {0}.

18

2 Functional analysis over cones

Note that the cone C satisfies condition (b) in the definition above if and only if C is a convex set. Example 2.1.12. (1) Let Rn+ := {x ∈ Rn | xi ≥ 0 ∀ i ∈ 1, n} = {x ∈ Rn | (0, x) ∈ Rn }.

(2.5)

Rn+ is obviously a cone in the linear space Rn , which fulfills all the conditions of Definition 2.1.11. (2) Let C[0, 1] be the linear space of all real functions defined and continuous on the interval [0, 1] ⊂ R; addition and multiplication by scalars are defined, naturally, by (x + y)(t) = x(t) + x(t),

(λx)(t) = λx(t)

∀ t ∈ [0, 1]

for x, y ∈ C[0, 1] and λ ∈ R. Then C+ [0, 1] := {x ∈ C[0, 1] | x(t) ≥ 0 ∀ t ∈ [0, 1]}

(2.6)

is a convex, nontrivial, pointed, and reproducing cone in C[0, 1]. Note that the set Q := {x ∈ C+ [0, 1] | x is nondecreasing} (2.7) is also a convex, nontrivial, and pointed cone in the space C[0, 1], but it doesn’t satisfy condition (d) from Definition 2.1.11: Q − Q is the proper linear subspace of all functions with bounded variation of C[0, 1]. (3) Consider the set C ⊂ Rn defined by C := {x = (x1 , . . . , xn )T ∈ Rn | x1 > 0, or x1 = 0, x2 > 0, or ... x1 = · · · = xn−1 = 0, xn > 0, or x = 0}; C satisfies all the conditions of Definition 2.1.11. With the help of cones we can characterize compatibility between linear and order structures: Theorem 2.1.13. Let X be a linear space and let C be a cone in X. Then the relation RC := {(x1 , x2 ) ∈ X × X | x2 − x1 ∈ C} (2.8) is reflexive and satisfies (2.3) and (2.4). Moreover, C is convex if and only if RC is transitive, and, respectively, C is pointed if and only if RC is antisymmetric. Conversely, if R is a reflexive relation on X satisfying (2.3) and (2.4), then C := {x ∈ X | 0Rx} is a cone and R = RC .

2.1 Order structures

19

Proof. Let us denote the fact that x1 RC x2 by x1 ≤ x2 . (i) RC is reflexive because 0 ∈ C. (ii) RC satisfies (2.3); indeed, if x1 , x2 ∈ X and λ ∈ R+ then x1 ≤ x2 =⇒ x2 − x1 ∈ C (2.8)

=⇒

Def. 2.1.9(e)

=⇒

Def. 2.1.11(a)

λ(x2 − x1 ) ∈ C

λx2 − λx1 ∈ C =⇒ λx1 ≤ λx2 . (2.8)

(iii) RC satisfies (2.4); indeed, if x1 , x2 , x ∈ X then x1 ≤ x2 =⇒ x2 − x1 ∈ C (2.8)

=⇒

Def. 2.1.9

(x2 + x) − (x1 + x) ∈ C =⇒ x1 + x ≤ x2 + x. (2.8)

(iv) If C is convex, then RC is transitive: x1 ≤ x2 , x2 ≤ x3 =⇒ x2 − x1 ∈ C, x3 − x2 ∈ C (2.8)

=⇒

Def. 2.1.11(b)

x3 − x1 ∈ C

=⇒ x1 ≤ x3 .

(2.8)

Conversely, assume that RC is transitive. Then x1 , x2 ∈ C =⇒ 0 ≤ x1 , x1 ≤ x1 + x2 (2.8)

=⇒

Def. 2.1.1(b)

0 ≤ x1 + x2 =⇒ x1 + x2 ∈ C. (2.8)

(v) If C is pointed, then RC is antisymmetric: x1 ≤ x2 , x2 ≤ x1 =⇒ x2 − x1 ∈ C ∩ (−C) = {0} =⇒ x2 = x1 . (2.8)

Conversely, assume that RC is antisymmetric. Then x ∈ C ∩ (−C) =⇒ 0 ≤ x, x ≤ 0 (2.8)

=⇒

Def. 2.1.1(c)

x = 0.

(vi) Let R be a reflexive compatible order structure in X and consider C := {x ∈ X | 0Rx}.

(2.9)

C is really a cone. Indeed, let λ ∈ R+ and x ∈ C; then 0Rx. Since R fulfills (2.3), we obtain that λ0Rλx, and so λx ∈ C. Moreover, x1 Rx2 ⇐⇒ (x1 − x1 )R(x2 − x1 ) ⇐⇒ x2 − x1 ∈ C ⇐⇒ x1 RC x2 . (2.4)

(2.9)

(2.8)

Therefore R = RC .
The preceding theorem shows that when ∅ = C ⊂ X, the relation RC defined by (2.8) is a reflexive preorder iff C is a convex cone, and RC is a partial order iff C is a pointed convex cone. Also note that RRn+ = Rn (defined in Example 2.1.5 (4)), while the relation RC with C ⊂ Rn defined in Example 2.1.12 (3) is a linear order, called the lexicographic order on Rn .

20

2 Functional analysis over cones

The cone Rn+ ⊂Rn (defined in (2.5)) has an interesting property: Taking n B := {x ∈ Rn+ | i=1 xi = 1}, for every x ∈ Rn+ \ {0} there exist unique elements b ∈ B and λ > 0 such that x = λb. Indeed, just take λ := x1 + · · · +xn (> 0) and b := λ−1 x. Taking into account this example, we introduce the following definition. Definition 2.1.14. Let X be a linear space and C a nontrivial convex cone in X. A nonempty convex subset B of C is called a base for C if each nonzero element x ∈ C has a unique representation of the form x = λb with λ > 0 and b ∈ B. Note that if B is a base of the nontrivial convex cone C, then 0 ∈ / B. Indeed, in the contrary case, taking b ∈ B \ {0} we have that b = 1 · b = 2 · ( 12 b) with b, 12 b ∈ B, contradicting the uniqueness of the representation of b ∈ C \ {0}. Theorem 2.1.15. Let C be a nontrivial convex cone in the linear space X and let B ⊂ X be a convex set. The following assertions are equivalent: (i) B is a base of C; (ii) C = R+ B and 0 ∈ / aff B; (iii) there exists a linear functional ϕ : X → R such that ϕ(x) > 0 for every x ∈ C \ {0} and B = {x ∈ C | ϕ(x) = 1}. Proof. (i) ⇒ (ii) Let B be a base for C; then, by Definition 2.1.14, C = R+ B. Since B is convex, aff B = {µb + (1 − µ)b
| b, b
∈ B, µ ∈ R}. Assume that 0 ∈ aff B, and so 0 = µb + (1 − µ)b
for some b, b
∈ B, µ ∈ R; since 0 ∈ / B, µ ∈ / [0, 1]. Therefore there exist µ0 > 1, b0 , b
0 ∈ B such that µ0 b0 = (µ0 −1)b
0 ∈ C, contradicting the definition of the base. Hence 0 ∈ / aff B. (ii) ⇒ (iii) Assume that C = R+ B and 0 ∈ / aff B. Consider b0 ∈ B and / X0 . Let L0 ⊂ X0 X0 := aff B −b0 . Then X0 is a linear subspace of X and b0 ∈ be a base of X0 . Then L0 ∪ {b0 } is linearly independent; let us complete it to a base L of X. There exists a unique linear function ϕ : X → R such that ϕ(x) = 0 for any x ∈ L \ {b0 } and ϕ(b0 ) = 1. Since aff B = b0 + X0 , we have that ϕ(x) = 1 for every x ∈ aff B, and so B ⊂ {x ∈ C | ϕ(x) = 1}. Conversely, let x ∈ C be such that ϕ(x) = 1. Then x = tb for some t > 0 and b ∈ B. It follows that 1 = ϕ(x) = tϕ(b) = t, and so x ∈ B. (iii) ⇒ (i) Assume that ϕ : X → R is linear, ϕ(x) > 0 for every x ∈ C \{0}, and B = {x ∈ C | ϕ(x) = 1}. Consider x ∈ C \{0} and take t := ϕ(x) > 0 and b := t−1 x; of course, x = tb. Because b ∈ C and ϕ(b) = 1 we have that b ∈ B. Assume that x = t
b
for some t
> 0 and b
∈ B. Then t = ϕ(x) = t
ϕ(b
) = t
, whence b = b
. Hence every nonnull element x of C has a unique representation tb with t > 0 and b ∈ B; i.e., B is a base of C.
Often it is useful to have also a topological structure on the set under consideration. Definition 2.1.16. Let X be a nonempty set. A topology τ on X is a family of subsets of X satisfying the following conditions:

2.1 Order structures

21

(a) every union of sets of τ belongs to τ , (b) every finite intersection of sets of τ belongs to τ , (c) the empty set ∅ and the whole set X belong to τ . The elements of τ are called open sets, and X equipped with τ is called a topological space and is denoted by (X, τ ). The dual notion for open set is that of closed set. So, the subset A ⊂ (X, τ ) is closed if X \ A is open; it follows easily that every intersection of closed sets is closed and every finite union of closed sets is closed. The interior and the closure of the subset A of the topological space (X, τ ) are defined, respectively, by  int A := {D ⊂ X | D ⊂ A, D open},  cl A := A := {F ⊂ X | A ⊂ F, F closed}. It is obvious that int A is open and cl A is closed. Important examples of topological spaces are furnished by metric spaces. Recall that a metric on a nonempty set X is a mapping ρ : X × X → R+ having the properties ∀ x, y ∈ X : ρ(x, y) = 0 ⇔ x = y, ∀ x, y ∈ X : ρ(x, y) = ρ(y, x), ∀ x, y, z ∈ X : ρ(x, z) ≤ ρ(x, y) + ρ(y, z). The set X equipped with the metric ρ is called a metric space and is denoted by (X, ρ). Having the metric space (X, ρ), x ∈ X and r > 0, consider the (closed) ball of center x and radius r : B(x, r) := Bρ (x, r) := {y ∈ X | ρ(x, y) ≤ r}. Define τρ := {∅} ∪ {D ⊂ X | ∀ x ∈ D, ∃ r > 0 : B(x, r) ⊂ D}. It is easy to verify that τρ is a topology on X, and so X becomes a topological space; a metric space (X, ρ) will always be endowed with the topology τρ if not stated explicitly otherwise. Note that B(x, r) is a closed set, while the open ball {y ∈ X | ρ(x, y) < r} is an open set. Taking X = Rn and ρ(x, y) the usual Euclidean distance between x ∈ Rn and y ∈ Rn , ρ is a metric on Rn , and the open sets w.r.t. ρ are the usual open sets of Rn . As in Rn , in an arbitrary topological space (X, τ ) we define other topological notions, that of neighborhood being very useful.

22

2 Functional analysis over cones

Definition 2.1.17. Let (X, τ ) be a topological space and x ∈ X. The subset U of X is a neighborhood of x (relative to τ ) if there exists an open set O ∈ τ such that x ∈ O ⊂ U . The class of all neighborhoods of x will be denoted by Nτ (x), or simply N (x). A subset B(x) of Nτ (x) is called a base of neighborhoods (filter base) of x relative to τ if for every U ∈ Nτ (x) there exists V ∈ B(x) such that V ⊂ U . One says that (X, τ ) is first-countable if every element x ∈ X has an (at most) countable neighborhood base. For the topological space (X, τ ) and a neighborhood base B(x) of x for every x ∈ X, the family {B(x) | x ∈ X} has the following properties: (NB1) ∀ x ∈ X, ∀ U ∈ B(x) : x ∈ U , (NB2) ∀ x ∈ X, ∀ U1 , U2 ∈ B(x), ∃ U3 ∈ B(x) : U3 ⊂ U1 ∩ U2 , (NB3) ∀ x ∈ X, ∀ U ∈ B(x), ∃ V ∈ B(x), ∀ y ∈ V, ∃ W ∈ B(y) : W ⊂ U . Conversely, if for every x ∈ X one has a nonempty family B(x) ⊂ P(X) such that {B(x) | x ∈ X} satisfies the conditions (NB1)–(NB3), then there exists a unique topology τ on X such that B(x) is a neighborhood base for x, for every x ∈ X; moreover, ∀ x ∈ X : Nτ (x) = {U ⊂ X | ∃ V ∈ B(x) : V ⊂ U }. In fact, the topology (X, ρ) of a metric space is constructed in this manner: The family {B0 (x) | x ∈ X} with B0 (x) := {Bρ (x, 1/n) | n ∈ N \ {0}} satisfies the conditions (NB1)–(NB3) above; the topology defined by {B0 (x) | x ∈ X} is nothing else but the topology τρ defined above. Since B0 (x) is countable for every x ∈ X, we obtain that every metric space is first-countable. As seen above (property (NB2)), taking B(x) a neighborhood base of x ∈ (X, τ ) and considering the relation  on B(x) defined by U1  U2 iff U1 ⊂ U2 , then for all U1 , U2 ∈ B(x) there exists U3 ∈ B(x) such that U3  U1 and U3  U2 . This is the prototype of a directed set. So, we say that the nonempty set I is directed by  if  is a partial order on I and ∀ i1 , i2 ∈ I, ∃ i3 ∈ I : i3  i1 and i3  i2 . Note that if (I
, 
) and (I

, 

) are directed sets, then I
× I

is directed by the relation  defined by (i
, i

)  (j
, j

) iff i

j
and i



j

. Another important example of directed set is N endowed with its usual order ≥. A subset J of a directed set (I, ) is called cofinal if for every i ∈ I there exists j ∈ J such that j  i; it follows immediately that (J, ) is directed if J is a cofinal subset of I. The above discussion shows that (B(x), ) is directed if B(x) is a neighborhood base for x ∈ (X, τ ). This consideration is the base of the following definition. Definition 2.1.18. A mapping ϕ : I → X, where (I, ) is a directed set, is called a net or a generalized sequence of X; generally, the net ϕ is denoted by (xi )i∈I , where xi := ϕ(i). When X is equipped with a topology τ we say that the net (xi )i∈I ⊂ X converges to x ∈ X if

2.1 Order structures

23

∀ V ∈ Nτ (x), ∃ iV ∈ I, ∀ i  iV : xi ∈ V. τ

We denote this fact by (xi ) → x or simply (xi ) → x, and x is called a limit of (xi ). Lemma 2.1.19. Let (X, τ ) be a topological space. Then every convergent net of X has a unique limit if and only if ∀ x, y ∈ X, x = y, ∃ U ∈ Nτ (x), V ∈ Nτ (y) : U ∩ V = ∅,

(2.10)

i.e., (X, τ ) is Hausdorff. Proof. Suppose that (2.10) does not hold. Then there exist x, y ∈ X, x = y, such that ∀ U ∈ Nτ (x), ∀ V ∈ Nτ (y), ∃ xU,V ∈ U ∩ V. As seen above, I := Nτ (x) × Nτ (y) is directed, and so (xU,V )(U,V )∈I is a net τ τ in X. It is obvious that (xU,V ) → x and (xU,V ) → y, which shows that the limit of nets is not unique in this case. Suppose now that (2.10) holds and assume that the net (xi )i∈I ⊂ X converges to x and y with x = y. Take U ∈ Nτ (x) and V ∈ Nτ (y) such that U ∩ V = ∅. Then there exist iU , iV ∈ I such that xi ∈ U for i  iU and xi ∈ V for i  iV . Because I is directed, there exists i0 ∈ I such that i0  iU and i0  iV . We thus obtain the contradiction that xi0 ∈ U ∩ V . The proof is complete.
A useful notion is that of subnet; so, we say that the net (yk )k∈K is a subnet of the net (xi )i∈I if there exists a mapping ψ : (K, ) → (I, ) such that yk = xψ(k) for every k ∈ K and for every i ∈ I there exists ki ∈ K such that ψ(k)  i for k  ki . If J is a cofinal subset of the directed set (I, ), then (xi )i∈J is a subnet of the net (xi )i∈I . With the help of nets and subnets it is possible to characterize several topological notions as the closure of a set, the cluster points, the compact subsets, and the continuity of mappings between topological spaces in the same manner as these notions are characterized in Rk by using sequences and subsequences. For example, a subset A of the topological space (X, τ ) is closed if and only if x ∈ A whenever the net (xi )i∈I ⊂ A converges to x ∈ X, but generally, taking sequences is not sufficient; we say that A is sequentially closed if x ∈ A whenever the sequence (xn )n∈N ⊂ A converges to x ∈ X. Recall that the function f : (X, τ ) → (Y, σ) is continuous at x ∈ X if ∀ V ∈ Nσ (f (x)), ∃ U ∈ Nτ (x), ∀ x
∈ U : f (x
) ∈ V ; f is continuous if f is continuous at every x ∈ X. When X is a linear space and τ is a topology on X, it is important to have compatibility between the linear and topological structures of X.

24

2 Functional analysis over cones

Definition 2.1.20. Let X be a linear space endowed with a topology τ . We say that (X, τ ) is a topological linear space or topological vector space (t.l.s. or t.v.s. for short) if both operations on X (the addition and the multiplication by scalars) are continuous; in this case τ is called a linear topology on X. Since these operations are defined on product spaces, we recall that for two topological spaces (X1 , τ1 ) and (X2 , τ2 ), there exists a unique topology on X1 × X2 , denoted by τ1 × τ2 , with the property that B(x1 , x2 ) := {U1 × U2 | U1 ∈ Nτ1 (x1 ), U2 ∈ Nτ2 (x2 )} is a neighborhood base of (x1 , x2 ) w.r.t. τ1 × τ2 for every (x1 , x2 ) ∈ X1 × X2 ; τ1 × τ2 is called the product topology on X1 × X2 . Of course, in Definition 2.1.20 the topology on X ×X is τ ×τ , and the topology on R×X is τ0 ×τ , where τ0 is the usual topology of R. It is easy to see that when (X, τ ) is a topological linear space, a ∈ X and λ ∈ R \ {0}, the mappings Ta , Hλ : X → X defined by Ta (x) = a + x, Hλ (x) := λx, are bijective and continuous with continuous inverses i.e., they are homeomorphisms). It follows, that V ∈ Nτ (0) iff a + V ∈ Nτ (a) for every a ∈ X and every neighborhood of 0 is absorbing (A ⊂ X is absorbing if for every x ∈ X there exists δ > 0 such that [−δ, δ] · x ⊂ A, or equivalently, for every x ∈ X there exists δ > 0 such that [0, δ] · x ⊂ A). When X is a t.v.s., the class of all neighborhoods of 0 ∈ X will be denoted also by NX . It is well known that the origin 0 of a t.v.s. (X, τ ) has a neighborhood base formed by balanced absorbing sets (∅ = A ⊂ X is balanced if [−1, 1] · A = A). Taking into account the formula  cl A = {A + V | V ∈ B} (2.11) for the closure of a subset A of the t.l.s. (X, τ ), where B is a neighborhood base for 0, one obtains that the origin of a topological linear space has a neighborhood base formed by closed, balanced, and absorbing sets. Note also that a topological linear space (X, τ ) is Hausdorff iff for every / V , or equivalently (see (2.11)), x ∈ X \ {0} there exists V ∈ NX such that x ∈ the set {0} is closed. Further examples of topological linear spaces are given in the next section. We discuss now the connections between topology and order. Contrary to the definition of an ordered linear space (i.e., a linear space endowed with a compatible reflexive preorder), the definition of an ordered topological linear space does not require any direct relation to exist between the order and the topology involved. However, because a compatible reflexive preorder on a linear space is defined by a convex cone, generally one asks that the cone defining the order be closed, have nonempty interior, or be normal. Before giving the definition of a normal cone let us say that the nonempty set A of the linear space X is full with respect to the convex cone C ⊂ X if A = [A]C , where [A]C := (A + C) ∩ (A − C); note that [A]C is full w.r.t. C for every set A ⊂ X.

2.1 Order structures

25

Definition 2.1.21. Let (X, τ ) be a t.v.s. and let C ⊂ X be a convex cone. Then C is called normal (relative to τ ) if the origin 0 ∈ X has a neighborhood base formed by full sets w.r.t. C. In the next result we give several characterizations of normal cones in a topological linear space. Theorem 2.1.22. Let (X, τ ) be a topological linear space and let C ⊂ X be a convex cone. Then the following statements are equivalent: (i) C is normal, (ii) ∀ V ∈ NX , ∃ U ∈ NX : [U ]C ⊂ V , (iii) for all nets (xi )i∈I , (yi )i∈I ⊂ X such that 0 ≤C xi ≤C yi for every i ∈ I one has (yi ) → 0 ⇒ (xi ) → 0, (iv) cl C is normal. Proof. Recall that x ≤C y means y − x ∈ C. (i) ⇔ (ii) is obvious from the definition of the normal cone. (ii) ⇒ (iii) Let V ∈ NX ; then there exists U ∈ NX such that [U ]C ⊂ V . Since (yi ) → 0, there exists iU ∈ I such that yi ∈ U for i  iU . Then xi ∈ C ∩ (yi − C) ⊂ (U + C) ∩ (U − C) = [U ]C ⊂ V for i  iU , and so (xi ) → 0. (iii) ⇒ (ii) Assume, by contradiction, that (ii) does not hold. Then there exists V0 ∈ NX such that [U ]C ⊂ V0 for every U ∈ NX . Hence, for
every U ∈ NX there exist yU , yU ∈ U and xU , x
U ∈ C such that zU :=



yU + xU = yU + xU ∈ / V0 . Of course, (yU ), (yU ) → 0, and so (yU − yU ) → 0,
too. Since 0 ≤C xU ≤C yU − yU , by hypothesis we obtain that (zU ) → 0. This contradicts the fact that zU ∈ / V0 for every U ∈ NX . (iv) ⇒ (ii) is obvious because A ⊂ [A]C ⊂ [A]cl C for ∅ = A ⊂ X. (ii) ⇒ (iv) Let V ∈ NX ; then there exists U ∈ NX such that [U ]C ⊂ V . By the continuity of “+”, there exists U0 ∈ NX such that U0 + U0 ⊂ U . Using (2.11), we obtain that [U0 ]cl C = (U0 + cl C) ∩ (U0 − cl C) ⊂ (U0 + U0 + C) ∩ (U0 − (U0 + C)) ⊂ (U + C) ∩ (U − C) = [U ]C ⊂ V. Therefore (ii) holds. The proof is complete.




Corollary 2.1.23. Let (X, τ ) be a Hausdorff t.v.s. and let C ⊂ X be a convex cone. If C is normal, then cl C is pointed, and so C is pointed, too. Proof. Indeed, if x ∈ cl C ∩ (− cl C), then x ∈ ({0} + cl C) ∩ ({0} − cl C) ⊂ (U + cl C) ∩ (U − cl C) = [U ]cl C for every U ∈ NX . Since cl C is normal (by the preceding theorem), the family {[U ]cl C | U ∈ NX } is a neighborhood base of 0. Because X is Hausdorff (see also (2.11)), x = 0.
In the next section we shall give several characterizations of normal cones in locally convex spaces. It is well known that a real nonincreasing and bounded below sequence (xn ) converges to inf{xn | n ∈ N}.

26

2 Functional analysis over cones

Let (X, τ ) be a Hausdorff t.v.s. partially ordered by the convex cone C; we say that the net (xi )i∈I ⊂ X is nonincreasing if ∀ i, j ∈ I : j  i ⇒ xj ≤C xi .

(2.12)

Let ∅ = A ⊂ X; we say that A is lower bounded with respect to C if A is lower bounded with respect to RC (see Definition 2.1.7). Similarly, a ∈ X is a lower bound (infimum) of A w.r.t. C if a is so for RC . Hence a ∈ X is a lower bound of A w.r.t. C if a ≤C x for every x ∈ A; a is the infimum of A w.r.t. C if a is a lower bound and for any lower bound a
of A we have that a
≤C a. The infimum of A w.r.t. C will be denoted by inf C A when it exists. Proposition 2.1.24. Let (X, τ ) be a Hausdorff t.v.s. partially ordered by the closed convex cone C. If the net (xi )i∈I ⊂ X is nonincreasing and convergent to x ∈ X, then {xi | i ∈ I} is bounded below and x = inf{xi | i ∈ I}. Proof. Let A := {xi | i ∈ I} and fix i0 ∈ I. Then xi ∈ xi0 − C for i  i0 , and so x ∈ cl(xi0 − C) = xi0 − C. Therefore x ≤C xi0 for every i0 ∈ I. Hence x is a lower bound of A. Let x
∈ X be an arbitrary lower bound of A. Then xi ∈ x
+ C for every i ∈ I, and so x ∈ cl(x
+ C) = x
+ C. Therefore x = inf C A.
In ordered topological linear spaces the result concerning bounded monotone sequences recalled above is not generally true. Consider the linear space ∞ of all bounded sequences x = (xk )k≥1 ⊂ R endowed with the norm

x = sup{|xk | | k = 1, 2, . . .}. In ∞ we consider the “usual” partial or∞ | xk ≥ 0 ∀ k ≥ 1}; ∞ der generated by the cone ∞ + := {x ∈ l + is a pointed closed convex cone (even reproducing and with nonempty interior). Example 2.1.25. (Peressini [294, p. 91]) The sequence (xn )n≥1 ⊂ ∞ , defined by (for n fixed)  −1 if 1 ≤ k ≤ n, xkn = 0 if k > n, is nonincreasing w.r.t. C, and inf{xn | n ≥ 1} = e
:= −e where e = (1, 1, 1, . . .) ∈ ∞ . But xn − e
= 1 for every n ≥ 1. Therefore (xn )n≥1 does not converge to its infimum. Sometimes a cone C that partially orders a Hausdorff linear topological space (X, τ ) is said to be Daniell if any nonincreasing net having a lower bound τ -converges to its infimum (see Jahn [195, p. 29], Luc [244, p. 47], Borwein [38]).

2.2 Functional Analysis and Convexity An important class of topological linear spaces (t.l.s.) is that of locally convex spaces. Usually this class of t.l.s. is introduced in two equivalent manners. The

2.2 Functional analysis and convexity

27

simplest is that the t.l.s. (X, τ ) is a locally convex space (l.c.s. for short) if the origin 0 ∈ X has a neighborhood base formed by convex sets. It follows then that the origin has a neighborhood base formed by balanced convex sets; these sets may be taken to be all closed or all open. Before discussing the second way of introducing locally convex spaces, let X be a linear space and p : X → R; we say that: • • • • • •

p is subadditive if ∀ x, y ∈ X : p(x + y) ≤ p(x) + p(y); p is positively homogeneous if ∀ x ∈ X, ∀ λ ∈ R+ : p(λx) = λp(x); p is symmetric if ∀ x ∈ X : p(−x) = p(x); p is sublinear if p is subadditive and positively homogeneous; p is a seminorm if p is sublinear and symmetric; note that in this case p(x) ≥ 0 for every x ∈ X. Indeed, 0 = p(0) ≤ p(x) + p(−x) = 2p(x); p is a norm if p is a seminorm and p(x) = 0 ⇔ x = 0.

Consider X a linear space and P a nonempty family of seminorms on X. For every nonempty finite subset P ⊂ P, x ∈ X, and ε > 0 let V (x; P, ε) := {y ∈ X | p(y − x) ≤ ε ∀ p ∈ P }. It is obvious that V (x; P, ε) is a convex set and V (x; P, ε) = x + V (0; P, ε); moreover, V (0; P, ε) is also symmetric (and so is balanced) and absorbing. The family {B(x) | x ∈ X}, where B(x) := {V (x; P, ε) | ε > 0, ∅ = P ⊂ P, P finite}, satisfies conditions (NB1)–(NB3) (see Section 2.1). Therefore there exists a unique topology τP on X such that B(x) is a neighborhood base of x w.r.t. τP for every x ∈ X. It is easy to show that (X, τP ) is a topological linear space, and so (X, τP ) is a locally convex space, denoted in the sequel by (X, P). In order to show that every l.c.s. can be introduced by using a family of seminorms, let us define the Minkowski functional. Consider A an absorbing subset of the linear space X; the Minkowski functional associated with A is the mapping pA : X → R,

pA (x) := inf{t ≥ 0 | x ∈ tA}.

In the next result we collect several properties of the Minkowski functional. Recall that the algebraic interior, or core, of A ⊂ X is the set aint A := Ai := {a ∈ X | A − a is absorbing}; of course, Ai ⊂ A. Lemma 2.2.1. Let X be a linear space; let A ⊂ B ⊂ X be such that A is absorbing and λ > 0. Then pA is positively homogeneous, pA ≥ pB , and pλA = λ−1 pA ; if A is symmetric, that is, A = −A, then pA is also symmetric. Assume that A is convex; then pA is sublinear and Ai = {x ∈ X | pA (x) < 1} ⊂ A ⊂ {x ∈ X | pA (x) ≤ 1}. If C ⊂ X is a convex cone and A = [A]C , then

(2.13)

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2 Functional analysis over cones

0 ≤C x ≤C y ⇒ pA (x) ≤ pA (y),

(2.14)

i.e., pA is monotone w.r.t. C. Proof. The fact that pA is positively homogeneous and the relations pA ≥ pB , pλA = λ−1 pA , and pA (−x) = pA (x) if A is symmetric are obvious. Assume that A is convex and take x, y ∈ X. Then for t, s ∈ R+ such that pA (x) < t and pA (y) < s there exist t
∈ [0, t] and s
∈ [0, s] such that x ∈ t
A and y ∈ s
A. It follows that x + y ∈ t
A + s
A = (t
+ s
)A, and so pA (x + y) ≤ t
+ s
≤ t + s. Letting t → pA (x) and s → pA (y), we obtain that pA (x + y) ≤ pA (x) + pA (y). Therefore pA is sublinear. The inclusions {x ∈ X | pA (x) < 1} ⊂ A ⊂ {x ∈ X | pA (x) ≤ 1}

(2.15)

are obvious. Let a ∈ Ai ; since A − a is absorbing, there exists δ > 0 such that a + δa ∈ A, i.e., a ∈ (1 + δ)−1 A. Hence pA (a) ≤ (1 + δ)−1 < 1. Now let a ∈ X be such that pA (a) < 1 and consider x ∈ X. Since pA (a+tx) ≤ pA (a)+tpA (x) for t ≥ 0, there exists δ > 0 such that pA (a + tx) < 1 for t ∈ [0, δ]. From (2.15) we obtain that a + tx ∈ A for t ∈ [0, δ], and so a ∈ Ai . Therefore (2.13) holds. To prove (2.14) consider x, y ∈ X such that 0 ≤C x ≤C y and take t > 0 such that y ∈ tA, i.e., y = ta with a ∈ A. Then x = y − (y − x) = t[a−t−1 (y −x)] ∈ t(A−C) and x = t(0+t−1 x) ∈ t(A+C). So x ∈ t[A]C = tA. Hence pA (x) ≤ pA (y).
The next theorem shows that by the second procedure one obtains all the locally convex spaces. Theorem 2.2.2. Let (X, τ ) be a locally convex space. Then there exists a nonempty family P of seminorms on X such that τ = τP . Proof. Let B be a neighborhood base of 0 ∈ X formed by balanced convex subsets of X. Consider P := {pU | U ∈ B}. By Lemma 2.2.1 every element of P is a seminorm. In order to show that τ = τP , consider first V ∈ NX . Then there exists U ∈ B such that U ⊂ V . Because V (0; {pU }, 1} ⊂ U ⊂ V , it follows that V ∈ NτP (0). Conversely, let V ∈ NτP (0). Then there exist U1 , . . . , Un ∈ B and ε > 0 such that {x ∈ X | pUi (x) ≤ ε ∀ i ∈ 1, n} ⊂ V . Since B is a neighborhood base (see (NB2) in Section 2.1), there exists U ∈ B such that U ⊂ U1 ∩ · · · ∩ Un , and so pU ≥ pUi for every i ∈ 1, n. It follows that {x ∈ X | pU (x) ≤ ε} ⊂ V , whence by (2.13), εU ⊂ V . Since εU ∈ NX , V ∈ NX , too. Therefore τ = τP .
It is useful to note that the Minkowski functional pU associated with a convex neighborhood U of the origin of the l.c.s. X is continuous and int U = {x ∈ X | pU (x) < 1},

cl U = {x ∈ X | pU (x) ≤ 1}.

(2.16)

In particular, every seminorm p ∈ P is τP -continuous on the l.c.s. (X, P). The proof of the preceding theorem also shows that the locally convex space

2.2 Functional analysis and convexity

29

(X, τ ) is first-countable if and only if the topology τ is defined by an (at most) countable family of seminorms. One can show that a Hausdorff and first-countable l.c.s. is metrizable, i.e., that there exists a metric ρ (even invariant under translations) such that τ = τρ . It is easy to show that the l.c.s. (X, P) is Hausdorff if and only if for every x ∈ X \ {0} there exists p ∈ P such that p(x) > 0. Using this characterization and the Hahn–Banach theorem below, one obtains a remarkable property of locally convex spaces;, that is, the continuous dual space of the nontrivial (X = {0}) Hausdorff l.c.s. (X, P), X ∗ := {x∗ : X → R | x∗ is linear and continuous}, does not reduce to {0}; more precisely, for every x ∈ X \ {0} there exists x∗ ∈ X ∗ such that x, x∗  := x∗ (x) > 0. Recall that the linear functional ϕ : (X, P) → R is continuous if and only if there exist M > 0 and p1 , . . . , pk ∈ P such that ϕ(x) ≤ M max{p1 (x), . . . , pk (x)} ∀ x ∈ X. We give now several examples of locally convex spaces. Example 2.2.3. (1) Let · : X → R be a norm on the linear space X. Taking P := { · }, the space (X, · ) := (X, P) is called a normed space. The normed space (X, · ) is also a metric space, where the metric ρ is defined by ρ(x, y) := x − y , and so is a Hausdorff first-countable topological space. Important examples of normed spaces are: (a) ∞ := {(xn )n≥1 ⊂ R | (xn )n≥1 is bounded} with the norm (xn )n≥1 ∞ := sup{|xn | | n ≥ 1} and its subspaces c := {(xn )n≥1 ⊂ R | (xn )n≥1 is convergent} and c0 = {(x n )n≥1 ⊂pR | (xn ) → 0} with the same norm; (b) p := {(xn )n≥1 ⊂ R | n≥1 |xn | < ∞}, for p ∈ [1, ∞[, with the norm 1/p  p

(xn )n≥1 p := ; n≥1 |xn | (c) C[0, 1] with the Chebyshev norm x := maxt∈[0,1]  |x(t)|;p (d) Lp (Ω) := {x : Ω → R | x is measurable and Ω |x(t)| dt < ∞} with  1/p p the norm x p := Ω |x(t)| dt , where p ∈ [1, ∞[ and Ω ⊂ Rn is an open set endowed with the Lebesgue measure (in fact, one identifies two functions if they coincide almost everywhere). The normed space (X, · ) is a Banach space if any Cauchy sequence (xn )n≥1 ⊂ X is convergent ((xn ) is Cauchy if ∀ ε > 0, ∃ nε ∈ N, ∀ n, m ≥ nε :

xn − xm < ε); the normed spaces mentioned in (a)–(d) are Banach spaces. The Banach space (X, · ) is a Hilbert space when the norm · is from an inner product or, equivalently, it satisfies the parallelogram law:

x + y 2 + x − y 2 = 2( x 2 + y 2 ) for all x, y ∈ X. (2) Let (X, P) be a locally convex space and let X0 be a linear subspace of X. Then p |X0 is a seminorm on X0 for every p ∈ P. Taking P0 := {p |X0 | p ∈ P}, the space (X0 , P0 ) becomes a locally convex space; the topology τP0 is the trace of the topology τP on X0 . If (X1 , P1 ) and (X2 , P2 ) are locally convex

30

2 Functional analysis over cones

spaces, then (X1 × X2 , P) is an l.c.s., where P := {p1 × p2 | p1 ∈ P1 , p2 ∈ P2 } and p1 × p2 : X1 × X2 → R, p1 × p2 (x1 , x2 ) := max{p1 (x1 ), p2 (x2 )}; the topology τP is exactly τP1 × τP2 . (3) Let X, Y be two linear spaces and let Φ : X × Y → R be a separating bilinear form, i.e., Φ(x, ·) and Φ(·, y) are linear on Y and X for all x ∈ X and y ∈ Y , respectively, and ∀ x ∈ X \ {0}, ∃ y ∈ Y such that Φ(x, y) = 0, and ∀ y ∈ Y \ {0}, ∃ x ∈ X such that Φ(x, y) = 0. It is obvious that py : X → R, py (x) := |Φ(x, y)|, and px : Y → R, px (y) := |Φ(x, y)|, are seminorms for all y ∈ Y and x ∈ X. Taking PX := {py | y ∈ Y }, the space (X, PX ) is a Hausdorff l.c.s.; similarly, taking PY := {px | x ∈ X}, (Y, PY ) is a Hausdorff l.c.s. One says that (X, Y, Φ) is a dual pair; the topology τPX is denoted by σ(X, Y ), while the topology τPY is denoted by σ(Y, X). (4) Let (X, τ ) be a Hausdorff locally convex space and X ∗ its topological dual. Then Φ : X × X ∗ → R defined by Φ(x, x∗ ) := x∗ (x) = x, x∗  is a separating bilinear form (see the discussion about X ∗ before the statement of this example). The topology w := σ(X, X ∗ ) is called the weak topology of X, while the topology w∗ := σ(X ∗ , X) is called the weak∗ topology of X ∗ ; the name weak topology for w is motivated by the fact that w ⊂ τ , i.e., w is weaker than τ (this means that there are fewer open sets w.r.t. w than open sets w.r.t. τ ). It is useful to mention that the topological dual of (X, w) is X ∗ , while the topological dual of (X ∗ , w∗ ) is X (in the sense that for every ϕ ∈ (X ∗ , w∗ )∗ there exists a unique xϕ ∈ X such that ϕ(x∗ ) = xϕ , x∗  for all x∗ ∈ X ∗ ; moreover, the mapping ϕ ∈ X ∗ → xϕ ∈ X is an isomorphism of linear spaces. Recall that two locally convex topologies τ1 and τ2 on the linear space X are called compatible if (X, τ1 ) and (X, τ2 ) have the same topological dual; the finest topology σ on the locally convex space (X, τ ) that is compatible with τ is called the Mackey topology of X (such a topology always exists!). (5) Let (X, · ) be a normed space. Then X ∗ is also a normed space,and with the norm (always) defined by x∗ := sup{x, x∗  | x ≤ 1}; (X ∗ , · ) is even a Banach space. As in (3), on X one has also the weak topology w, while on X ∗ one has the topology w∗ ; one has that w = τ· iff X is finite-dimensional. Since (X ∗ , · ) is a normed space, on X ∗ one has also the topology σ(X ∗ , X ∗∗ ); note that w∗ ⊂ σ(X ∗ , X ∗∗ ) ⊂ τ· . Taking x ∈ X, the mapping φx : (X ∗ , · ) → R, φx (x∗ ) := x∗ (x), is linear and continuous; moreover, φx = x . The mapping JX : (X, · ) → (X ∗∗ , · ) defined by JX (x) := φx is a linear operator having the property that JX (x) = x for every x ∈ X. One says that (X, · ) is reflexive if JX is onto. Because (X ∗∗ , · ) is a Banach space, every reflexive normed space is a Banach space. One gets that w∗ = σ(X ∗ , X ∗∗ ) iff (X, · ) is a reflexive Banach space. Among the usual Banach spaces mentioned in (1), those that are reflexive are p and Lp (Ω) for p ∈ ]1, ∞[. An important result in the theory of locally convex spaces is the Alaoglu– Bourbaki theorem: If U is a neighborhood of the origin 0 of the Hausdorff

2.2 Functional analysis and convexity

31

l.c.s. (X, τ ), then the polar set U 0 of U is w∗ -compact (i.e., every net (x∗i )i∈I ⊂ U 0 contains a subnet (x∗ψ(k) )k∈K converging to x∗ ∈ U 0 ), where the polar set of ∅ = A ⊂ X is A0 := {x∗ | x, x∗  ≥ −1 ∀ x ∈ A}; A0 is, obviously, w∗ -closed, convex and contains 0 ∈ X ∗ . If (X, · ) is a normed space and UX := {x ∈ X | x ≤ 1} is the closed unit ball of X, then (UX )0 = UX ∗ . Therefore UX ∗ is w∗ -compact. It follows that the normed space (X, · ) is reflexive iff UX is w-compact. Another famous characterization of reflexive Banach spaces is due to R.C. James: (X, · ) is reflexive iff any x∗ ∈ X ∗ attains its supremum on UX . The dual cone and its quasi-interior are defined analogously to polar sets. Definition 2.2.4. Let C be a cone in the l.c.s. X. Then C + := {x∗ ∈ X ∗ | x, x∗  ≥ 0 ∀ x ∈ C} is called the dual cone of C, while C # := {x∗ ∈ C + | x, x∗  > 0 ∀ x ∈ C \ {0}} is called the quasi-interior of C + . Obviously, C + is a w∗ -closed convex cone as the intersection of a family of closed half-spaces. As mentioned above, a fundamental tool in functional analysis is the (algebraic) Hahn–Banach theorem. Theorem 2.2.5. Let X be a linear space, X0 ⊂ X a linear subspace, p : X → R a sublinear functional, and ϕ0 : X0 → R a linear functional such that ϕ0 (x) ≤ p(x) for every x ∈ X0 . Then there exists a linear functional ϕ : X → R such that ϕ |X0 = ϕ0 and ϕ(x) ≤ p(x) for every x ∈ X. The proof of this result can be found in any book on functional analysis, so it is omitted; just note that for the proof one uses the Zorn axiom mentioned in the first section. If (X, · ) is a normed space, X0 is a linear subspace of X, and ϕ0 ∈ (X0 , · )∗ , then p := ϕ0 · · is sublinear, and ϕ0 (x) ≤ p(x) for every x ∈ X0 . By the Hahn–Banach theorem, there exists a linear functional ϕ : X → R such that ϕ |X0 = ϕ0 and ϕ(x) ≤ p(x) for every x ∈ X. It follows that ϕ is continuous, and so ϕ ∈ X ∗ , and ϕ = ϕ0 . If (X, P) is an l.c.s., a similar procedure is possible. Theorem 2.2.6. Let (X, P) be a locally convex space, X0 ⊂ X a linear subspace, and ϕ0 : X0 → R a continuous linear functional. Then there exists ϕ ∈ X ∗ such that ϕ |X0 = ϕ0 . Proof. Because ϕ0 is a continuous linear functional, there exist M > 0 and p1 , . . . , pk ∈ P such that ϕ0 (x) ≤ M · max{p1 (x), . . . , pk (x)} for every x ∈ X0 .

32

2 Functional analysis over cones

Since p := M · max{p1 , . . . , pk } is a seminorm, there exists ϕ : X → R such that ϕ |X0 = ϕ0 and ϕ(x) ≤ p(x) for every x ∈ X. The last inequality shows that ϕ is continuous, and so ϕ ∈ X ∗ .
Another important application of the Hahn–Banach theorem is to the separation of convex sets. The following result is met in the literature as the geometric form of the Hahn–Banach theorem. Theorem 2.2.7. Let (X, τ ) be a locally convex space, A ⊂ X a convex set with nonempty interior and x0 ∈ X \ int A. Then there exists x∗ ∈ X ∗ \ {0} such that x, x∗  ≤ x0 , x∗  ∀ x ∈ A. (2.17) Proof. Replacing, if necessary, A by A − a with a ∈ int A, we assume that 0 ∈ int A. Thus A is an absorbing convex set, and so, by Lemma 2.2.1, the Minkowski functional pA is sublinear. Because x0 ∈ / int A we have that pA (x0 ) ≥ 1 (see (2.16)). Let X0 := lin{x0 } = Rx0 and ϕ0 : X0 → R defined by ϕ0 (tx0 ) := tpA (x0 ) for every t ∈ R. Since ϕ0 (−x0 ) = −pA (x0 ) ≤ pA (−x0 ), we obtain that ϕ0 (x) ≤ pA (x) for every x ∈ X0 . Using the Hahn–Banach theorem, there exists a linear functional ϕ : X → R such that ϕ |X0 = ϕ0 and ϕ(x) ≤ p(x) for every x ∈ X. The last inequality shows that ϕ is continuous. Denoting ϕ by x∗ and taking into account (2.16), we obtain that x, x∗  ≤ pA (x) ≤ 1 ≤ pA (x0 ) = ϕ0 (x0 ) = x0 , x∗ 

∀ x ∈ A,

and so the conclusion holds.
Another useful variant of the separation theorem is given in the next result. Theorem 2.2.8. Let (X, τ ) be a locally convex space, A ⊂ X a nonempty closed convex set, and x0 ∈ X \ A. Then there exists a continuous linear functional x∗ ∈ X ∗ such that sup{x, x∗  | x ∈ A} < x0 , x∗  .

(2.18)

Proof. Since x0 ∈ / A = cl A, from (2.11) we get the existence of an open / A + U . Since A + U is convex and convex neighborhood U of 0 such that x0 ∈ open, by the preceding theorem there exists x∗ ∈ X ∗ \ {0} such that x0 , x∗  ≥ sup{x, x∗  + u, x∗  | x ∈ A, u ∈ U } = sup{x, x∗  | x ∈ A} + sup{u, x∗  | u ∈ U }. But sup{u, x∗  | u ∈ U } > 0 (otherwise, x∗ = 0, because U is absorbing), and so (2.18) holds.
∗ Using the preceding theorem in the topological dual X of the l.c.s. X endowed with the weak∗ topology we obtain that    0  (2.19) λA + (1 − λ)B 0 (A ∩ B)0 = conv(A0 ∪ B 0 ) = cl λ∈[0,1]

2.2 Functional analysis and convexity

33

for all closed convex subsets A, B of X with 0 ∈ A∩B, where convE := conv E is the closed convex hull of the subset E of the topological vector space (t.v.s.) Y . The last property of the Minkowski functional in Lemma 2.2.1 turns out to be useful in characterizing normal cones in locally convex spaces. Proposition 2.2.9. Let (X, τ ) be an l.c.s. and C ⊂ X a convex cone. (i) C is normal if and only if there exists a family P of monotone seminorms w.r.t. C such τ = τP . (ii ) If C is normal, then C + is reproducing; i.e., X ∗ = C + − C + . (iii ) C is w-normal if and only if C + is reproducing. Proof. (i) Assume that C is normal; since X is an l.c.s., there exists a base B of symmetric convex and full neighborhoods of 0 ∈ X. Let P = {pU | U ∈ B}. By (the proof of) Theorem 2.2.2, τ = τP . From Lemma 2.2.1 we have that every pU is monotone w.r.t. C for every U ∈ B. Conversely, suppose that the topology τ is determined by the family P of monotone seminorms w.r.t. C. Let V ∈ NX . Then there exist ε > 0 and p1 , . . . , pn ∈ P such that {x ∈ X | pi (x) ≤ ε ∀ i ∈ 1, n} ⊂ V. Consider U = {x ∈ X | pi (x) ≤ ε/3 ∀ i ∈ 1, n} ⊂ V and take y ∈ [U ]C . It follows that y = x1 + c1 = x2 − c2 with x1 , x2 ∈ U and c1 , c2 ∈ C. Since 0 ≤C c1 = x2 − x1 − c2 ≤C x2 − x1 , we obtain that pi (c1 ) ≤ pi (x2 − x1 ) ≤ pi (x2 ) + pi (x1 ) ≤ 2ε/3 for 1 ≤ i ≤ n, whence pi (y) = pi (x1 + c1 ) ≤ pi (x1 ) + pi (c1 ) ≤ ε/3 + 2ε/3 = ε for 1 ≤ i ≤ n. Therefore y ∈ V . Hence [U ]C ⊂ V . From Theorem 2.1.22 we obtain that C is normal. (ii) Let x∗ ∈ X ∗ ; then U := {x ∈ X | x, x∗  ≥ −1} ∈ NX , and so, by Theorem 2.1.22, there exists a convex neighborhood V of 0 ∈ X such that (V + C) ∩ (V − C) ⊂ U . Therefore  0 x∗ ∈ (V + C) ∩ (V − C)     = conv (V + C)0 ∪ (V − C)0 = conv (V + C)0 ∪ (V − C)0 =

⊂



 λ∈[0,1]



 λ∈[0,1]

λ(V + C)0 + (1 − λ)(V − C)0



 λC 0 + (1 − λ)(−C)0 ⊂ (C + − C + ).

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2 Functional analysis over cones

  We have used the formulae cl (V + C) ∩ (V − C) = cl(V + C) ∩ cl(V − C), (2.19), and the fact that (V + C)0 and (V − C)0 are w∗ -compact by the Alaoglu–Bourbaki theorem. Therefore C + is reproducing. (iii) If C is w-normal, by (ii) we obtain that C + is reproducing. Suppose now that C + is reproducing and the nets (xi )i∈I , (yi )i∈I ⊂ X are such that w 0 ≤C xi ≤C yi for all i ∈ I and (yi )i∈I −→ 0. Let x∗ ∈ X. Since C + is reproducing, x∗ = x∗1 − x∗2 with x∗1 , x∗2 ∈ C + . It follows that 0 ≤ xi , x∗1  ≤ yi , x∗1  for every i ∈ I, and so (xi , x∗1 ) → 0. Similarly, (xi , x∗2 ) → 0, and w so (xi , x∗ ) → 0. Therefore (xi )i∈I −→ 0. By Theorem 2.1.22 we obtain that C is w-normal.
The statements (ii) and (iii) of the preceding proposition can be found in Peressini [294, Prop. 2.1.21] Some further characterizations of normal cones are possible, if we restrict to normed spaces. We recall that the subset B of the topological linear space (X, τ ) is bounded if for every V ∈ NX there exists λ > 0 such that λB ⊂ V ; if (X, P) is an l.c.s., the set B is bounded (in the sense of the preceding definition) if p(B) is bounded in R for every p ∈ P. Theorem 2.2.10. Let (X, · ) be a normed vector space and C ⊂ X a convex cone. The following statements are equivalent: (i) C is normal; (ii) [UX ]C is bounded; (iii) there exists an equivalent monotone (w.r.t. C) norm · 0 ; (iv) there exists α > 0 such that y ≥ α x whenever x, y ∈ X with 0 ≤C x ≤C y; (v) there exists β > 0 such that x + y ≥ β whenever x, y ∈ C with x =

y = 1. Proof. (i) ⇒ (ii) Let V := UX ∈ N· (0). Using Theorem 2.1.22, there exists r > 0 such that [rUX ]C = r[UX ]C ⊂ UX . Therefore [UX ]C is bounded. (ii) ⇒ (i) is immediate: {[rUX ]C | r > 0} is a base of neighborhoods of 0. (ii) ⇒ (iii) There exists r > 0 such that rUX ⊂ [rUX ]C =: U ⊂ UX . It follows that

· = pUX ≤ pU ≤ prUX = r−1 pUX = r−1 · . Since U is a convex, symmetric, and full neighborhood, from the above inequalities we get that · 0 := pU is a monotone norm, equivalent to · . (iii) ⇒ (iv) There exist r1 , r2 > 0 such that ∀ x ∈ X : r1 x ≤ x 0 ≤ r2 x . Let 0 ≤C x ≤C y. Then r1 x ≤ x 0 ≤ y 0 ≤ r2 y ,

2.2 Functional analysis and convexity

35

whence y ≥ α x , where α = r1 /r2 > 0. (iv) ⇒ (ii) Let y ∈ [UX ]C. Then y = x1 + c1 = x2 − c2 with x1 , x2 ∈ UX and c1 , c2 ∈ C. It follows that 0 ≤C c1 = x2 − x1 − c2 ≤C x2 − x1 , whence α c1 ≤ x2 − x1 ≤ 2. Therefore

y = x1 + c1 ≤ x1 + c1 ≤ 1 + 2/α. Hence [UX ]C is bounded. (iv) ⇒ (v) Let x, y ∈ C with x = y = 1. Then 0 ≤C x ≤C x+y, whence

x + y ≥ α x = α. (v) ⇒ (iv) Suppose that (iv) does not hold. Then there exist (xn ), (yn ) ⊂ X such that 0 ≤C xn ≤C yn and yn < n−1 xn for every n ∈ N∗ . We may suppose that xn = 1 for every n. Consider un = (yn − xn )/ yn − xn ∈ C. Of course, un = 1. From the hypothesis we have that xn + un ≥ β for every n. On the other hand,   yn − xn 1 1 xn + un = xn + xn + = 1− yn . (2.20)

yn − xn

yn − xn

yn − xn But 1 − 1/n ≤ 1 − yn ≤ xn − yn ≤ 1 + yn ≤ 1 + 1/n, whence ( yn − xn ) → 1. From (2.20) we get the contradiction (xn +un ) → 0.
In finite-dimensional spaces the normal cones are recognized easily. Corollary 2.2.11. Let (X, τ ) be a finite-dimensional Hausdorff t.l.s. and C ⊂ X a convex cone. Then C is normal if and only if cl C is pointed. Proof. The necessity is valid without the assumption dim X < ∞, as seen in Corollary 2.1.23. Suppose that cl C is pointed. It is well known that all the Hausdorff linear topologies on a finite-dimensional vector space coincide. Therefore we may suppose that X is Rn (n = dim X) endowed with the Euclidean norm: 1/2  . Since the mapping X × X  (x, y) −→

(x1 , . . . , xn ) := x21 + · · · + x2n x + y ∈ X = Rn is continuous and S = {(x, y) ∈ cl C × cl C | x = y = 1} is compact, there exist (x, y) ∈ S such that β := x + y ≤ x + y for all (x, y) ∈ S. Because cl C is pointed, β > 0. Therefore condition (v) of the preceding theorem is satisfied. It follows that C is normal.
Let X be a Hausdorff l.c.s. and C ⊂ X a nontrivial convex cone having a base B; then 0 ∈ / B. If 0 ∈ / cl B, Theorem 2.2.8 gives an element x∗0 ∈ X ∗ ∗ such that b, x0  > 0 for every b ∈ cl B. Since every x ∈ C \ {0} has a (unique) representation x = λb with λ > 0 and b ∈ B, we get x, x∗0  = λ b, x∗0  > 0, and so x∗0 ∈ C # . Conversely, if x∗0 ∈ C # , the set B := {x ∈ C | x, x∗0  = 1} is a base of C with 0 ∈ / cl B (since x∗0 is continuous). So we have proved the next result. Theorem 2.2.12. Let X be a Hausdorff l.c.s. and C ⊂ X a nontrivial convex cone. Then C has a base B with 0 ∈ / cl B if and only if C # = ∅. A classical theorem is that of Krein and Rutman: If C is a nontrivial closed convex and pointed cone of a separable normed space X, then C # = ∅. So

36

2 Functional analysis over cones

each such cone has a base. Now we assume that int C + = ∅ (the interior being taken w.r.t. a linear topology compatible with w∗ ). Of course, int C + ⊂ C # (equality holds if int C + = ∅). Then Theorem 2.2.12 can be refined, as shown in [199, Th. 3.8.4]. Theorem 2.2.13. Let X be a Hausdorff l.c.s. and C ⊂ X a nontrivial closed convex cone. Then C has a closed and bounded base if and only if int C + = ∅, where the interior is considered w.r.t. the strong topology on X ∗ . Recall that the strong topology on X ∗ is that topology for which the family {B 0 | B is bounded} is a base of neighborhoods of the origin of X ∗ . Many times, the existence of a base for the convex cone C is too weak. In the following definition we strengthen this condition. Definition 2.2.14. Let X be a Hausdorff t.v.s. and C ⊂ X a nontrivial convex cone. (i) We say that C is based if there exists a convex set B such that C = R+ B and 0 ∈ / cl B. (ii) We say that C is well-based if there exists a bounded convex set B such that C = R+ B and 0 ∈ / cl B. (iii) Let the topology of X be defined by the family P of seminorms. We say that C is supernormal or nuclear if for every p ∈ P there exists x∗ ∈ X ∗ such that p(x) ≤ x, x∗  for all x ∈ C; of course, x∗ ∈ C + in this case. When X is an H.l.c.s., in Definition 2.2.14 (i), (ii) one may ask the uniqueness of the representation of any x ∈ C \ {0} as λb with λ > 0 and b ∈ B; that is, B is also a base in the sense of Definition 2.1.14, but this assumption is not very useful in what follows. Moreover, when X is not an l.c.s., it is not certain that C has a base (in the sense of Definition 2.1.14) when C is based or well-based. Sometimes a convex set B ⊂ X such that C = R+ B and 0 ∈ / cl B is called also a base of C. When C = R+ B with B a bounded convex set and 0 ∈ / cl B, then cl C = R+ · cl B; hence, for a well-based closed convex cone C, the set B in Definition 2.2.14(ii) can be chosen to be closed. Moreover, it follows that C is well-based iff cl C is well-based. The notion of nuclear cones in Definition 2.2.14(iii) was introduced by Isac in [175]. Isac [175, 177, 178] and Postolic˘ a [303, 304] gave several examples of supernormal cones. From the definition of supernormal cones it is obvious that C is supernormal iff cl C is supernormal. When (X, · ) is a normed space, the convex cone C is supernormal iff there exists x∗ ∈ X ∗ such that C ⊂ {x ∈ X | x ≤ x, x∗ },

(2.21)

that is, iff C has the angle property in the sense of Cesari and Suryanarayana [60, Def. 4.2] (in fact, this is equivalent to their definition). Of course, x∗ from (2.21) is in C # .

2.2 Functional analysis and convexity

37

Proposition 2.2.15. (Isac [175]) Let (X, P) be an H.l.c.s. and C ⊂ X a proper convex cone. Then C well-based ⇒ C supernormal ⇒ C normal. If X is a normed space, then C supernormal ⇒ C well-based. Proof. Suppose that C is well-based and B is a bounded base of C. Since 0 ∈ / cl B, there exists x∗ ∈ X ∗ such that 1 ≤ x, x∗  for every x ∈ B. Let p ∈ P; since B is bounded, there exists µ ∈ R+ such that p(x) ≤ µ for every x ∈ B. It follows that p(x) ≤ x, µx∗  for every x ∈ C, and so C is supernormal. Suppose now that C is supernormal and consider the nets (xi )i∈I , (yi )i∈I ⊂ X such that 0 ≤C xi ≤C yi for all i ∈ I and (yi )i∈I → 0. Let p ∈ P. By hypothesis, there exists x∗ ∈ C + such that p(x) ≤ x, x∗  for all x ∈ C. It follows that p(xi ) ≤ xi , x∗  ≤ yi , x∗  for every i ∈ I. As (yi )i∈I → 0, w (yi )i∈I → 0, and so (yi , x∗ )i∈I → 0. It follows that (p(xi ))i∈I → 0, and so (xi )i∈I → 0. Using Theorem 2.1.22 we obtain that C is normal. Assume now that X is a normed space and C is supernormal. Then there exists x∗ ∈ C + such that x ≤ x, x∗  for all x ∈ C. Take B := {x ∈ C | x, x∗  = 1}; it is obvious that B is a bounded base of C.
The proof above shows that C satisfies a stronger condition than normality when C is supernormal; more exactly, if C is supernormal, then w

0 ≤C xi ≤C yi ∀ i ∈ I, (yi )i∈I → 0 ⇒ (xi )i∈I → 0. w

In particular, for every net (xi )∈I ⊂ C we have (xi )i∈I → 0 ⇒ (xi )i∈I → 0. Also note (see Isac [177]) that C is w-supernormal ⇔ C is w-normal ⇔ C + − C + = X ∗ . The last equivalence is stated in Proposition 2.2.9(iii). The implication ⇒ of the first equivalence follows from the preceding proposition. Assume that C is w-normal. Let y ∗ ∈ X ∗ ; since X ∗ = C + − C + , there exist y1∗ , y2∗ ∈ C + with y ∗ = y1∗ − y2∗ . Consider x∗ := y1∗ + y2∗ ∈ C + . It follows that |x, y ∗ | ≤ |x, y1∗ | + |x, y2∗ | = x, y1∗  + x, y2∗  = x, x∗  for all x ∈ C, and so C is w-supernormal. Example 2.2.16. (1) The convex cone C ⊂ Rn is well-based if and only if cl C is pointed; hence C is well-based iff C is normal. Indeed, assume that cl C is pointed and take S := {x ∈ cl C | x = 1}. Because S is compact, conv S is also  compact. But 0 ∈ / conv S; otherwise, 0 = λ1 x1 + · · · + λk xk k with λi > 0, i=1 λi = 1, and xi ∈ S for i ∈ 1, k (k ≥ 2). This implies the contradiction 0 = −λk xk ∈ cl C. It is obvious that B := C ∩ conv S is a bounded convex base of C. (2) In any nontrivial normed space (X, · ) there exist well-based convex cones, even with nonempty interior: C := R+ (x+rUX ), where x ∈ X \{0} and 0 < r < x .

38

2 Functional analysis over cones

(3) Among the classical Banach spaces (mentioned in Example 2.2.3) their usual positive cones are well-based only in 1 and L1 (Ω). Lemma 2.2.17. ([385, Prop. 5]) Let X be a Hausdorff l.c.s., X ∗ its topological dual, and let C be a closed proper convex cone in X with int C = ∅. If x ∈ int C, then the set B = {x∗ ∈ C + | x, x∗  = 1} is a weak∗ -compact base for C + . Proof. Since x ∈ int C, the set V := C − x is a neighborhood of 0 ∈ X. Therefore its polar set V 0 := {x∗ ∈ X ∗ | x, x∗  ≥ −1 ∀ x ∈ C} = {x∗ ∈ X ∗ | c, x∗  ≥ x, x∗  − 1 ∀ c ∈ C} = {x∗ ∈ C + | x, x∗  ≤ 1} is weak∗ -compact, by the Alaoglu–Bourbaki theorem. Consequently, B is a weak∗ -closed subset of V 0 and so weak∗ -compact. Therefore B is a weak∗ compact convex set not containing the origin. Let x∗ ∈ C + \ {0}. Then γ := x, x∗  > 0; otherwise, x∗ is nonnegative on the neighborhood V of 0 ∈ X, and so x∗ = 0. It follows that γ −1 x∗ ∈ B, which proves that B is a weak∗ -compact
base of C + . Finally, we prove a characterization of well-based cones in normed vector spaces. Proposition 2.2.18. ([143, Prop. 3]) Let Y be a normed vector space and C ⊂ Y a proper convex cone. Then C is well-based if and only if there exist c0 ∈ C and y ∗ ∈ C + such that c0 , y ∗  > 0 and C ∩ SY ⊂ c0 + {y ∈ Y | y, y ∗  > 0}, where SY = {y ∈ Y | y = 1} is the unit sphere in Y . Proof. Suppose first that C is well-based with bounded base S; therefore 0∈ / cl S and C = [0, ∞[·S. Then there exists y ∗ ∈ Y ∗ such that 1 ≤ y, y ∗  for all y ∈ S. Consider S := {y ∈ C | y, y ∗  = 1}. It follows that S is a base of C; moreover, since S ⊂ [0, 1] · S, S is also bounded. Taking c1 ∈ C \ {0} we have C ∩ SY ⊂ λc1 + B+ for some λ > 0, where B+ = {y ∈ Y | y, y ∗  > 0}. Otherwise, ∀ n ∈ N∗ , ∃ cn ∈ C ∩ SY : cn ∈ / n1 c1 + B+ . Therefore, cn , y ∗  ≤ n1 c1 , y ∗  for every n ≥ 1. But because S is a base,  it follows that 1 = cn = λn bn ≤ λn M cn = λn bn with λn > 0 and bn ∈ S; with M > 0 (because S is bounded). Therefore M −1 ≤ λn = λn bn , y ∗  = cn , y ∗  ≤ n−1 c1 , y ∗ 

∀ n ∈ N∗ ,

2.3 Separation theorems for not necessarily convex sets

39

whence M −1 ≤ 0, a contradiction. Thus there exists λ > 0 such that C ∩SY ⊂ λc1 + B+ . Taking c0 := λc1 the conclusion follows. Suppose now that C ∩ SY ⊂ c0 + B+ for some c0 ∈ C and y ∗ ∈ C + with c0 , y ∗  =: γ > 0, where B+ is defined as above. Consider S = {y ∈ −1 C | y, y ∗  = 1}. Let c ∈ C \ {0}; then c c = c0 + y for some y ∈ B+ . It follows that c, y ∗  > γ c > 0; therefore y ∗ ∈ C # , and so c ∈ ]0, ∞[·S. Since cl S ⊂ {y ∈ X | y, y ∗  = 1}, we have that S is a base of C. Now −1 let y ∈ S (⊂ C). Then y y ∈ C ∩ SY . There exists z ∈ B+ such that −1 0

y y = c + z. We get 1 = y, y ∗  = y c0 + z, y ∗  ≥ γ y , whence y ≤ γ −1 . Therefore S is bounded, and so C is well-based.




2.3 Separation Theorems for Not Necessarily Convex Sets Throughout this section Y is a t.v.s. We shall use some usual notions and notation from convex analysis. So, having the function f : Y → R, its domain and epigraph are defined, respectively, by dom f := {y ∈ Y | f (y) < ∞},

epi f := {(y, t) ∈ Y × R | f (y) ≤ t};

f is said to be convex if epi f is a convex set, and f is said to be proper if dom f = ∅ and f does not take the value −∞. Of course, f is lower semicontinuous if epi f is closed. The aim of this section is to find a suitable functional ϕ : Y → R and conditions such that two given nonempty sets A and D can be separated by ϕ. Provided that D contains the rays generated by k 0 ∈ Y \ {0}, i.e., D + [0, ∞) · k 0 ⊂ D,

(2.22)

we move −D along this ray and consider the set D
:= {(y, t) ∈ Y × R | y ∈ tk 0 − D}. The assumption on D shows that D
is of epigraph type; i.e., if (y, t) ∈ D
and t
≥ t, then (y, t
) ∈ D
. Indeed, if y ∈ tk 0 − D and t
≥ t, since tk 0 − D = t
k 0 − [D + (t
− t)k 0 ] ⊂ t
k 0 − D, we obtain that (y, t
) ∈ D
. Also observe that D
= T −1 (D), where T : Y × R → Y is the continuous linear operator defined by T (y, t) := tk 0 − y. So, if D is closed (convex, cone), then D
is closed (convex, cone). Since D
is of epigraph type, we associate with D and k 0 the function ϕ := ϕD,k0 : Y → R defined by ϕ(y) := inf{t ∈ R | (y, t) ∈ D
} = inf{t ∈ R | y ∈ tk 0 − D}.

(2.23)

40

2 Functional analysis over cones

An illustration of the function ϕ is furnished by Figure 2.3.1. Obviously, the domain of ϕ is the set Rk 0 − D and D
⊂ epi ϕ ⊂ cl D
, from which it follows that if D is closed, we have that D
= epi ϕ, and so ϕ is a lower semicontinuous (l.s.c.) function. In the next results we collect several useful properties of ϕ = ϕD,k0 . Theorem 2.3.1. Let D ⊂ Y be a closed proper set and k 0 ∈ Y be such that (2.22) holds. Then ϕ is l.s.c., dom ϕ = Rk 0 − D, {y ∈ Y | ϕ(y) ≤ λ} = λk 0 − D and

ϕ(y + λk 0 ) = ϕ(y) + λ

∀ λ ∈ R,

(2.24)

∀ y ∈ Y, ∀ λ ∈ R.

(2.25)

Moreover, (a) ϕ is convex if and only if D is convex; ϕ(λy) = λϕ(y) for all λ > 0 and y ∈ Y if and only if D is a cone. (b) ϕ is proper if and only if D does not contain lines parallel to k 0 , i.e., ∀ y ∈ Y, ∃ t ∈ R : y + tk 0 ∈ / D.

(2.26)

(c) ϕ is finite-valued if and only if D does not contain lines parallel to k 0 and Rk 0 − D = Y.

(2.27)

(d) Let B ⊂ Y ; ϕ is B-monotone (i.e., y2 − y1 ∈ B ⇒ ϕ(y1 ) ≤ ϕ(y2 )) if and only if D + B ⊂ D. (e) ϕ is subadditive if and only if D + D ⊂ D. Suppose, furthermore, that D + (0, ∞) · k 0 ⊂ int D.

(2.28)

Then (f) ϕ is continuous and {y ∈ Y | ϕ(y) < λ} = λk 0 − int D, 0

{y ∈ Y | ϕ(y) = λ} = λk − bd D,

∀ λ ∈ R, ∀ λ ∈ R.

(2.29) (2.30)

(g) If ϕ is proper, then ϕ is B-monotone ⇔ D + B ⊂ D ⇔ bd D + B ⊂ D. Moreover, if ϕ is finite-valued, then ϕ strictly B-monotone (i.e., y2 − y1 ∈ B \ {0} ⇒ ϕ(y1 ) < ϕ(y2 )) ⇔ D + (B \ {0}) ⊂ int D ⇔ bd D + (B \ {0}) ⊂ int D. (h) Assume that ϕ is proper; then ϕ is subadditive ⇔ D + D ⊂ D ⇔ bd D + bd D ⊂ D.

2.3 Separation theorems for not necessarily convex sets

41

Proof. We have already observed that dom ϕ = Rk 0 − D and ϕ is l.s.c. when D is closed. From the definition of ϕ the inclusion ⊃ in (2.24) is obvious, while the converse inclusion is immediate, taking into account the closedness of D. Formula (2.25) follows easily from (2.24). (a) Since the operator T defined above is onto and epi ϕ = T −1 (D), we have that epi ϕ is convex (cone) if and only if D = T (epi ϕ) is so. The conclusion follows. (b) We have ϕ(y) = −∞ ⇔ y ∈ tk 0 − D for every t ∈ R ⇔ {y + tk 0 | t ∈ R} ⊂ D. The conclusion follows. (c) The conclusion follows from (b) and the fact that dom ϕ = Rk 0 − D. (d) Suppose first that D+B ⊂ D and take y1 , y2 ∈ Y with y2 −y1 ∈ B. Let t ∈ R be such that y2 ∈ tk 0 − D. Then y1 ∈ y2 − B ⊂ tk 0 − (D + B) ⊂ tk 0 − D, and so ϕ(y1 ) ≤ t. Hence ϕ(y1 ) ≤ ϕ(y2 ). Assume now that ϕ is B-monotone and take y ∈ D and b ∈ B. From (2.24) we have that ϕ(−y) ≤ 0. Since (−y) − (−y − b) ∈ B, we obtain that ϕ(−y − b) ≤ ϕ(−y) ≤ 0, and so, using again (2.24), we obtain that −y − b ∈ −D, i.e., y + b ∈ D. (e) Suppose first that D + D ⊂ D and take y1 , y2 ∈ Y . Let ti ∈ R be such that yi ∈ ti k 0 − D for i ∈ {1, 2}. Then y1 + y2 ∈ (t1 + t2 )k 0 − (D + D) ⊂ (t1 + t2 )k 0 − D, and so ϕ(y1 + y2 ) ≤ t1 + t2 . It follows that ϕ(y1 + y2 ) ≤ ϕ(y1 ) + ϕ(y2 ). Assume now that ϕ is subadditive and take y1 , y2 ∈ D. From (2.24) we have that ϕ(−y1 ), ϕ(−y2 ) ≤ 0. Since ϕ is subadditive, we obtain that ϕ(−y1 − y2 ) ≤ ϕ(−y1 ) + ϕ(−y2 ) ≤ 0, and so, using again (2.24), we obtain that −y1 − y2 ∈ −D, i.e. y1 + y2 ∈ D. Suppose now that (2.28) holds. (f) Let λ ∈ R and take y ∈ λk 0 − int D. Since λk 0 − y ∈ int D, there exists ε > 0 such that λk 0 − y − εk 0 ∈ int D ⊂ D. Therefore ϕ(y) ≤ λ − ε < λ, which shows that the inclusion ⊃ always holds in (2.29) when int D = ∅. Let λ ∈ R and y ∈ Y be such that ϕ(y) < λ. There exists t ∈ R, t < λ, such that y ∈ tk 0 − D. It follows that y ∈ λk 0 − (D + (λ − t)k 0 ) ⊂ λk 0 − int D. Therefore (2.29) holds, and so ϕ is upper semicontinuous. Because ϕ is also lower semicontinuous, we have that ϕ is continuous. From (2.24) and (2.29) we obtain immediately that (2.30) holds. (g) Let us prove the second part, the first one being similar (and partially proved in (d)). So, let ϕ be finite-valued. Assume that ϕ is strictly B-monotone and take y ∈ D and b ∈ B \ {0}. From (2.24) we have that ϕ(−y) ≤ 0, and so, by hypothesis, ϕ(−y − b) < 0. Using (2.29) we obtain that y +b ∈ int D. Assume now that bd D +(B \{0}) ⊂ int D. Consider y1 , y2 ∈ Y with y2 − y1 ∈ B \ {0}. From (2.30) we have that y2 ∈ ϕ(y2 )k 0 −bd D, and so y1 ∈ ϕ(y2 )k 0 −(bd D+(B\{0})) ⊂ ϕ(y2 )k 0 −int D. From (2.29) we obtain that ϕ(y1 ) < ϕ(y2 ). The remaining implication is obvious. (h) Let ϕ be proper. One has to prove bd D + bd D ⊂ D ⇒ ϕ is subadditive. Consider y1 , y2 ∈ Y . If {y1 , y2 } ⊂ dom ϕ, there is nothing to prove; hence let y1 , y2 ∈ dom ϕ. Then, by (2.30), yi ∈ ϕ(yi )k 0 − bd D for i ∈ {1, 2},

42

2 Functional analysis over cones

and so y1 + y2 ∈ (ϕ(y1 ) + ϕ(y2 ))k 0 − (bd D + bd D) ⊂ (ϕ(y1 ) + ϕ(y2 ))k 0 − D. Therefore ϕ(y1 + y2 ) ≤ ϕ(y1 ) + ϕ(y2 ).


6 HH

H

A A A

HH H

H

A A

HH

s y ∈ tk0 − bd D H *   B  B  0 A k B  * A  B  A B  A PP BB PP PP PP PP PP D PP P

-

Figure 2.3.1. Level sets of the functional ϕ in (2.23).

Remark 2.3.2. The conditions (2.22), (2.26), (2.27), and (2.28) are invariant under translations of D, but the conditions D + D ⊂ D and bd D + bd D ⊂ D are not. Note also that R ⊂ Im ϕD,k0 if ϕD,k0 is finite somewhere. Related to condition (2.28) we have the following remark. Remark 2.3.3. If for D ⊂ Y and k 0 ∈ Y one has that if cl D+(0, ∞)·k 0 ⊂ int D then cl(int D) = cl D and int(cl D) = int D. Indeed, if y ∈ D, then y + n−1 k 0 ∈ int D for any n ∈ N∗ , and so y ∈ cl(int D); this proves the first equality. Now let y ∈ int(cl D); then there exists t > 0 such that y − tk 0 ∈ cl D. Using the hypothesis we obtain that y ∈ int D. Therefore the second equality holds, too. It is obvious that (2.28) ⇒ (2.22). Other relations among several conditions used in Theorem 2.3.1 are established in the next result. Proposition 2.3.4. Let D ⊂ Y be a closed proper set and k 0 ∈ Y . (i) If there exists a cone C ⊂ Y such that k 0 ∈ int C and D + int C ⊂ D, then (2.26), (2.27), and (2.28) hold.

2.3 Separation theorems for not necessarily convex sets

43

(ii) If D is convex, int D = ∅, and (2.22), (2.27) are satisfied, then (2.26) and (2.28) hold. In particular, if the hypotheses of (i) or (ii) hold, then ϕD,k0 is finite-valued and continuous, being also convex in case (ii). Proof. (i) Let y ∈ Y . Since k 0 ∈ int C, int C − k 0 is a neighborhood of 0, and so there exists t > 0 such that ty ∈ int C − k 0 . It follows that y ∈ int C − (0, ∞)k 0 . Therefore C + Rk 0 = C − (0, ∞) · k 0 = int C + Rk 0 = int C − (0, ∞) · k 0 = Y. Taking y0 ∈ D, from the inclusion D + int C ⊂ D, we have that D + Rk 0 ⊃ y0 + int C + Rk 0 = y0 + Y = Y ; i.e., (2.27) holds. Suppose that the line Rk 0 + y is contained in D; then Y = y + Rk 0 + int C ⊂ D + int C ⊂ D, contradicting the properness of D. Since D + (0, ∞) · k 0 ⊂ D + int C ⊂ D, it is obvious that (2.28) holds, too. (ii) Let us show that (2.28) holds in our hypotheses. In the contrary case there exist y0 ∈ D and t0 ∈ (0, ∞) such that y0 + t0 k 0 ∈ / int D. Since D is convex, by a separation theorem, there exists y ∗ ∈ Y ∗ \ {0} such that  y0 + t0 k 0 , y ∗ ≤ y, y ∗  ∀ y ∈ D.   From (2.22) we obtain that  y0 + t0 k 0 , y ∗ ≤ y0 + tk 0 , y ∗ for every t ≥ 0. Since t0 > 0, it follows that k 0 , y ∗ = 0, and so  ∀ y ∈ D, ∀ t ∈ R. y0 , y ∗  ≤ y + tk 0 , y ∗ From (2.27) we obtain that y0 , y ∗  ≤ y, y ∗  for all y ∈ Y , which shows that y ∗ = 0. This contradiction proves that (2.28) holds. Assume now that y + Rk 0 ⊂ D for some y ∈ Y . Let d ∈ D and t ∈ R. Since 1 0 D is convex, for every n ∈ N∗ we have that n−1 n d + n (y + tnk ) ∈ D. Taking 0 the limit we obtain that d + tk ∈ cl D = D. Therefore, using also (2.27), we get the contradiction Y = D + Rk 0 ⊂ D. Because in both cases conditions (2.26), (2.27), and (2.28) hold, from Theorem 2.3.1(c,f) we have that ϕ is finite-valued and continuous; moreover, ϕ is convex in case (ii), D being so.
Using the preceding result we obtain the following important particular case of Theorem 2.3.1. Corollary 2.3.5. Let C ⊂ Y be a proper closed convex cone and k 0 ∈ int C. Then ϕ : Y → R, ϕ(y) := inf{t ∈ R | y ∈ tk 0 − C} is a well-defined continuous sublinear function such that for every λ ∈ R, {y ∈ Y | ϕ(y) ≤ λ} = λk 0 − C,

{y ∈ Y | ϕ(y) < λ} = λk 0 − int C.

Moreover, ϕ is strictly int C-monotone.

44

2 Functional analysis over cones

Proof. Just take D = C in Theorem 2.3.1 and use Proposition 2.3.4(ii). For the last part note that C + int C = int C.
Now all preliminaries are done, and we can prove the following nonconvex separation theorem. Theorem 2.3.6. Nonconvex Separation Theorem. Let D ⊂ Y be a closed proper set with nonempty interior, A ⊂ Y a nonempty set such that A∩(− int D) = ∅ and k 0 ∈ Y . Assume that one of the following two conditions holds: (i) there exists a cone C ⊂ Y such that k 0 ∈ int C and D + int C ⊂ D; (ii) D is convex, (2.22) and (2.27) are satisfied. Then ϕD,k0 is a finite-valued continuous function such that ϕD,k0 (x) ≥ 0 > ϕD,k0 (−y)

∀ x ∈ A, ∀ y ∈ int D;

(2.31)

moreover, ϕD,k0 (x) > 0 for every x ∈ int A. Proof. By Proposition 2.3.4, ϕD,k0 is a finite-valued continuous function. By Theorem 2.3.1(f) we have that − int D = {y ∈ Y | ϕD,k0 (y) < 0}, and so (2.31) obviouslyholds. It is evident that in our conditions int A∩(− int D) = ∅,  whence int A ∩ − cl(int D) = ∅. From Remark 2.23 we obtain that int A ∩ (−D) = ∅. Using now (2.24) we obtain that ϕD,k0 (x) > 0 for every x ∈ int A.
Of course, if we impose additional conditions on D, we have additional properties of the separating functional ϕD,k0 (see Theorem 2.3.1). As we observed in Proposition 2.3.4, when condition (i) of the preceding theorem holds, condition (2.27) holds, too. When D is a pointed convex cone the converse implication is valid if one replaces the interior with the algebraic interior. To be more precise, we have the following result. Proposition 2.3.7. Let D ⊂ Y be a convex cone such that D = −D (i.e., D is not a linear subspace) and k 0 ∈ Y . Then D + Rk 0 = Y if and only if {k 0 , −k 0 } ∩ Di = ∅. Proof. Assume first that k 0 ∈ Di (the proof for −k 0 ∈ Di being the same). Let y ∈ Y . Because D − k 0 is absorbing, there exists t > 0 such that ty ∈ D − k 0 , and so y ∈ D − t−1 k 0 ⊂ D + Rk 0 . Hence Y = D + Rk 0 . Assume now that Y = D + Rk 0 . Of course, k 0 = 0 and Y0 := D − D is a linear subspace. Moreover, Y = Y0 + Rk 0 . Suppose that Y0 = Y . Let y ∈ Y0 ⊂ Y . Then y = v + λk 0 for some v ∈ D and λ ∈ R, whence λk 0 = y − v ∈ Y0 − D ⊂ Y0 . Since k 0 ∈ / Y0 (otherwise, Y0 + Rk 0 = Y0 ), we obtain that λ = 0, and so Y0 ⊂ D ⊂ Y0 . This contradicts the fact that D is not a linear subspace. Hence Y0 = Y . Let us show now that Di = ∅. First, because k 0 ∈ D −D, k 0 = v1 −v2 with v1 , v2 ∈ D. Let v := v1 + v2 ∈ D. Consider λ, µ ≥ 0. Then

2.4 Convexity notions for sets and multifunctions

D + λk 0 = D + 2λv1 − λv ⊂ D − R+ v,

45

D − µk 0 = D + 2µv2 − µv ⊂ D − R+ v.

Therefore Y = D − R+ v. Consider y ∈ Y ; then y + v ∈ Y = D − R+ v, and so there exists λ ≥ 0 such that (1 + λ)v + y ∈ D, whence (1 + λ)−1 y + v ∈ D. Since D is convex, this implies that v ∈ Di . Hence Di = ∅. Suppose by contradiction that {k 0 , −k 0 } ∩ Di = ∅. By the algebraic separation theorem, there exist ϕ, ψ ∈ X
\ {0} (linear functionals on X) such that ϕ(k 0 ) ≤ 0 ≤ ϕ(y),

ψ(−k 0 ) ≤ 0 ≤ ψ(y)

∀ y ∈ D.

(2.32)

If ϕ(k 0 ) = 0, then ϕ(y + λk 0 ) ≥ 0 for all y ∈ D and λ ∈ R, and so we get the contradiction ϕ = 0 because D + Rk 0 = Y . Hence ϕ(k 0 ) < 0. Similarly, ψ(−k 0 ) < 0. Therefore there exists α > 0 such that (αϕ + ψ)(k 0 ) = 0. From (2.32) we obtain that (αϕ + ψ)(y) ≥ 0 for every y ∈ D. As above, it follows that αϕ+ψ = 0. Hence ϕ(y−y
) = ϕ(y)+α−1 ψ(y
) ≥ 0 for all y, y
∈ D. Since D − D = Y , we obtain the contradiction ϕ = 0. Hence {k 0 , −k 0 } ∩ Di = ∅.
Remark 2.3.8. By a similar proof, if D ⊂ Y is a convex cone and k 0 ∈ Y , then D − R+ k 0 = Y ⇔ k 0 ∈ D i . The function ϕD,k0 defined by (2.23) was introduced by Gerstewitz (Tammer) and Iwanow [129]. The most part of the properties of this function established in Theorem 2.3.1 and its corollaries were stated by Z˘alinescu [388], Gerth (Tammer), and Weidner [131], Tammer [337] and G¨opfert, Tammer, and Z˘ alinescu [144]. Theorem 2.3.6 is stated by Gerth (Tammer), and Weidner [131]; here one can find versions of nonconvex separation theorems without interiority conditions. Proposition 2.3.7 can be found in Z˘alinescu [388].

2.4 Convexity Notions for Sets and Multifunctions Let X be a topological vector space over the reals. Definition 2.4.1. Let A ⊂ X be a nonempty set. We say that A is α-convex, where α ∈ ]0, 1[, if αx + (1 − α)y ∈ A for all x, y ∈ A. The set A is midconvex if A is 12 -convex. The set A is nearly convex if A is α-convex for some α ∈ ]0, 1[. The empty set is α-convex for all α ∈ ]0, 1[ (and so nearly convex). Of course, A is convex if!and only if A is α-convex for every α ∈ ]0, 1[. Let α ∈ ]0, 1[ and Λα := n≥0 Λα n , where Λα 0 := {0, 1},

α Λα n+1 := {αt + (1 − α)s | t, s ∈ Λn }, ∀ n ≥ 0.

α α It is obvious that Λα n ⊂ Λn+1 for every n ≥ 0 and Λ is α-convex. Moreover, if A is α-convex, then λx + (1 − λ)y ∈ A for all x, y ∈ A and λ ∈ Λα . Indeed, fixing x, y ∈ A and taking Λx,y := {λ ∈ [0, 1] | λx + (1 − λ)y ∈ A}, one obtains easily, by induction, that Λα n ⊂ Λx,y for every n ≥ 0.

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2 Functional analysis over cones

Lemma 2.4.2. Let Λ ⊂ [0, 1] be such that 0, 1 ∈ Λ and ∃ δ ∈ ]0, 1/2], ∀ t, s ∈ Λ, ∃ λ ∈ [δ, 1 − δ] : λt + (1 − λ)s ∈ Λ. Then cl Λ = [0, 1]. In particular, cl Λα = [0, 1]. Proof. Suppose that cl Λ = [0, 1]. Then there exists t ∈ [0, 1] \ cl Λ = ]0, 1[ \ cl Λ. Since the last set is open in R, it follows that α := sup{t ∈ [0, t] | t ∈ cl Λ} < t,

β := inf{t ∈ [t, 1] | t ∈ cl Λ} > t

and α, β ∈ cl Λ. Let δ := δ(β − α) > 0. From the definitions of α and β, there exist α, β ∈ Λ such that α − δ < α ≤ α, β ≤ β < β + δ. Since α, β ∈ Λ, there exists λ ∈ [δ0 , 1 − δ0 ] with λα + (1 − λ)β ∈ Λ. But λα + (1 − λ)β ≤ λα + (1 − λ)(β + δ) = β + δ − λ(β − α + δ) ≤ β + δ − δ(β − α + δ) = β − δδ < β, λα + (1 − λ)β ≥ λ(α − δ) + (1 − λ)β = α − δ + (1 − λ)(β − α + δ) ≥ α − δ + δ(β − α + δ) = α + δδ > α, contradicting the fact that Λ∩ ]α, β[ = ∅. Therefore cl Λ = [0, 1]. It is obvious that Λ = Λα satisfies the hypothesis with δ = min{α, 1 − α}.
The first statement of the next result can be found in Z˘alinescu [390, Prop. 2.4]. Proposition 2.4.3. Let X be a topological vector space and A ⊂ X a nonempty nearly convex set. Then (i) cl A is convex. (ii) If x ∈ aint A and y ∈ A, then [x, y] ⊂ A. Moreover, if x ∈ int A and y ∈ A, then [x, y[ ⊂ int A. (iii) If int A = ∅, then int A is convex and aint A = int A. Proof. By hypothesis there exists α ∈ ]0, 1[ such that A is α-convex. Note first that for all γ ∈ Λα and x, y ∈ A we have that γx + (1 − γ)y ∈ A. (i) Consider x, y ∈ cl A and λ ∈ ]0, 1[. Since λ ∈ cl Λα , there exist the nets (xi ), (yi ) ⊂ A, and (λi ) ⊂ Λα converging to x, y, and λ, respectively. Since λi xi +(1−λi )yi ∈ A for every i, we obtain that λx+(1−λ)y ∈ cl A. Therefore cl A is convex. (ii) Let x ∈ aint A and y ∈ A. Suppose that there exists λ0 ∈ ]0, 1[ such / A; of course, x = y. Since x ∈ aint A, there exists that λ0 x + (1 − λ0 )y ∈ δ ∈ ]0, 1[ such that (1 − λ)x + λy ∈ A for every λ ∈ [−δ, δ]. Let γ := sup{γ ∈ ]0, 1[ | (1 − λ)x + λy ∈ A ∀ λ ∈ [0, γ[}. Of course, δ ≤ γ ≤ λ0 and (1 − λ)x + λy ∈ A for every λ ∈ [0, γ[. Let γ ∈ Λα ∩ [0, γ[. It follows that for all λ ∈ [0, γ[ we have

2.4 Convexity notions for sets and multifunctions

47

(1 − γ) ((1 − λ)x + λy) + γy = (1 − (λ + γ − λγ)) x + (λ + γ − λγ)y ∈ A. But γ ∈ [γ, γ + γ(1 − γ)[ = {λ + γ − λγ | λ ∈ [0, γ[}. From the above relation it follows that (1 − λ)x + λy ∈ A for every λ ∈ [0, γ + γ(1 − γ)[, contradicting the choice of γ. Now let x ∈ int A, y ∈ A, and λ ∈ ]0, 1[. There exists a neighborhood V of the origin such that x + V ⊂ int A ⊂ aint A. From the first part we have λx + (1 − λ)y + λV ⊂ A, which shows that λx + (1 − λ)y ∈ int A. (iii) Suppose that int A = ∅ and fix x0 ∈ int A. Consider x ∈ aint A. There exist y ∈ A and λ ∈ ]0, 1[ such that x = (1 − λ)x0 + λy. From the second part of (ii) we get that x ∈ int A.
An immediate consequence of the preceding proposition is the next corollary. Corollary 2.4.4. Let X be a topological vector space and A ⊂ X a nonempty nearly convex set. If A is open or closed, then A is convex.
Remark 2.4.5. If the nonempty subset A of the t.v.s. X satisfies the condition ∃ δ ∈ ]0, 1/2], ∀ x, y ∈ A, ∃ λ ∈ [δ, 1 − δ] : λx + (1 − λ)y ∈ A, then cl A is convex, but int A is not necessarily convex. Indeed, the proof of the fact that cl A is convex is similar to that of Proposition 2.4.3, taking into account Lemma 2.4.2. The set A = R \ {0} satisfies the above condition, is open, but is not convex. Note that the intersection of an arbitrary family of α-convex sets is also α-convex, but the intersection of two nearly convex sets may not be nearly 1 1 convex; indeed, Λ 2 ∩ Λ 3 = {0, 1} is not nearly convex. If X, Y are real vector spaces, T : X → Y is a linear operator and A ⊂ X, B ⊂ Y are α-convex, then T (A) and T −1 (B) are α-convex, too. Let X, Y be arbitrary nonempty sets. A function Γ : X → 2Y is called a multifunction, and is denoted by Γ : X ⇒ Y . So, if Γ : X ⇒ Y is a multifunction, the image of Γ at x ∈ X is a (possibly empty) subset Γ (x) of Y . The domain of Γ is dom Γ := {x ∈ X | Γ (x) = ∅}, while its image is Im Γ := {y ∈ Y | ∃ x ∈ X : y ∈ Γ (x)}. The multifunction Γ : X ⇒ Y is usually identified with its graph, gr Γ := {(x, y) ∈ X × Y | y ∈ Γ (x)}. In this way, with each multifunction one associates a relation and vice versa. It is obvious that dom Γ = projX (gr! Γ ) and Im Γ = projY (gr Γ ). The image by Γ of the set A ⊂ X is Γ (A) := x∈A Γ (x); so, Im Γ = Γ (X). The inverse image by Γ of the set B ⊂ Y is Γ −1 (B) := {x ∈ X | Γ (x) ∩ B = ∅}. In fact, Γ −1 (B) is the image of B by the inverse multifunction Γ −1 : Y ⇒ X defined by Γ −1 (y) := {x ∈ X | y ∈ Γ (x)}; so Γ −1 (y) = Γ −1 ({y}). Of course, dom Γ −1 = Im Γ and Im Γ −1 = dom Γ . We shall use in the sequel also another type of inverse image: Γ +1 (B) := {x ∈ X | Γ (x) ⊂ B}. When ∆ : Y ⇒ Z is another multifunction, the composition of ∆ and Γ is the

48

2 Functional analysis over cones

multifunction ∆ ◦ Γ : X ⇒ Z defined by ∆ ◦ Γ (x) := {z ∈ Z | ∃ y ∈ Γ (x) with z ∈ ∆(y)}; note that gr(∆ ◦ Γ ) = PrX×Z gr Γ × Z ∩ X × gr ∆ . When Y is a linear space and Γ, Γ1 , Γ2 : X ⇒ Y , we define the sum Γ1 + Γ2 and multiplication by a scalar γΓ as the multifunctions Γ1 + Γ2 , γΓ : X ⇒ Y defined by (Γ1 + Γ2 )(x) := Γ1 (x) + Γ2 (x), (γΓ )(x) := γ · Γ (x) with the usual conventions that A + ∅ := ∅ + A := ∅, γ · ∅ = ∅ for A ⊂ Y . Suppose now that X, Y are real vector spaces and Γ : X ⇒ Y . We say that Γ is α-convex (mid-convex, nearly convex, convex) if gr Γ is αconvex (mid-convex, nearly convex, convex). It is obvious that if Γ is α-convex (mid-convex, nearly convex, convex), so are dom Γ and Im Γ . It is easy to see that Γ is α-convex if and only if αΓ (x) + (1 − α)Γ (y) ⊂ Γ (αx + (1 − α)y)

∀ x, y ∈ dom Γ.

Let C ⊂ Y be a convex cone. We say that Γ is C-α-convex (C-midconvex, C-nearly convex, C-convex) if the multifunction ΓC : X ⇒ Y,

ΓC (x) := Γ (x) + C,

is α-convex (mid-convex, nearly convex, convex). Of course, Γ is C-α-convex if and only if αΓ (x) + (1 − α)Γ (y) ⊂ Γ (αx + (1 − α)y) + C

∀ x, y ∈ dom Γ.

By inverting the inclusion in these definitions we get concavity notions for multifunctions; so Γ is α-concave or C-α-concave if αΓ (x) + (1 − α)Γ (y) ⊃ Γ (αx + (1 − α)y)

∀ x, y ∈ dom Γ

or αΓ (x) + (1 − α)Γ (y) + C ⊃ Γ (αx + (1 − α)y)

∀ x, y ∈ dom Γ,

respectively. Note that sometimes gr ΓC is denoted by epiC Γ , or simply epi Γ , and is called the epigraph of Γ . The sublevel set of Γ of height y (w.r.t. C) is the set levΓ (y) := {x ∈ X | Γ (x) ∩ (y − C) = ∅}; when Y is a topological vector space and int C = ∅ we also consider the strict sublevel set of Γ of height y (w.r.t. C) defined by lev< Γ (y) := {x ∈ X | Γ (x) ∩ (y − int C) = ∅}. In this way we get the sublevel and strict sublevel multifunctions levΓ , lev< Γ : Y ⇒ X. We say that Γ is C-α-quasiconvex (C-mid-quasiconvex, C-nearly quasiconvex, C-quasiconvex) if for every z ∈ Y the sublevel set levΓ (z)

2.4 Convexity notions for sets and multifunctions

49

is α-convex (mid-convex, nearly convex, convex). An equivalent definition of C-α-quasiconvexity is that (Γ (x) + C) ∩ (Γ (y) + C) ⊂ Γ (αx + (1 − α)y) + C

∀ x, y ∈ dom Γ.

Notice that Γ is C-α-quasiconvex if Γ (x) ⊂ Γ (αx + (1 − α)y) + C or Γ (y) ⊂ Γ (αx + (1 − α)y) + C for all x, y ∈ dom Γ . Note also that Γ is C-α-quasiconvex (C-mid-quasiconvex, C-nearly quasiconvex, C-quasiconvex) whenever Γ is C-α-convex (C-midconvex, C-nearly convex, C-convex). In order to characterize the C-α-quasiconvexity of Γ we use the scalarization function ϕ introduced in relation (2.23) (for D replaced by C). Proposition 2.4.6. ([246, Prop. 2.3], [69, Lemma 2.3]) Let X, Y be topological vector spaces, Γ : X ⇒ Y have nonempty domain, C ⊂ Y be a convex cone, and k 0 ∈ int C be fixed. Then Γ is C-α-quasiconvex if and only if ϕz ◦ Γ : X ⇒ R is R+ -α-quasiconvex for every z ∈ Y , where ϕz (y) := ϕ(y − z) and ϕz ◦ Γ (x) := {ϕz (y) | y ∈ Γ (x)}. Proof. For t ∈ R and z ∈ Y we have that levϕz ◦Γ (t) = {x ∈ X | ϕz ◦ Γ (x) ∩ (t − R+ ) = ∅} = {x ∈ M | ∃ y ∈ Γ (x) : ϕz (y) ≤ t} = {x ∈ M | ∃ y ∈ Γ (x) : ϕ(y − z) ≤ t} = {x ∈ M | ∃ y ∈ Γ (x) : y − z ∈ tk 0 − C} (see Corollary 2.3.5) = {x ∈ M | Γ (x) ∩ (z + tk 0 − C) = ∅} = levΓ (z + tk 0 ). The conclusion follows.
We introduce some useful notions and notation related to vector-valued functions. To Y we adjoin a greatest element ∞ (∈ / Y ), thereby obtaining Y • := Y ∪ {∞} and C • := C ∪ {∞}. We consider that y + ∞ = ∞, λ · ∞ = ∞, and y ≤C ∞ for all y ∈ Y • and λ ∈ R+ . Of course, if f : X → Y • , the domain of f is dom f := {x ∈ X | f (x) ∈ Y }, the sublevel and strict sublevel sets of f of height y are levf (y) := {x ∈ X | f (x) ≤C y} and lev< f (y) := {x ∈ X | y −f (x) ∈ int C}, and the epigraph of f is epi f := {(x, y) ∈ X ×Y | f (x) ≤C y}. With such an f we associate the multifunction Γf,C : X ⇒ Y whose graph is epi f . So, Γf,C (x) = f (x) + C for every x ∈ dom Γf,C = dom f , epi f = < epi Γf,C , levf (y) = levΓf,C (y), and lev< f (y) = levΓf,C (y) for every y ∈ Y ; in particular, Γf,C = (Γf,C )C . We say that f : X → Y • is C-α-convex (C-midconvex, C-nearly convex, C-convex, C-α-quasiconvex, C-mid-quasiconvex, C-nearly quasiconvex, C-quasiconvex) if the multifunction Γf,C is C-α-convex (C-mid-convex, C-nearly convex, C-convex, C-α-quasiconvex,

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2 Functional analysis over cones

C-mid-quasiconvex, C-nearly quasiconvex, C-quasiconvex); in particular, f is C-convex if and only if f (αx1 + (1 − α)x2 ) ≤C αf (x1 ) + (1 − α)f (x2 )

∀ x1 , x2 ∈ X, ∀ α ∈ [0, 1].

If f is C-α-convex (C-mid-convex, C-nearly convex, C-convex), then dom f is so, and f is C-α-quasiconvex (C-mid-quasiconvex, C-nearly quasiconvex, C-quasiconvex). We also remark that f is C-α-quasiconvex, provided that   {f (x), f (y)} ∩ f (αx + (1 − α)y) + C = ∅ ∀ x, y ∈ dom f. For the converse we need the condition C∪(−C) = Y . Indeed, let x, y ∈ dom f ; suppose that f (x) ∈ / f (αx + (1 − α)y)+C, that is, αx+(1−α)y ∈ / levf (f (x)). / levf (f (x)), which means Since levf (f (x)) is α-convex, we deduce that y ∈ that f (x) − f (y) ∈ / C. This fact, together with C ∪ (−C) = Y , implies that f (y) − f (x) ∈ C, that is, x ∈ lef f (f (y)). Since lef f (f (y)) is α-convex, we obtain that f (y) ∈ f (αx + (1 − α)y) + C, which completes the proof. This remark shows that R+ -(α-)quasiconvexity of real-valued functions reduces to usual (α-)quasiconvexity. So, in Proposition 2.4.6 we may replace the R+ -α-quasiconvexity of ϕz ◦ f with the usual α-quasiconvexity. Let Y be ordered by the proper convex cone C and f : X → Y • . Definition 2.4.7. We call the subdifferential of f at x0 ∈ dom f the set ∂ ≤C f (x0 ) := {T ∈ L(X, Y ) | T (x − x0 ) ≤C f (x) − f (x0 ) ∀ x ∈ X}, where L(X, Y ) denotes the class of all continuous linear operators from X into Y . It is obvious that ∂ ≤C f (x0 ) is a convex subset of L(X, Y ). When f : X → Y is a sublinear operator; i.e., f (x1 + x2 ) ≤C f (x1 ) + f (x2 ), f (0) = 0, and f (αx) = αf (x) for all x, x1 , x2 ∈ X and α ∈ (0, ∞), and C is pointed, the formula ∂ ≤C f (x0 ) = {T ∈ ∂ ≤C f (0) | T (x0 ) = f (x0 )} •

holds for all x0 ∈ dom f . The following important result was stated by Valadier [360]. Theorem 2.4.8. Let (X, · ) and (Y, · ) be real reflexive Banach spaces and C ⊂ Y a proper convex cone with a weak compact base. If f : X → Y • is a C-convex operator, continuous at some point of its domain, then y ∗ ◦ ∂ ≤C f (x) = ∂(y ∗ ◦ f )(x)

∀ x ∈ int(dom f ), ∀ y ∗ ∈ C + ,

where we use the convention that y ∗ (∞) = ∞ for y ∗ ∈ C + .

2.5 Continuity notions for multifunctions

51

2.5 Continuity Notions for Multifunctions In this section X and Y are separated (in the sense of Hausdorff) topological spaces and Γ : X ⇒ Y a multifunction. When mentioned explicitly, Y is a separated topological vector space (s.t.v.s.). Definition 2.5.1. Let x0 ∈ X. We say that (a) Γ is upper continuous (u.c.) at x0 if ∀ D ⊂ Y, D open, Γ (x0 ) ⊂ D, ∃ U ∈ VX (x0 ) ∀ x ∈ U : Γ (x) ⊂ D, (2.33) i.e., Γ +1 (D) is a neighborhood of x0 for each open set D ⊂ Y such that Γ (x0 ) ⊂ D; (b) Γ is lower continuous (l.c.) at x0 if ∀ D ⊂ Y, D open, Γ (x0 )∩D = ∅, ∃ U ∈ VX (x0 ), ∀ x ∈ U : Γ (x)∩D = ∅, (2.34) i.e., Γ −1 (D) is a neighborhood of x0 for each open set D ⊂ Y such that Γ (x0 ) ∩ D = ∅. (c) Γ is continuous at x0 if Γ is u.c. and l.c. at x0 . (d) Γ is upper continuous (lower continuous, continuous) if Γ is so at every x ∈ X; (e) Γ is lower continuous at (x0 , y0 ) ∈ X × Y if ∀ V ∈ VY (y0 ), ∃ U ∈ VX (x0 ), ∀ x ∈ U : Γ (x) ∩ V = ∅. It follows from the definition that x0 ∈ int(dom Γ ) and y0 ∈ cl (Γ (x0 )) if Γ is l.c. at (x0 , y0 ) and Γ is l.c. at x0 ∈ dom Γ if and only if Γ is l.c. at every (x0 , y) with y ∈ Γ (x0 ); moreover, Γ is l.c. at every x0 ∈ X \ dom Γ . If x0 ∈ X \ dom Γ , then Γ is u.c. at x0 if and only if x0 ∈ int(X \ dom Γ ). So, if Γ is u.c., then dom Γ is closed, while if Γ is l.c., then dom Γ is open. The next result follows immediately from the definitions. Proposition 2.5.2. (i) Γ is upper continuous if and only if Γ +1 (D) is open for every open set D ⊂ Y ; (ii) Γ is lower continuous if and only if Γ −1 (D) is open for every open set D ⊂Y. The limit inferior of Γ at x0 ∈ X is defined by lim inf Γ (x) := {y ∈ Y | ∀ V ∈ VY (y), ∃ U ∈ VX (x0 ), x→x0

∀ x ∈ U • : Γ (x) ∩ V = ∅}, while the limit superior of Γ at x0 ∈ X is defined by

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2 Functional analysis over cones

lim sup Γ (x) :={y ∈ Y | ∀ V ∈ VY (y), ∀ U ∈ VX (x0 ), x→x0

=

∃ x ∈ U • : Γ (x) ∩ V = ∅}

 U ∈VX (x0 )

cl (Γ (U • )) ,

where for U ∈ VX (x0 ), U • := U \ {x0 }. Note that if lim supx→x0 Γ (x) = ∅, then x0 is an accumulation point of dom Γ , while if lim inf x→x0 Γ (x) = ∅, then x0 ∈ int (dom Γ ∪ {x0 }); if x0 is an isolated point of X (i.e., {x0 } ∈ VX (x0 )), then lim inf x→x0 Γ (x) = Y . Of course, if x0 is an accumulation point of dom Γ , then lim inf Γ (x) ⊂ lim sup Γ (x), x→x0

x→x0

both of them being closed sets. Sometimes in the definitions of lim inf x→x0 Γ (x) and lim supx→x0 Γ (x) one takes x ∈ dom Γ . Note that this situation reduces to the preceding one by considering the restriction of Γ at {x0 } ∪ dom Γ . Recall first that if (X, d) is a metric space and A, B ⊂ X, then the excess of A over B is e(A, B) = sup dist(x, B) if A, B = ∅,

e(∅, B) = 0,

x∈A

e(A, ∅) = ∞ if A = ∅,

where dist(x, A) := inf a∈A d(x, a); in particular, dist(x, ∅) = ∞. It is simple to show that for a nonempty and compact set A ⊂ X and an open set D ⊂ X, if A ⊂ D, then there exists ε > 0 such that Aε := {x ∈ X | dist(x, A) < ε} = {x | A ∩ B(x, ε) = ∅} ⊂ D, where B(x, ε) := {x
∈ X | d(x, x
) < ε}. In particular, cases for X or/and Y one has useful characterizations for the elements of lim inf x→x0 Γ (x) and lim supx→x0 Γ (x). Proposition 2.5.3. Let x0 ∈ X and y ∈ Y . (i) y ∈ lim inf x→x0 Γ (x) if and only if ∀ X \ {x0 } ⊃ (xi )i∈I → x0 , ∃ (xϕ(j) )j∈J , ∃ Y ⊃ (yj )j∈J → y, ∀ j ∈ J : yj ∈ Γ (xϕ(j) ), (2.35) and y ∈ lim supx→x0 Γ (x) if and only if ∃ X \ {x0 } ⊃ (xi )i∈I → x0 , ∃ Y ⊃ (yi )i ∈I → y, ∀ i ∈ I : yi ∈ Γ (xi ). (2.36) (ii) If X and Y are first countable, then y ∈ lim inf x→x0 Γ (x) if and only if ∀ X \ {x0 } ⊃ (xn ) → x0 , ∃ Y ⊃ (yn ) → y, ∃ n0 ∈ N, ∀ n ≥ n0 : yn ∈ Γ (xn ), (2.37) and y ∈ lim supx→x0 Γ (x) if and only if ∃ X \ {x0 } ⊃ (xn ) → x0 , ∃ Y ⊃ (yn ) → y, ∀ n ∈ N : yn ∈ Γ (xn ). (2.38)

2.5 Continuity notions for multifunctions

53

(iii) If (Y, ρ) is a metric space, then y ∈ lim inf Γ (x) ⇔ lim dist(y, Γ (x)) = 0,

(2.39)

y ∈ lim sup Γ (x) ⇔ lim inf dist(y, Γ (x)) = 0.

(2.40)

x→x0

x→x0

x→x0

x→x0

Proof. (i) Suppose that y ∈ lim inf x→x0 Γ (x) and take X \ {x0 } ⊃ (xi )i∈I → x0 . Then for every V ∈ VY (y) there exists iV ∈ I such that Γ (xi ) ∩ V = ∅ for i  iV . Consider J := I × VY (y) ordered by (i, V )  (i
, V
) if and only if i  i
and V ⊂ V
. Consider ϕ : J → I such that ϕ(i, V )  i, iV and take yj ∈ Γ (xϕ(j) ) ∩ V for j = (i, V ). It is obvious that (xϕ(j) )j∈J is a subnet of (xi )i∈I . Moreover, (yj )j∈J → y. Therefore (2.35) holds. Suppose now that (2.35) holds, but y ∈ / lim inf x→x0 Γ (x). Then there exists V0 ∈ VY (y) such that for every U ∈ VX (x0 ) there exists xU ∈ U • with Γ (xU )∩ V = ∅. Of course, (xU ) → x0 . Therefore there exists a subnet (xϕ(j) )j∈J and a net (yj )j∈J → y such that yj ∈ Γ (xϕ(j) ) for j ∈ J. Since V0 ∈ VY (y), there exists j0 ∈ J with yj ∈ V0 for j  j0 . It follows that yj0 ∈ Γ (xϕ(j0 ) ) ∩ V0 = ∅, a contradiction. Suppose now that y ∈ lim supx→x0 Γ (x). Then for every (U, V ) ∈ VX (x0 )× VY (y) =: I there exist xU,V ∈ U • and yU,V ∈ Γ (xU,V ) ∩ V . Defining, as usual, (U, V )  (U
, V
) iff U ⊂ U
and V ⊂ V
, it is obvious that (2.36) holds. The converse implication is obtained easily by contradiction. (ii) Consider (Un )n∈N and (Vn )n∈N decreasing bases of neighborhoods for x0 and y, respectively. Suppose first that y ∈ lim inf x→x0 Γ (x). Then for k ∈ N there exists • mk ∈ N such that Γ (x) ∩ Vk = ∅ for x ∈ Um ; without loss of generality we k may suppose that (mk ) is increasing. Take X \ {x0 } ⊃ (xn ) → x0 . For every k ∈ N there exists nk ∈ N such that xn ∈ Umk for n ≥ nk ; once again, we may suppose that nk+1 > nk for every k. For k ∈ N and nk ≤ n < nk+1 take yn ∈ Γ (xn ) ∩ Vk . Of course, (yn ) → y and yn ∈ Γ (xn ) for n ≥ n0 . Conversely, suppose that the right-hand side of (2.37) holds, but y ∈ / lim inf x→x0 Γ (x). Then there exists W0 ∈ VY (y) such that for every U ∈ VX (x0 ) there exists xU ∈ U • with Γ (xU ) ∩ W0 = ∅. Therefore, for every n ∈ N there exists xn ∈ Un such that Γ (xn ) ∩ W0 = ∅. By hypothesis, there exists Y ⊃ (yn ) → y and n0 ∈ N such that yn ∈ Γ (xn ) for n ≥ n0 . Since (yn ) → y, yn ∈ W0 for every n ≥ n1 . Taking n = max{n0 , n1 }, we obtain that Γ (xn ) ∩ W0 = ∅, a contradiction. The proof of (2.38) is similar. (iii) Let Y be a metric space. Recall that for f : X → R and x0 ∈ X, lim inf f (x) := x→x0

So,

sup

inf • f (x).

U ∈VX (x0 ) x∈U

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2 Functional analysis over cones

y ∈ lim sup Γ (x) ⇔ ∀ ε > 0, ∀ U ∈ VX (x0 ), ∃ x ∈ U • : Γ (x) ∩ B(y, ε) = ∅ x→x0

⇔ ∀ ε > 0, ∀ U ∈ VX (x0 ), ∃ x ∈ U • : dist(y, Γ (x)) < ε ⇔ ∀ U ∈ VX (x0 ), ∀ ε > 0 : inf • dist(y, Γ (x)) < ε x∈U

⇔ lim inf dist(y, Γ (x)) = 0, x→x0

i.e., (2.40) holds. The proof of (2.39) is similar.
Using the above remark, we have the following characterization of upper continuity in a special case. Proposition 2.5.4. Suppose that (Y, ρ) is a metric space and Γ (x0 ) is compact. Then Γ is u.c. at x0 if and only if limx→x0 e(Γ (x), Γ (x0 )) = 0. Proof. The (easy) proof is left to the reader.
In the next results we give (other) characterizations for upper and lower continuity at a point. Proposition 2.5.5. Let X, Y be first countable (in particular, let X, Y be metric spaces) and x ∈ dom Γ . The following statements are equivalent: (i) the multifunction Γ is u.c. at x; (ii) for every closed set F ⊂ Y with Γ (x) ∩ F = ∅, there exists U ∈ VX (x) such that Γ (U ) ∩ F = ∅; (iii) for every closed set F ⊂ Y and every sequence X ⊃ (xn ) → x with Γ (xn ) ∩ F = ∅ for n ∈ N, we have Γ (x) ∩ F = ∅; (iv) for every open set D ⊂ Y with Γ (x) ⊂ D, and every sequence X ⊃ (xn ) → x there exists nD ∈ N such that Γ (xn ) ⊂ D for n ≥ nD ; (v) for all sequences (xn ) ⊂ X, (yn ) ⊂ Y with (xn ) → x and yn ∈ Γ (xn ) \ Γ (x) for n ∈ N, there exists a subsequence (ynk ) → y ∈ Γ (x). If X, Y are not first countable, the conditions (i)–(iv) remain equivalent by replacing sequences by nets. Proof. The equivalence of conditions (i)–(iv) is quite simple (and known), so it is left to the reader. (iii) ⇒ (v) Consider (Vn )n∈N a countable basis of neighborhoods for y ∈ Y ; we suppose (without loss of generality) that Vn+1 ⊂ Vn for every n ∈ N. Let (xn ) ⊂ X, (yn ) ⊂ Y with (xn ) → x and yn ∈ Γ (xn ) \ Γ (x) for n ∈ N, and consider F := cl{yn | n ∈ N}. Since F is closed and Γ (xn ) ∩ F = ∅ for every n ∈ N, there exists y ∈ Γ (x)∩F . Of course, yn = y for every n ∈ N. Since y ∈ F , there exists yn1 ∈ V1. Since Y is separated, there exists m1 ∈ N such that Vm1 ∩ {yk | 0 ≤ k ≤ n1 } = ∅. There exists n2 ∈ N such that yn2 ∈ Vm1 . Similarly, there exists m2 ∈ N such that Vm2 ∩ {yk | 0 ≤ k ≤ n2 } = ∅. There exists n3 ∈ N such that yn3 ∈ Vm2 . The choice of m2 shows that m2 > m1 and n3 > n2 . Continuing in this way we find increasing sequences (mk ), (nk ) ⊂ N such that ynk ∈ Vmk for every k ∈ N. It is obvious that (ynk ) → y.

2.5 Continuity notions for multifunctions

55

(v) ⇒ (iv) In the contrary case, there exist an open set D ⊂ Y with Γ (x) ⊂ D and a sequence X ⊃ (xn ) → x such that P := {n ∈ N | Γ (xn ) ⊂ D} is infinite. For each n ∈ P take yn ∈ Γ (xn ) \ D ⊂ Γ (xn ) \ Γ (x). Letting P = {n0 , n1 , . . . , nk , . . .} with n0 < n1 < · · · < nk < · · · , by (v) there exists the subsequence (ynkp ) → y ∈ Γ (x) \ D, a contradiction.
Proposition 2.5.6. Let (x0 , y0 ) ∈ X ×Y . The following statements are equivalent: (i) the multifunction Γ is l.c. at (x0 , y0 ); (ii) for every net (xi )i∈I → x0 there exist a subnet (xϕ(j) )j∈J of (xi ) and a net (yj )j∈J → y0 such that yj ∈ Γ (xϕ(j) ) for j ∈ J. (iii) y0 ∈ lim inf x→x0 Γ (x) ∩ cl Γ (x0 ). Suppose now that X, Y are first countable. Then (i) is equivalent to (iv) for every sequence X ⊃ (xn ) → x0 there exist a sequence Y ⊃ (yn ) → y0 and n0 ∈ N such that yn ∈ Γ (xn ) for every n ≥ n0 . Proof. The proofs of the equivalences (i) ⇔ (ii) and (i) ⇔ (iv) are, mainly, the same as those of the first parts of Proposition 2.5.3 (i) and (ii), respectively. The equivalence of (i) and (iii) is immediate.
From the preceding result we obtain that Γ is l.c. at x0 ⇔ Γ (x0 ) ⊂ lim inf Γ (x). x→x0

Definition 2.5.7. We say that (i) Γ is closed if gr Γ is a closed subset of X × Y ; (ii) Γ is closed at x ∈ X if for every net ((xi , yi ))i∈I ⊂ gr Γ converging to (x, y) we have that y ∈ Γ (x); (iii) Γ is closed-valued if Γ (x) is closed for every x ∈ X; (iv) Γ is compact at x ∈ X if for every net ((xi , yi ))i∈I ⊂ gr Γ with (xi ) → x, there exists a subnet (yϕ(j) )j∈J converging to some y ∈ Γ (x). Of course, Γ is closed if and only if Γ is closed at every x ∈ X; moreover, if Γ is closed, then Γ is closed-valued. If Γ is compact at x, then Γ is closed at x; moreover, if Γ is closed (compact) at x, then Γ (x) is closed (compact). Note also that Γ is compact at x ∈ X \ dom Γ if and only if x ∈ int(X \ dom Γ ), but Γ may be closed at some x ∈ cl(dom Γ ) \ dom Γ . Indeed, consider Γ : R ⇒ R with dom Γ = ]0, 1[ and Γ (x) = {x−1 } for x ∈ dom Γ ; Γ is closed at 0 (but not compact). Proposition 2.5.8. Assume that ∅ = A ⊂ dom Γ and Γ is u.c. at every x ∈ A. (i) If A is compact and Γ (x) is compact for every x ∈ A, then Γ (A) is compact.

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2 Functional analysis over cones

(ii) If A is connected and Γ (x) is connected for every x ∈ A, then Γ (A) is connected. Recall that the nonempty subset A of the topological space (X, τ ) is connected if A ∩ D1 ∩ D2 = ∅ whenever D1 , D2 are open subsets of X such that A ∩ D1 = ∅, A ∩ D2 = ∅ and A ⊂ D1 ∪ D2 . Recall that the connected subsets of R are the intervals, while in a topological vector space every convex set is connected. Proof. Replacing Γ by ΓA : A ⇒ Y defined by ΓA (x) := Γ (x) for x ∈ A, we may assume that A = X = dom Γ . (i) Let (Di )i∈I be an open cover of Γ (X). Then for every x ∈ X we have that Γ (x) ⊂ ∪i∈I Di . Since Γ (x) is compact, there exists a finite set Ix ⊂ I such that Γ (x) ⊂ ∪i∈Ix Di =: Dx ; of course, x ∈ Γ +1 (Dx). Hence X  = ∪x∈X Γ +1 (Dx ). Since Dx is open, Γ +1 (Dx ) is open, and so Γ +1 (Dx ) x∈X is an open cover of the compact set X. Therefore there exists a finite set X0 ⊂ X such that X = ∪x∈X0 Γ +1 (Dx ). Let I0 := ∪x∈X0 Ix . Of course, I0 is finite. Let y ∈ Γ (X); hence y ∈ Γ (x) for some x ∈ X = ∪x ∈X0 Γ +1 (Dx ). Therefore there exists x0 ∈ X0 such that x ∈ Γ +1 (Dx0 ). It follows that y ∈ Γ (x) ⊂ Dx0 = ∪i∈Ix Di ⊂ ∪i∈I0 Di . Hence Γ (X) is compact. (ii) Now let D1 , D2 ⊂ Y be open sets such that Γ (X) ∩ D1 = ∅, Γ (X) ∩ D2 = ∅ and Γ (X) ⊂ D1 ∪ D2 . We have to show that Γ (X) ∩ D1 ∩ D2 = ∅. Assume that Γ (x) ∩ D1 ∩ D2 = ∅ for every x ∈ X. Because Γ (x) is connected and Γ (x) ⊂ D1 ∪ D2 we have that Γ (x) ∩ D1 = ∅ or Γ (x) ∩ D2 = ∅; hence Γ (x) ⊂ D2 or Γ (x) ⊂ D1 . It follows that X = Γ +1 (D1 ) ∪ Γ +1 (D2 ). From our hypotheses we have that Γ +1 (D1 ) and Γ +1 (D2 ) are nonempty open sets. Because X = A is connected we obtain that there is some x0 ∈ Γ +1 (D1 ) ∩ Γ +1 (D2 ). Then Γ (x0 ) ⊂ D1 ∩D2 , whence the contradiction Γ (x0 )∩D1 ∩D2 = ∅. Therefore Γ (x) ∩ D1 ∩ D2 = ∅ for some x ∈ X, and so Γ (X) ∩ D1 ∩ D2 = ∅. Hence Γ (X) is connected.
When Y is a metric space or a topological vector space, Proposition 2.5.8(i) follows also from the next result. Proposition 2.5.9. Let x ∈ X. The following assertions hold: (i) Γ is compact at x if and only if Γ (x) is compact and Γ is u.c. at x. (ii) If (Y, ρ) is a metric space, then Γ is compact at x if and only if Γ (x) is compact and limx →x e(Γ (x
), Γ (x)) = 0. (iii) Let Y be first countable. If Γ is compact at x, then for every sequence ((xn , yn ))n∈N ⊂ gr Γ with (xn ) → x, there exists a subsequence (ynk ) → y ∈ Γ (x). If X is first countable and Y is a metric space, the converse is true. Proof. If x ∈ / dom Γ , the hypotheses of every assertion of the proposition imply that x ∈ int(X \ dom Γ ); so the conclusion is obvious. So we suppose, during the proof, that x ∈ dom Γ . (i) We noted already that Γ (x) is compact. Suppose that Γ is not u.c. at x. Then there exists an open set D ⊂ Y such that for every U ∈ VX (x) there

2.5 Continuity notions for multifunctions

57

exists xU ∈ U and yU ∈ Γ (xU ) \ D. Since Γ is compact at x, there exists a subnet (yϕ(j) )j∈J converging to y ∈ Γ (x). Since (yϕ(j) )j∈J ⊂ Y \ D and D is open, it follows that y ∈ Y \ D, contradicting Γ (x) ⊂ D. (ii) Let ((xi , yi ))i∈I ⊂ gr Γ with (xi ) → x. For every j ∈ I consider the set Fj := cl{yi | i  j}. Suppose that for some j0 ∈ I, Fj0 ∩ Γ (x) = ∅. Then Γ (x) ⊂ Y \ Fj0 . Since Y \ Fj0 is open and Γ is u.c. at x, there exists i0 ∈ I such that Γ (xi ) ⊂ Y \ Fj0 for every i  i0 . Taking i ∈ I such that i  i0 and i  j0 we get the contradiction yi ∈ Fj0 ∩(Y \Fj0 ). Therefore Fi ∩Γ (x) = ∅ for every i ∈ I. Since Γ (x) is compact and the " family (Fi ∩ Γ (x))i∈I has "the finite intersection property, there exists y ∈ i∈I (Fi ∩ Γ (x)) = Γ (x) ∩ i∈I Fi . It follows that for every i ∈ I and every V ∈ VY (y) there exists ϕ(i, V ) ∈ I such that yϕ(i,V ) ∈ V and ϕ(i, V )  i. Let J := I × VY (y) in which (i, V )  (i
, V
) iff i  i
and V ⊂ V
. Then (yϕ(j) )j∈J is a subnet of (yi )i∈I converging to y. The conclusion follows. (iii) Suppose that Γ is compact at x, but limx →x e(Γ (x
), Γ (x)) does not exist or is different from 0. Then there exists ε0 > 0 such that for every U ∈ VX (x) there exist xU ∈ U and yU ∈ Γ (xU ) such that d(yU , Γ (x)) := inf y∈Γ (x) ρ(yU , y) > ε0 . By hypothesis, there exists a subnet (yϕ(j) )j∈J → y ∈ Γ (x). Since d(·, Γ (x)) is continuous, we obtain the contradiction 0 = d(y, Γ (x)) ≥ ε0 . Conversely, suppose that Γ (x) is compact and limx →x e(Γ (x
), Γ (x)) = 0, and take the net ((xi , yi ))i∈I ⊂ gr Γ with (xi ) → x. It follows that limi∈I d(yi , Γ (x)) = 0. Therefore, for every ε > 0 there exists iε ∈ I such that d(yi , Γ (x)) < ε for i  iε . Let J := I× ]0, ∞[ ordered by (i, ε)  (i
, ε
) iff i  i
and ε ≤ ε
. For every j = (i, ε) ∈ J consider ϕ(j) ∈ I such that ϕ(j)  i, iε . Consider also y ϕ(j) ∈ Γ (x) such that ρ(yϕ(j) , y ϕ(j) ) < ε. Since Γ (x) is com    pact, y ϕ(j) contains a subnet y ϕ◦ψ(k) converging to y ∈ Γ (x). It k∈K   follows that yϕ◦ψ(k) k∈K → y. (iv) Let Y be first countable and Γ compact at x. Consider the sequence ((xn , yn ))n∈N ⊂ gr Γ such that (xn ) → x. By hypothesis there exists a subnet (yϕ(j) )j∈J → y ∈ Γ (x). There exists a base (Vk )k∈N of neighborhoods of y. For every k ∈ N there exists jk ∈ J such that yϕ(j) ∈ Vk for j  jk . Of course, taking n0 = ϕ(j0 ), for every k ∈ N there exists nk+1 ∈ N such that nk+1 ≥ max{ϕ(jk+1 ), nk + 1}. It is obvious that (ynk ) → y. For the converse implication we must show that limx →x e(Γ (x
), Γ (x)) = 0. Otherwise, since X is first countable, there exist ε0 > 0 and (xn ) → x such that e(Γ (xn ), Γ (x)) > ε0 for every n ∈ N. Proceeding as in the proof of the first part of (iii), we get a contradiction.
Related to closedness of multifunctions we have the following result. Proposition 2.5.10. Let x ∈ X. The following assertions hold: (i) Γ is closed at x if and only if for every y ∈ Y \Γ (x) there exist U ∈ VX (x) and V ∈ VY (y) such that Γ (U ) ∩ V = ∅. (ii) Γ is closed at x if and only and lim supx →x Γ (x
) ⊂ Γ (x) = cl Γ (x).

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2 Functional analysis over cones

(iii) Suppose that X, Y are first countable. Then Γ is closed at x if and only if for every sequence ((xn , yn ))n∈N → (x, y), y ∈ Γ (x). (iv) If Γ is u.c. at x and Γ (x) is compact, then Γ is closed at x. (v) If Y is regular (in particular, if Y is a metric space or a topological vector space), Γ is u.c. at x, and Γ (x) is closed, then Γ is closed at x. Proof. The proof, which is not difficult, is left to the reader.
The next result gives sufficient conditions for the upper continuity of the intersection of two multifunctions. Proposition 2.5.11. Consider Γ1 , Γ2 : X ⇒ Y and Γ (x) = Γ1 (x) ∩ Γ2 (x) for every x ∈ X. Then Γ is u.c. at x0 ∈ X if one of the following conditions holds: (i) Γ1 is closed at x0 , Γ2 is u.c. at x0 , and Γ2 (x0 ) is compact; (ii) Y is normal (in particular, Y is a metric space), Γ1 , Γ2 are u.c. at x0 , and Γ1 (x0 ), Γ2 (x0 ) are closed. Proof. (i) Let D ⊂ Y be an open set such that Γ (x0 ) ⊂ D. Since Γ1 is closed at x0 , for every y ∈ Γ2 (x0 ) \ Γ1 (x0 ) there exist Uy ∈ V! X (x0 ) and Vy ∈ VY (y) such that Γ1 (Uy )∩Vy = ∅. It follows that Γ2 (x0 ) ⊂ D ∪ y∈Γ2 (x0 )\Γ1 (x0 ) int Vy . Since Γ2 (x0 ) is!compact, there exist y1 , . . . , yn ∈ Γ2 (x0 ) \ Γ1 (x0 ) such that n Γ2 (x0 ) ⊂ D ∪ i=1 int ! Vyi . Since Γ2 is u.c. at x0 , there "n exists U0 ∈ VX (x0 ) n such that Γ2 (U0 ) ⊂ D∪ i=1 int Vyi . Let x ∈ U := U0 ∩ i=1 Uyi and y ∈ Γ (x). Since x ∈ Γ1 (Uyi ), y ∈ / Vyi for every i, 1 ≤ i ≤ n. It follows that, necessarily, y ∈ D. Therefore Γ is u.c. at x0 . (ii) Let D ⊂ Y be an open set such that Γ (x0 ) ⊂ D. Then (Γ1 (x0 ) \ D) ∩ (Γ2 (x) \ D) = ∅. Since the sets Γ1 (x0 ) \ D and Γ2 (x0 ) \ D are closed and Y is normal, there exist the disjoint open sets D1 , D2 such that Γi (x0 )\D ⊂ D1 for i = 1, 2. Since Γi is u.c. at x0 and Γi (x0 ) ⊂ D ∪ Di , there exists Ui ∈ VX (x0 ) such that Γi (Ui ) ⊂ D ∪ Di for i = 1, 2. Let x ∈ U1 ∩ U2 and y ∈ Γ (x). It follows that y ∈ (D ∪ D1 ) ∩ (D ∪ D2 ) = D. Therefore Γ is u.c. at x0 .
When Y is a topological vector space, VY denotes the class of balanced neighborhoods of 0 ∈ Y . Definition 2.5.12. Let Y be a topological vector space and x0 ∈ X. We say that (a) Γ is Hausdorff upper continuous (H-u.c.) at x0 if ∀ V ∈ VY , ∃ U ∈ VX (x0 ), ∀ x ∈ U : Γ (x) ⊂ Γ (x0 ) + V.

(2.41)

(b) Γ is Hausdorff lower continuous (H-l.c.) at x0 if ∀ V ∈ VY , ∃ U ∈ VX (x0 ), ∀ x ∈ U : Γ (x0 ) ⊂ Γ (x) + V.

(2.42)

(c) Γ is Hausdorff continuous at x0 if Γ is H-u.c. and H-l.c. at x0 .

2.5 Continuity notions for multifunctions

59

(d) Γ is Hausdorff upper continuous (Hausdorff lower continuous, Hausdorff continuous) if Γ is so at every x ∈ X. The above definition can be given when Y is a metric space, too; just replace V ∈ VY by ε > 0 and Γ (x0 ) + V by (Γ (x0 ))ε . Of course, if Y is a metric space, then Γ is H-u.c. at x0 if and only if limx→x0 e (Γ (x), Γ (x0 )) = 0, and Γ is H-l.c. at x0 if and only if limx→x0 e (Γ (x0 ), Γ (x)) = 0. Concerning the continuity of the sum of multifunctions and of the multiplication by scalars, we have the following result. Proposition 2.5.13. Let Y be a topological vector space, Γ, Γ1 , Γ2 : X ⇒ Y , x0 ∈ X, and α ∈ R. (i) If Γ is u.c. (resp. l.c., H-u.c., H-l.c.) at x0 , then αΓ is u.c. (resp. l.c., H-u.c., H-l.c.) at x0 . (ii) If Γ1 and Γ2 are l.c. (resp. H-u.c., H-l.c.) at x0 , then Γ1 + Γ2 is l.c. (resp. H-u.c., H-l.c.) at x0 . Proof. The proof of (i) is immediate. (ii) Assume that Γ1 and Γ2 are l.c. at x0 and let D ⊂ Y be an open set  such that Γ1 (x0 ) + Γ2 (x0 ) ∩ D = ∅. Let y1 ∈ Γ1 (x0 ) and y2 ∈ Γ2 (x0 ) be such that y1 + y2 ∈ D. Let V ∈ VY be such that y1 + y2 + V ⊂ D. There exists V0 ∈ VY with V0 + V0 ⊂ V . Because Γ1 and Γ2 are l.c. at x0 , there exist U1 , U2 ∈ VX (x0 ) suchthat Γi (x) ∩ (y  i + V0 ) = ∅ for i ∈ {1, 2} and all x ∈ Ui . It follows that Γ1 (x) + Γ2 (x) ∩ (y1 + V0 + y2 + V0 ) = ∅ for x ∈ U := U1 ∩ U2 ∈ VX (x0 ). Hence (Γ1 + Γ2 )(x) ∩ D = ∅ for x ∈ U , and so Γ1 + Γ2 is l.c. at x0 . Assume now that Γ1 and Γ2 are H-u.c. at x0 and take V ∈ VY . Consider V0 ∈ VY with V0 + V0 ⊂ V . Then there exist U1 , U2 ∈ VX (x0 ) such that Γi (x) ⊂ Γi (x0 ) + V0 for i ∈ {1, 2} and x ∈ Ui . It follows that (Γ1 + Γ2 )(x) ⊂ Γ1 (x0 ) + V0 + Γ2 (x0 ) + V0 ⊂ (Γ1 + Γ2 )(x0 ) + V for x ∈ U := U1 ∩ U2 ∈ VX (x0 ), and so Γ1 + Γ2 is H-u.c. at x0 . The proof for H-lower continuity is similar.
Note that we have not a similar result to Proposition 2.5.13(ii) for upper continuity. To see this, consider the multifunctions Γ1 , Γ2 : R ⇒ R2 defined by Γ1 (x) := R × {0} and Γ2 (x) := {(0, x)}. It is obvious that Γ1 and Γ2 are u.c. at every x ∈ R, but Γ1 + Γ2 is not u.c. at every x ∈ R. Indeed, (Γ1 + Γ2 )(0) = R×{0} ⊂ D := {(u, v) ∈ R2 | |v| < exp(−u)}, but (Γ1 +Γ2 )(x) = R×{x} ⊂ D for every x ∈ R \ {0}. Note that if Γ is u.c. at x0 , then Γ is H-u.c. at x0 ; the converse implication is true when Γ (x0 ) is compact. On the other hand, if Γ is H-l.c. at x0 , then Γ is l.c. at x0 , the converse being true if Γ (x0 ) is compact. We can characterize Hausdorff upper and lower continuities by using nets, and even sequences when X and Y are first countable. Proposition 2.5.14. Suppose that Y is a topological vector space and x ∈ X. Then:

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(i) Γ is H-l.c. at x if and only if for all nets (xi )i∈I ⊂ X with (xi ) → x and (y i )i∈I ⊂ Γ (x) there exist a subnet (xϕ(j) ) and a net (yj )j∈J such that yj − y ϕ(j) → 0 and yj ∈ Γ (xϕ(j) ) for all j ∈ J; (ii) Γ is H-u.c. at x if and only if for every net ((xi , yi ))i∈I ⊂ gr Γ with (xi ) → x there exists a subnet (yϕ(j) )j∈J and a net (y j )j∈J ⊂ Γ (x) such that yϕ(j) − y j → 0. Suppose now that X and Y are first countable. (iii) Γ is H-l.c. at x if and only if for every sequence (xn )n∈N ⊂ X with (xn ) → x and every sequence (y n ) ⊂ Γ (x) there exists a sequence (yn ) such that yn − y n → 0 and yn ∈ Γ (xn ) for all n ≥ n0 ; (iv) Γ is H-u.c. at x if and only if for every sequence ((xn , yn ))n∈N ⊂ gr Γ with (xn ) → x there exists a sequence (y n ) ⊂ Γ (x) such that yn − y n → 0. Proof. We prove (i) and (iv), the proofs of (ii) and (iii) being similar. (i) Suppose that Γ is H-l.c. at x and consider the nets (xi ) → x and (y i )i∈I ⊂ Γ (x). For every V ∈ VY there exists iV ∈ I such that Γ (x) ⊂ Γ (xi ) + V for all i  iV . Ordering J := I × VY in the usual way, consider ϕ : J → I with ϕ(i, V )  i, iV and take yj ∈ Γ (xϕ(j) ) ∩ (y ϕ(j) + V ) for j = (i, V ). It is obvious that (yj )j∈J satisfies the needed conditions. We prove by contradiction the converse implication. So, assume that Γ is not H-l.c. at (x, y). Then there exists V0 ∈ VY such that for every U ∈ VX (x) there exist xU ∈ U and y U ∈ Γ (x) with y U ∈ / Γ (xU ) + V0 . By hypothesis, there exist a subnet (xϕ(j) ) and a net (yj )j∈J such that yj − y ϕ(j) → 0 and yj ∈ Γ (xϕ(j) ) for all j ∈ J. Since V0 ∈ VY , there exists j0 ∈ J such that yj − y ϕ(j) ∈ V0 for j  j0 . It follows that y ϕ(j0 ) ∈ yj0 + V0 ⊂ Γ (xϕ(j0 ) ) + V0 , a contradiction. (iv) Suppose that Γ is H-u.c. at x and consider ((xn , yn ))n∈N ⊂ gr Γ with (xn ) → x. Let (Vk )k∈N be a base of neighborhoods of 0 ∈ Y . For every k ∈ N there exists Uk ∈ VX (x0 ) such that Γ (x
) ⊂ Γ (x) + Vk for x
∈ Uk . Since (xn ) → x, there exists nk ∈ N such that xn ∈ Uk for every n ≥ nk . Without loss of generality we may suppose that (nk ) is increasing. For every n such that nk ≤ n < nk+1 we take y n ∈ Γ (x) such that yn −y n ∈ Vk . The conclusion follows. The converse part follows immediately by contradiction.
Proposition 2.5.15. Suppose that Γ is upper continuous and closed-valued. Then gr Γ is closed. The same conclusion holds if Y is a topological vector space and Γ is H-u.c. instead of being u.c. Proof. We give the proof for the second case. So, let Γ be H-u.c. and closedvalued. Consider (x, y) ∈ X × Y \ gr Γ . If x ∈ / dom Γ , since Γ is H-u.c., there exists U ∈ VX (x) such that U ⊂ dom Γ . Hence U × Y ∩ gr Γ = ∅. It follows that (x, y) ∈ / cl(gr Γ ). Suppose now that x ∈ dom Γ , and so y ∈ / Γ (x). Since Γ (x) is closed, there exists V ∈ VY such that (y + V ) ∩ Γ (x) = ∅. Let W ∈ VY with W + W ⊂ V . Since Γ is H-u.c. at x, there exists U ∈ VX (x)

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61

such that Γ (U ) ⊂ Γ (x) + W . It follows that (U × (y + W )) ∩ gr Γ = ∅, and so (x, y) ∈ / cl(gr Γ ). Therefore gr Γ is closed in X × Y .
Note that requiring the upper continuity of Γ only on dom Γ is not sufficient for the conclusion of the preceding result. Taking for example X = Y = R and Γ (x) = R+ for x ∈ ]0, 1[ and Γ (x) = ∅ otherwise, we have that Γ is u.c. (and so H-u.c.) at every x ∈ dom Γ , is closed-valued, but gr Γ = ]0, 1[ ×R+ is not closed in X × Y . Let (An )n∈N ⊂ P(Y ). Recall that lim inf An := {y ∈ Y | ∃ (yn ) → y such that yn ∈ An for n ≥ n0 }, n→∞

lim sup An := {y ∈ Y | ∃ (ynk ) → y such that ynk ∈ Ank for k ∈ N}; n→∞

of course, lim inf n→∞ An ⊂ lim supn→∞ An . We say that (An ) converges in the sense of Kuratowski–Painlev´ e to A ⊂ Y if lim supn→∞ An ⊂ A ⊂ lim inf n→∞ An . Having A, An ⊂ Y , n ∈ N, and taking X := N ∪ {∞} ⊂ R (X endowed with the topology induced by that of R) we may consider the multifunction Γ : X ⇒ Y defined by Γ (n) = An for n ∈ N and Γ (∞) = A. When Y is first countable, lim inf n→∞ An is exactly lim inf n→∞ Γ (n), and lim supn→∞ An is lim supn→∞ Γ (n). Definition 2.5.16. Suppose that Y is a separated topological vector space and C ⊂ Y is a convex cone. We say that Γ is C-u.c., C-l.c., H-C-u.c., or HC-l.c. at x0 ∈ X if relation (2.33), (2.34), (2.41) or (2.42) holds with Γ (x) ⊂ D + C, Γ (x) ∩ (D − C) = ∅, Γ (x) ⊂ Γ (x0 ) + V + C, Γ (x0 ) ⊂ Γ (x) + V + C instead of Γ (x) ⊂ D, Γ (x) ∩ D = ∅, Γ (x) ⊂ Γ (x0 ) + V , Γ (x0 ) ⊂ Γ (x) + V , respectively. Γ is C-continuous (C-Hausdorff continuous) at x0 if Γ is C-u.c. and C-l.c. (H-C-u.c. and H-C-l.c.) at x0 . Remark 2.5.17. As in Proposition 2.5.13 one can prove that having the multifunctions Γ, Γ1 , Γ2 : X ⇒ Y that are C-l.c. (resp. C-H-u.c., C-H-l.c.) at x0 ∈ X, Y being a t.v.s., C ⊂ Y a convex cone, and α ∈ R+ , then Γ1 + Γ2 and αΓ are C-l.c. (resp. C-H-u.c., C-H-l.c.) at x0 . When P ⊂ C is another convex cone, if Γ is P -l.c. (P -u.c., H-P -l.c., or H-P -u.c.) at x0 , then Γ is C-l.c. (C-u.c., H-C-l.c., or H-C-u.c.) at x0 . Note also that Γ is C-l.c., H-C-u.c., or H-C-l.c. if and only if ΓC is l.c., H-u.c., or H-l.c., respectively, but such an equivalence is not true for upper continuity. In fact, we have that Γ is C-u.c. at x0 if ΓC is u.c. at x0 , but the converse is not true even if Γ = ΓC , as shown by the following example: Γ : [0, ∞[ ⇒ R2 , Γ (x) = [0, ∞[ ×[0, x], and C = [0, ∞[ ×{0}. Definition 2.5.18. When Y is a topological vector space, we say that Γ is uniformly C-l.c. at x0 on A if A ⊂ Γ (x0 ) and ∀ W ∈ VY , ∃ U ∈ VX (x0 ), ∀ x ∈ U : A ⊂ Γ (x) + W + C.

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Remark 2.5.19. If Γ is C-l.c. at (x0 , y) for every y ∈ A ⊂ Γ (x0 ) and A is compact, then Γ is uniformly C-l.c. at x0 on A. Indeed, let W ∈ VY and consider W1 ∈ VY open with W1 + W1 ⊂ W . For y ∈ A, since Γ is C-l.c. at (x0 , y), there exists Uy ∈ VX (x !0 ) such that y ∈ Γ (x)+W1 +C for every x ∈ Uy . Since A is compact and A ⊂ y∈A (y+W1 ), !n "n there exist y1 , . . . , yn ∈ A such that A ⊂ i=1 (yi +W1 ). Let x ∈ U := i=1 Uyi and y ∈ A. There exists i, 1 ≤ i ≤ n, such that y ∈ yi + W1 . Since x ∈ Uyi , yi ∈ Γ (x) + W1 + C, whence y ∈ Γ (x) + W + C. Therefore A ⊂ Γ (x) + W + C for every x ∈ U . With any multifunction Γ : X ⇒ Y we associate its closure Γ : X ⇒ Y defined by Γ (x) := cl (Γ (x)). Of course, dom Γ = dom Γ . Continuity properties of Γ and Γ are deeply related. Proposition 2.5.20. Let x ∈ X and y ∈ Y . The following assertions hold: (i) Γ is l.c. at (x, y) if and only if Γ is l.c. at (x, y). (ii) Γ is l.c. at x if and only if Γ is l.c. at x. (iii) If Y is a topological vector space and C ⊂ Y is a convex cone, then Γ H-C-u.c. (H-C-l.c.) at x if and only if Γ is H-C-u.c. (H-C-l.c.) at x. (iv) If Γ is u.c. at x and either Y is normal or Y is regular and cl Γ (x) compact, then Γ is u.c. at x. (v) If Γ is u.c. at x and either Y is regular or cl Γ (x) is compact, then Γ closed at x. (vi) Γ is closed at x if and only if Γ (x) is closed and Γ is closed at x. (vii) If Y is a metric space, then Γ is compact at x if and only if Γ (x) closed and Γ is compact at x.

is is is

is

Proof. The proofs of assertions (i)–(iii) are not complicated. One uses that for A, D ⊂ Y with D open, D " ∩ cl A = ∅ ⇒ D ∩ A = ∅, and, when Y is a topological vector space, cl A = W ∈VY (A + W ). For (vi) take into consideration Proposition 2.5.10 (i), while for (vii) take into account Proposition 2.5.9 (ii). (iv) Let D ⊂ Y be an open set such that Γ (x) ⊂ D. If Y is normal, there exists D0 ⊂ Y open such that cl Γ (x) ⊂ D0 ⊂ cl D0 ⊂ D. If Y is regular and cl Γ (x) is compact, for every y ∈ cl Γ (x) there exist disjoint open sets Vy , Dy ⊂ Y such that y ∈ Vy and Y \ D ⊂ Dy . Since {Vy | y ∈ cl Γ (x)} is an open cover of cl Γ (x), there exist y1 , . . . , yn ∈ cl Γ (x) such that cl Γ (x) ⊂ D0 := Vy1 ∪ · · · ∪ Vyn . Of course, D0 and D1 := Dy1 ∩ · · · ∩ Dyn (⊃ Y \ D) are disjoint open sets. Therefore D0 ⊂ cl D0 ⊂ Y \ D1 ⊂ D. Since Γ is u.c. at x, there exists U ∈ VX (x) such that Γ (x
) ⊂ D0 , whence Γ (x
) ⊂ cl D0 ⊂ D, for every x
∈ U . (v) Let y ∈ Y \Γ (x). If Y is regular, there exist V, D ⊂ Y disjoint open sets such that y ∈ V and Γ (x) ⊂ D. When Γ (x) is compact, for every z ∈ Γ (x) there exist disjoint open sets Vz , Dz ⊂ Y such that y ∈ Vz and z ∈ Dz . Since Γ (x) is compact, there exist z1 , . . . , zn ∈ Γ (x) such that D := Dz1 ∪ · · · ∪Dzn . Taking V := Vz1 ∩ · · · ∩ Vzn , we have again that V, D are open and disjoint,

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63

y ∈ V , and Γ (x) ⊂ D. Since Γ is u.c. at x and Γ (x) ⊂ D, there exists U ∈ VX (x) with Γ (U ) ⊂ D. Because V is open, we have even that Γ (U ) ∩ V = ∅. Therefore Γ is closed at x.
Note that if Γ is u.c. at x, it does not follow that Γ is u.c. at x (but the implication is true if Γ (x) is closed). Take Γ : R ⇒ R, Γ (x) = ]x, x + 1[; Γ is not u.c. at 0, but Γ is. From now on (in this section), Y is a separated topological vector space and C ⊂ Y is a convex cone. Recall that the convex cone C determines a preorder ≤C defined by y1 ≤C y2 ⇔ y2 − y1 ∈ C. Definition 2.5.21. We say that (i) Γ is C-lower semicontinuous (C-l.s.c. for short) if levΓ (y) := {x ∈ X | Γ (x) ∩ (y − C) = ∅} is closed for every y ∈ Y ; (ii) Γ is C-upper semicontinuous (C-u.s.c. for short) if lev< Γ (y) := {x ∈ X | Γ (x) ∩ (y − int C) = ∅} is open in X for every y ∈ Y . −1 (y) for every Note that levΓ (y) = (ΓC )−1 (y) and lev< Γ (y) = (Γint C ) −1 y ∈ Y , so that Γ is C-l.s.c. iff (ΓC ) is closed-valued, and Γ is C-u.s.c. iff (Γint C )−1 is open valued. Moreover, if y1 ≤C y2 , then levΓ (y1 ) ⊂ levΓ (y2 ) < and lev< Γ (y1 ) ⊂ levΓ (y2 ).   When int C = ∅ we have that int dom(levΓ ) = dom(lev< Γ ). First observe < ) is open. Indeed, let y ∈ dom(lev that dom(lev< ). Then there exist x ∈ X Γ Γ and z ∈ Γ (x) ∩ (y − int C). Hence z = y − k for some k ∈ int C. Taking v ∈ V := int C − k, we have that z = y − k = (y + v) − (v + k) ⊂ (y + v) − int C, which shows that Γ (x) ∩ (y − int C) = ∅. Therefore y + V ⊂ dom(lev< Γ ), and < ) is open. From the obvious inclusion dom(lev ) ⊂ dom(lev so dom(lev< Γ ) we Γ Γ   < ) ⊂ int dom(lev ) . For the converse inclusion consider obtain that dom(lev< Γ Γ y ∈ int (dom(levΓ )). Let k ∈ int C; there exists t > 0 such that y − tk ∈ dom(levΓ ). Therefore, there exists x ∈ X such that Γ (x) ∩ (y − tk − C) = ∅. Since tk + C ⊂ int C, we have that y ∈ dom(lev< Γ ). The following result holds.

Proposition 2.5.22. The following assertions hold: (i) If epi Γ is closed, then Γ is C-lower semicontinuous. (ii) Suppose that Y is a locally convex space, C is closed, int C = ∅, Γ is C-lower semicontinuous, and either Γ (x) is weakly compact or Γ (x) + C is closed for every x ∈ X. Then epi Γ is closed. Moreover, if y0 , y1 ∈ Y are such that y1 − y0 ∈ int C and levΓ (y1 ) is compact, then levΓ (y0 ) is compact and levΓ is upper continuous at y0 . Proof. (i) Suppose that epi Γ = gr ΓC is closed. Then gr(ΓC )−1 is also closed, whence (ΓC )−1 is closed-valued. Hence Γ is C-l.s.c. (ii) Suppose now that Γ is C-l.s.c., Y is a locally convex space, C is closed, int C = ∅ and either Γ (x) is weakly compact or Γ (x) + C is closed for every x ∈ X. Let us consider the net ((xi , yi ))i∈I ⊂ epi Γ converging to (x, y) ∈ X × Y . Let k ∈ int C be fixed. Since k − C ∈ VY (0) and (yi ) → y, there exists

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2 Functional analysis over cones

ik ∈ I such that yi −y ∈ k −C for i  ik . It follows that (xi )i ik ⊂ levΓ (y +k), whence x ∈ levΓ (y + k) for every k ∈ int C. Fixing k0 ∈ int C we obtain that x ∈ levΓ (y + n−1 k0 ) for every n ∈ N∗ . Therefore, for every n there exists zn ∈ Γ (x) ∩ (y + n−1 k0 − C); in particular, y + n−1 k0 ∈ Γ (x) + C. If Γ (x) + C is closed, we get y ∈ Γ (x) + C, and so (x, y) ∈ epi Γ . If Γ (x) is weakly compact, the sequence (zn )n∈N∗ contains a subnet converging weakly to some z ∈ Γ (x). Since C is (weakly) closed, we obtain that z ∈ y − C, which shows that (x, y) ∈ epi Γ in this case, too. Now let y0 , y1 ∈ Y be such that y1 −y0 ∈ int C and levΓ (y1 ) is compact. Of course, levΓ (y0 ) ⊂ levΓ (y1 ). Since levΓ (y0 ) is closed and levΓ (y1 ) is compact, we have that levΓ (y0 ) is compact. Suppose that levΓ is not u.c. at y0 . Then there exist an open set D ⊂ X and the nets (yi )i∈I ⊂ Y , (xi )i∈I ⊂ X such that levΓ (y0 ) ⊂ D, (yi )i∈I converges to y0 , and xi ∈ levΓ (yi ) \ D for every i ∈ I. Since y0 ∈ y1 − int C, there exists i1 ∈ I such that yi ∈ y1 − C; i.e., yi ≤C y1 , for every i  i1 . It follows that xi ∈ levΓ (y1 ) for every i  i1 . Since levΓ (y1 ) is compact, there exists a subnet (xϕ(j) )j∈J of (xi ) converging to x. Of course, x ∈ / D. Since (xi , yi ) ∈ epi Γ for every i ∈ I and epi Γ is closed in our conditions (as seen above), we obtain that (x, y0 ) ∈ epi Γ , whence x ∈ levΓ (y0 ) ⊂ D. This is a contradiction because x ∈ / D.
Proposition 2.5.23. Consider the multifunction Γ : X ⇒ Y . (i) Suppose that C is closed and P ⊂ C is another convex cone. If Γ is P -upper continuous, then Γ is C-lower semicontinuous. (ii) Suppose that int C = ∅; then Γ is C-lower continuous if and only if Γ is C-upper semicontinuous. Proof. (i) Let y ∈ Y and x ∈ / levΓ (y). It follows that Γ (x) ⊂ Y \ (y − C). Since Γ is P -upper continuous at x, there exists U ∈ VX (x) such that Γ (x
) ⊂ (Y \ (y − C)) + P , for every x
∈ U . Since (Y \ (y − C)) + P ⊂ Y \ (y − C), we obtain that Γ (x
) ∩ (y − C) = ∅ for each x
∈ U . Therefore U ∩ levΓ (y) = ∅, which shows that x ∈ / cl (levΓ (y)). Hence levΓ (y) is closed for every y ∈ Y . (ii) Suppose first that Γ is C-upper semicontinuous and fix x0 ∈ X. Let D be an open set in Y such that Γ (x0 )∩D = ∅; then D−Γ (x0 ) is a neighborhood of 0 in Y . Since 0 ∈ cl(int C), we deduce (D − Γ (x0 )) ∩ int C = ∅. Let y0 ∈ D be such that (y0 − Γ (x0 )) ∩ int C = ∅; then x0 ∈ U := lev< Γ (y0 ), and U is open because Γ is C-u.s.c.; hence U ∈ VX (x0 ) and ∅ = Γ (x) ∩ (y0 − int C) ⊂ Γ (x) ∩ (D − C) for each x ∈ U . It follows that Γ is C-l.c. at x0 . Suppose now that Γ is C-l.c. and fix y ∈ Y . Let x0 ∈ lev< Γ (y); this means that Γ (x0 ) ∩ D = ∅, where D := y − int C is an open set by our hypothesis. Because Γ is C-l.c. at x0 , there exists U ∈ VX (x0 ) such that ∅ = Γ (x) ∩ (D − C) = Γ (x) ∩ (y − int C − C) = Γ (x) ∩ (y − int C) < for all x ∈ U . It follows that U ⊂ lev< Γ (y). Therefore levΓ (y) is open, and so Γ is C-u.s.c. The proof is complete.


2.5 Continuity notions for multifunctions

65

From the preceding result we obtain that Γ is C-l.s.c. and −C-l.s.c. when Γ is upper continuous (and C is closed). The following interesting result holds. Proposition 2.5.24. Suppose that int C = ∅ and y ∈ dom(levΓ ). If levΓ is lower continuous at y, then levΓ (y) ⊂ cl lev< Γ (y) . Conversely, if X is a topological vector space (or a metric space), lev Γ (y) is totally bounded, and   levΓ (y) ⊂ cl lev< Γ (y) , then levΓ is Hausdorff lower continuous at y. Moreover, if levΓ (y) is compact, then levΓ is lower continuous at y. Proof. Suppose first that levΓ is lower continuous at y and consider x ∈ levΓ (y). Fix k ∈ int C and take yn := y − n−1 k for n ∈ N∗ . By Proposition 2.5.6 there exist a subnet (yϕ(j) )j∈J of (yn )n∈N∗ and a net (xj )j∈J convergent to x such that xj ∈ levΓ (yϕ(j) ) for every j ∈ J. Since yϕ(j) −  C <⊂ y −  int C, we obtain that xj ∈ lev< (y) for every j ∈ J. Therefore x ∈ cl lev (y) Γ Γ . Suppose  now that X is a t.v.s., levΓ (y) is totally bounded, and levΓ (y) ⊂ cl lev< Γ (y) . We show first that ∀ U ∈ VX , ∃ k ∈ int C : levΓ (y) ⊂ levΓ (y − k) + U.

(2.43)

So, let U ∈ VX (0); there exists a balanced neighborhood W of 0 ∈ X such that W +W ⊂ U . Since levΓ (y) is totally bounded, there exist x1 , . . . , xp ∈ levΓ (y) such that p  levΓ (y) ⊂ (xl + W ). (2.44) 



l=1

, for every l ∈ {1, . . . , p} there exists x
l ∈ (xl + Since levΓ (y) ⊂ cl < W ) ∩ levΓ (y). It follows that for every l there exists kl
∈ int C such that yl := y − kl
∈ Γ (x
l ). Let t1 := 1 and k1 := t1 k1
. Since k1 − tk2
→ k1 ∈ int C for t → 0, there exists t2 ∈ ]0, 1] such that k1 − t2 k2
∈ int C. Let k2 := t2 k2
. Similarly, there exists t3 ∈ ]0, 1] such that k2 − t3 k2
∈ int C. Let k3 := t3 k3
. Continuing in this way, for every l ∈ N∗ , we find tl ∈ ]0, 1] and kl := tl kl
such that kl −kl+1 ∈ int C for 1 ≤ l ≤ p−1. Since yl = y−kl −(1−tl )kl
⊂ y−kl −C, we have that x
l ∈ levΓ (y − kl ) for every l. Moreover, lev< Γ (y)

levΓ (y − k1 ) ⊂ levΓ (y − k2 ) ⊂ . . . ⊂ levΓ (y − kp ). Now let x ∈ levΓ (y). From (2.44) there exists 1 ≤ l ≤ p such that x ∈ xl + W . Since x
l ∈ xl + W , we obtain that x ∈ x
l + W − W ⊂ x
l + U ⊂ levΓ (y − kl ) + U ⊂ levΓ (y − kp ) + U. Therefore levΓ (y) ⊂ levΓ (y − k) + U with k := kp ∈ int C; i.e., (2.43) holds. Taking V = y + C − k, V is a neighborhood of y and levΓ (y) ⊂ levΓ (y − k) + U ⊂ levΓ (y
) + U

∀ y
∈ V.

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Therefore levΓ is H-l.c. at y. When X is a metric space instead of U one takes ε > 0, and instead of xl + W one takes B(xl , ε/2). When levΓ (y) is compact the conclusion is obvious.
It is quite interesting that continuity properties for vector-valued functions and those of multifunctions are related. We give first some continuity notions for extended vector-valued functions. Definition 2.5.25. Let f : X → Y • and x0 ∈ dom f . (i) f is C-lower continuous (C-l.c.) at x0 if ∀ V ∈ VY , ∃ U ∈ VX (x0 ), ∀ x ∈ U : f (x) ∈ f (x0 ) + V − C • ; (ii) f is C-upper continuous (C-u.c.) at x0 if ∀ V ∈ VY , ∃ U ∈ VX (x0 ), ∀ x ∈ U : f (x) ∈ f (x0 ) + V + C; (iii) f is C-lower semicontinuous if levf (y) is closed for every y ∈ Y ; (iv) f is C-upper semicontinuous if lev< f (y) is open for every y ∈ Y . From Proposition 2.5.23(ii) we obtain that the function f : X → Y is C-upper semicontinuous if and only if f is C-lower continuous. Consider x0 ∈ dom f . If f is C-u.c. at x0 , then x0 ∈ int(dom f ). On the other hand, if Γf,C is closed at x0 , then C is closed; if C is closed and Γf,C is u.c. at x0 , then Γf,C is closed at x0 . Moreover, f is {0}-u.c. at x0 ⇔ if f is continuous at x0 ⇔ x0 ∈ int(dom f ) and f is {0}-l.c. at x0 . Proposition 2.5.26. Let f : X → Y • , Γf,C : X ⇒ Y its associated multifunction, and x0 ∈ dom f . Then: (i) f is C-l.c. at x0 ⇔ Γf,C is C-u.c. at x0 ⇔ Γf,C is H-u.c. at x0 . (ii) f is C-u.c. at x0 ⇔ Γf,C is l.c. at x0 ⇔ Γf,C is H-l.c. at x0 . Proof. (i) Suppose that f is C-l.c. at x0 and consider D ⊂ Y an open set such that Γf,C (x0 ) = f (x0 ) + C ⊂ D. Since f (x0 ) ∈ D, there exists U ∈ VX (x0 ) such that f (x) ∈ D + C • for every x ∈ U . Hence Γf,C (x) ⊂ D + C for every x ∈ U , and so Γf,C is C-u.c. at x0 . Suppose now that Γf,C is H-u.c. at x0 and consider V ∈ VY . By definition, there exists U ∈ VX (x0 ) such that Γf,C (x) ⊂ Γf,C (x0 ) + V = f (x0 ) + V + C for all x ∈ U . Therefore f (x) ∈ f (x0 ) + V + C • for every x ∈ U ; i.e., f is C-l.c. at x0 . Since the other implication is immediate, the proof of (i) is complete. (ii) Suppose that f is C-u.c. at x0 and consider V ∈ VY ; there exists U ∈ VX (x0 ) such that f (x) ∈ f (x0 ) + V − C for every x ∈ U . It follows that f (x0 ) ∈ Γf,C (x) + V , whence Γf,C (x0 ) ⊂ Γf,C (x) + V for every x ∈ U . Therefore Γf,C is H-l.c. at x0 . Suppose now that Γf,C is l.c. at x0 and consider V ∈ VY . Since f (x0 ) ∈ (f (x0 ) + int V ) ∩ Γ (x0 ), there exists U ∈ VX (x0 ) such that (f (x0 ) + int V ) ∩ Γ (x) = ∅ for all x ∈ U . Therefore f (x) ∈ f (x0 ) + V − C for every x ∈ U ; i.e.,

2.5 Continuity notions for multifunctions

67

f is C-u.c. at x0 . Since Hausdorff lower continuity implies lower continuity, the proof is complete.
When Y = R and C = R+ we have more refined statements; Γf := Γf,R+ in this case. Proposition 2.5.27. Let f : X → R := R ∪ {−∞, ∞} with the associated multifunction Γf : X ⇒ R whose graph is epi f . (i) Suppose that x0 ∈ dom f (= dom Γf ). Then f is l.c. at x0 ⇔ Γf is u.c. at x0 ⇔ Γf is H-u.c. at x0 . (ii) Suppose that x0 ∈ X. Then Γf is H-l.c. at x0 ⇒ f is u.c. at x0 ⇔ Γf is l.c. at x0 ; moreover, if either f (x0 ) > −∞ or X is a topological vector space and f is convex, then f is u.c. at x0 ⇔ Γf is H-l.c. at x0 . Proof. When f (x0 ) ∈ R the conclusion follows from the preceding proposition; one must only mention that in this case upper continuity and C = R+ upper continuity for Γf coincide. If f (x0 ) = ∞, it is obvious that f is u.c. at x0 and Γf is l.c. and H-l.c. at x0 (since Γf (x0 ) = ∅); therefore (ii) holds in this case. Suppose now that f (x0 ) = −∞; in this case Γf (x0 ) = R. Obviously, f is l.c. at x0 and Γf is u.c. and H-u.c. at x0 . Therefore (i) holds. Assume that Γf is H-l.c. at x0 . Since Γf (x0 ) = R, taking V = [−1, 1] ∈ VR (0), there exists U ∈ VX (x0 ) such that Γf (x) = R for every x ∈ U . Therefore f (x) = −∞ for every x ∈ U , whence f is u.c. at x0 . Assume that f is u.c. at x0 and consider D ⊂ R an open set; take y0 ∈ Γf (x0 ) ∩ D = D. There exists ε > 0 such that ]y0 − ε, y0 + ε[ ⊂ D. Since f is u.c. at x0 , there exists U ∈ VX (x0 ) such that f (x) < y0 for every x ∈ U . It follows that y0 ∈ Γf (x) ∩ D for every x ∈ U . Therefore Γf is l.c. at x0 . Assume that Γf is l.c. at x0 and take λ ∈ R. Since Γf (x0 )∩ ] − ∞, λ[ = ∅, there exists U ∈ VX (x0 ) such that Γf (x)∩ ] − ∞, λ[ = ∅ for every x ∈ U . It follows that f (x) < λ for every x ∈ U . Therefore f is u.c. at x0 . Of course, if f takes the value −∞ on a neighborhood of x0 , then Γf is constant on that neighborhood, and so it is H-l.c. at x0 . This is the case in which f is convex, f (x0 ) = −∞, and f is u.c. at x0 . The proof is complete.
We are now interested in continuity properties of the composition of two multifunctions or of a function with a multifunction. So consider another topological space U, the functions f : X × U → Y , g : X → Y , and the multifunctions Λ : U ⇒ X, F : X × U ⇒ Y and G : X ⇒ Y ; we associate the multifunctions f Λ, gΛ, F Λ, GΛ : U ⇒ Y defined by (f Λ)(u) := f (Λ(u) × {u}), (gΛ)(u) := g (Λ(u)), (F Λ)(u) = F (Λ(u) × {u}), and (GΛ)(u) := G (Λ(u)). Proposition 2.5.28. Let f, g and Λ, F, G be as above and u0 ∈ dom Λ. (i) If Λ is u.c. at u0 and g is C-l.c. on Λ(u0 ), then gΛ is C-u.c. at u0 . Moreover, if Λ(u0 ) is compact, then gΛ is H-C-u.c. at u0 .

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(ii) If Λ is u.c. at u0 , F is C-u.c. at (x, u0 ) for all x ∈ Λ(u0 ), and Λ(u0 ) is compact, then F Λ is C-u.c. at u0 . If f is C-l.c. on Λ(u0 ), then f Λ is H-C-u.c. at u0 . (iii) If Λ is l.c. at (x0 , u0 ) ∈ gr Λ and F is C-l.c. at (x0 , u0 ; y0 ), then F Λ is C-l.c. at (u0 , y0 ). In particular, if f is C-u.c. at (x0 , u0 ), then f Λ is C-l.c. at (u0 , y0 ), where y0 = f (u0 , x0 ). (iv) If Λ is compact at u0 and F is closed at (x, u0 ) for all x ∈ Λ(u0 ), then F Λ is closed at u0 . (v) If Λ is compact at every u ∈ U and G is C-l.s.c., then GΛ is C-l.s.c., too. Suppose now that X is a topological vector space, too. (vi) If Λ is H-u.c. at u0 and f is equi-C-l.c. on Λ(u0 ) × {u0 }, then f Λ is H-C-u.c. at u0 . (vii) If Λ is H-l.c. at u0 and f is equi-C-u.c. on Λ(u0 ) × {u0 }, then f Λ is H-C-l.c. at u0 . Proof. (i) Let D ⊂ Y be an open set such that g (Λ(u0 )) ⊂ D. For every x ∈ Λ(u0 ), since g is C-l.c. at x and D ∈ VY (g(x)), there exists!an open neighborhood Vx of x in X such that g(Vx ) ⊂ D+C. The set D
:= x∈Λ(u0 ) Vx is open and contains Λ(u0 ). Therefore there exists U0 ∈ VU (u0 ) such that Λ(u) ⊂ D
for every u ∈ U0 . It follows that (gΛ)(u) = g (Λ(u)) ⊂ g(D
) ⊂ D + C for u ∈ U0 . Hence gΛ is C-u.c. at u0 . If Λ(u0 ) is compact, then (gΛ)(u0 ) is also compact, and so gΛ is H-C-u.c. at u0 . (ii) Let D ⊂ Y be an open set such that F (Λ(u0 ) × {u0 }) ⊂ D. Then for every x ∈ Λ(u0 ), F (x, u0 ) ⊂ D. Since F is C-u.c. at (x, u0 ), there exists an open neighborhood Vx of x in X and Ux ∈ VU (u0 ) such that F (Vx × Ux ) ⊂ D + C. Since {Vx | x ∈ Λ(u0 )} is an open cover for the compact set Λ(u0 ), there exist x1 , . . . , xn ∈ Λ(u0 ) such that Λ(u0 ) ⊂ Vx1 ∪ · · · ∪ Vxn =: V ; of course, V is an open set. Since Λ is u.c. at u0 , there exists U0 ∈ VU (u0 ) such that Λ(U0 ) ⊂ V . Let U := U0 ∩ Ux1 ∩ · · · ∩ Uxn , u ∈ U , and x ∈ Λ(u) (⊂ V ). It follows that x ∈ Vxi for some 1 ≤ i ≤ n. Since u ∈ Uxi , F (x, u) ⊂ D + C. Hence F Λ(u) ⊂ D + C. Therefore F Λ is C-u.c. If f is C-l.c. on Λ(u0 ) × {u0 }, by applying what precedes for the multifunction F := gr f , one obtains that f Λ is C-u.c. at u0 . To obtain the stronger  result note that the multifunction Λ : U ⇒ X × U defined by Λ(u) := Λ(u) × {u} is l.c. at u0 ; by applying then (i), the conclusion follows. The fact that Λ is l.c. at u0 is obtained by using a similar argument to that used above, so that the proof is left to the reader. (iii) Let W ∈ VY (y0 ). Since F is C-u.c. at (x0 , u0 ; y0 ), there exist V ∈ VX (x0 ) and U1 ∈ VU (u0 ) such that F (x, u) ∩ (W − C) = ∅ for every (x, u) ∈ V ×U1 . Since Λ is l.c. at (u0 , x0 ), there exists U2 ∈ VU (u0 ) such that Λ(u)∩V = ∅ for all u ∈ U2 . Let u ∈ U1 ∩ U2 and take x ∈ Λ(u) ∩ V . It follows that there exists y ∈ F (x, u) ∩ (W − C). Therefore y ∈ F Λ(u) ∩ (W − C). Hence F Λ is C-l.c. at (u0 , y0 ).

2.5 Continuity notions for multifunctions

69

When f is C-u.c. at (x0 , u0 ), by applying what precedes to gr f , we get the desired conclusion. (iv) Let gr F Λ ⊃ ((ui , yi ))i∈I → (u0 , y0 ). There exists X ⊃ (xi )i∈I such that ((ui , xi ))i∈I ⊂ gr Λ and ((xi , ui ; yi ))i∈I ⊂ gr F . Since Λ is compact at u0 , there exists a subnet (xϕ(j) )j∈J → x0 ∈ Λ(u0 ). Since F is closed at (x0 , u0 ), (x0 , u0 ; y0 ) ∈ gr F . It follows that (u0 , y0 ) ∈ gr F Λ. Therefore F Λ is closed at u0 . (v) Let y ∈ Y and let (ui )i∈I be a net in levGΛ (y) converging to u ∈ U. There exists (xi )i∈I ⊂ levG (y) such that xi ∈ Λ(ui ) for every i ∈ I. Since Λ is compact at u, there exists the subnet (xϕ(j) )j∈J of (xi )i∈I convergent to x ∈ Λ(u). Since G is C-l.s.c., x ∈ levG (y), whence x ∈ levGΛ (y). Consider now that X is a topological vector space. (vi) Let W ∈ VY . Since f is equi-C-l.c. on Λ(u0 ) × {u0 }, there exist V ∈ VX and U1 ∈ VU (u0 ) such that f ((x + V ) × U1 ) ⊂ f (x, u0 ) + W + C for every x ∈ Λ(u0 ). Since Λ is H-u.c. at u0 , there exists U2 ∈ VU (u0 ) such that Λ(U2 ) ⊂ Λ(u0 ) + V . Consider u ∈ U1 ∩ U2 and y ∈ (f Λ)(u). Then y = f (x, u) for some x ∈ Λ(u). It follows that x = x0 + v with x0 ∈ Λ(u0 ) and v ∈ V . So y = f (x0 + v, u) ∈ f (x0 , u0 ) + W + C ⊂ (f Λ)(u0 ) + W + C, whence (f Λ)(U1 ∩ U2 ) ⊂ (f Λ)(u0 ) + W + C. Thus f Λ is H-C-u.c. at u0 . (vii) Let W ∈ VY . Since f is equi-C-u.c. on Λ(u0 ) × {u0 }, there exist V ∈ VX and U1 ∈ VU (u0 ) such that f (x
, u) ∈ f (x, u0 ) + W − C for every x ∈ Λ(u0 ), x
∈ x+V , and u ∈ U1 , or equivalently, f (x, u0 ) ∈ f (x
, u)+W +C for every x ∈ Λ(u0 ), x
∈ x + V , and u ∈ U1 . Since Λ is H-l.c. at u0 , there exists a neighborhood U2 of u0 such that Λ(u0 ) ⊂ Λ(u) + V for every u ∈ U2 . Let u ∈ U1 ∩ U2 and y ∈ (f Λ)(u0 ). Then y = f (x, u0 ) for some x ∈ Λ(u0 ) ⊂ Λ(u) + V . Hence there exists some x
∈ Λ(u) ∩ (x + V ). It follows that y ∈ f (x
, u) + W + C ⊂ f (Λ(u)) + W + C, whence (f Λ)(u0 ) ⊂ (f Λ)(u) + W + C for every u ∈ U1 ∩ U2 . The proof is complete.
The implication “⇒” of assertion (i) of Proposition 2.5.9 was obtained by Smithson [324, Th. 3.1], while the implication “⇒” was established by Ferro [115, Lemma 2.2]. The assertion (iv) of Proposition 2.5.10 is stated in Ferro [115, Lemma 2.1], while assertion (v) is stated in Smithson [324, Prop. 1.1.2]. The assertion (i) of Proposition 2.5.11 can be found in [26], the assertion (ii) in [224], while assertions (iv) and (v) of Proposition 2.5.20 in [291], where one can also find the statement (i) of Proposition 2.5.10. The continuity notions for vector-valued functions are in accordance with those of Penot and Th´era [293]. Remark 2.5.8 and Proposition 2.5.28 (ii)–(iv) may be found in [244] for C = {0}. Note that the statements (ii), (iii), (vi), and (vii), for g instead of f , of Proposition 2.5.28 can be found in [17], [18], and [19]. Other results on continuity of multifunctions can be found in the books by Aubin and Frankowska [13] and Cˆ arj˘ a [52].

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2.6 Continuity Properties of Multifunctions Under Convexity Assumptions Throughout this section X and Y are topological vector spaces, and C ⊂ Y is a convex cone and Γ : X ⇒ Y is a multifunction. The next result is the analogue of a known one for convex functions. Theorem 2.6.1. Let Γ be C-nearly convex. Suppose that Γ is C-l.c. at (x0 , y0 ) ∈ gr Γ . Then Γ is C-l.c. at every x ∈ int(dom Γ ) = aint(dom Γ ). Moreover, Γ is H-C-l.c. at every x ∈ int(dom Γ ) for which there exists a bounded set B ⊂ Y such that Γ (x) ⊂ B + C. Proof. Let α ∈ ]0, 1[ be such that Γ is C-α-convex. Consider V ∈ VY ; there exists W ∈ VY with W + W ⊂ V . By hypothesis, there exists U0 ∈ VX such that Γ (x0 + u) ∩ (y0 + V ) = ∅, or, equivalently, y0 ∈ Γ (x0 + u) + W + C

∀ u ∈ U0 .

(2.45)

Let (x, y) ∈ gr Γ with x ∈ int(dom Γ ). Of course, dom Γ is α-convex, whence, by Proposition 2.4.3 (iii), int(dom Γ ) is a convex set. Then there exist λ ∈ Λα \ {0, 1} and (x1 , y1 ) ∈ gr Γ , with x1 ∈ int(dom Γ ), such that x = λx0 +(1−λ)x1 ; take y := λy0 + (1 − λ)y1 . It follows that y ∈ λ (Γ (x0 + u) + W + C) + (1 − λ)Γ (x1 ) ⊂ Γ (x + λu) + λW + C ∀ u ∈ U0 .

(2.46)

Consider µ ∈ Λα \ {0, 1} such that µ(y − y) ∈ W . Then, by (2.46), the C-αconvexity of Γ and the choice of µ, y ∈ (1 − µ)Γ (x) + µy + µ(y − y) ⊂ (1 − µ)Γ (x) + µΓ (x + λu) + λµW + C + W ∀ u ∈ U0 .

⊂ Γ (x + λµu) + V + C

(2.47)

Taking U := λµU0 , we see that Γ is l.c. at (x, y). Consider now x ∈ int(dom Γ ) and a bounded set B ⊂ Y such that Γ (x) ⊂ B + C. For V ∈ VY , with the same construction as above, we obtain W and U0 for which (2.45) holds. Then we take (x1 , y1 ), λ and y as above; so relation (2.46) holds, too. Taking now µ ∈ Λα \ {0, 1} such that µ(B − y) ⊂ W , we find that (2.47) holds for every y ∈ Γ (x). Taking again U := λµU0 , we obtain that Γ is H-C-l.c. at x.
The next result is related to the boundedness condition of the preceding theorem. Proposition 2.6.2. Let Γ be C-nearly convex. Suppose that x0 ∈ int(dom Γ ) and Γ (x0 ) ⊂ B + C for some bounded set B ⊂ Y . Then for every x ∈ dom Γ there exists a bounded set Bx ⊂ Y such that Γ (x) ⊂ Bx + C.

2.6 Continuity properties of multifunctions under convexity assumptions

71

Proof. By hypothesis there exists α ∈ ]0, 1[ such that Γ is C-α-convex. Let x ∈ dom Γ ; there exist x1 ∈ dom Γ and λ ∈ Λα \ {0, 1} such that x0 = λx + (1 − λ)x1 . Since Γ is α-convex, λΓ (x) + (1 − λ)Γ (x1 ) ⊂ Γ (x0 ) + C ⊂ B + C. Taking y1 ∈ Γ (x1 ), we obtain that Γ (x) ⊂ λ−1 (B − (1 − λ)y1 ) + C = Bx + C, where Bx := λ−1 (B − (1 − λ)y1 ).
As an application of the preceding result we obtain that if f : X → R is nearly convex and finite-valued at x0 ∈ int(dom f ), then f is finite-valued on dom f . Proposition 2.6.3. Assume that Γ is C-l.c. at (x0 , y0 ) ∈ gr Γ and C-nearly convex. If Γ (x0 ) ⊂ B + C for some bounded set B ⊂ Y , then Γ is H-C-u.c. at x0 . Proof. Let α ∈ ]0, 1[ be such that Γ is α-convex. Without loss of generality we may suppose that x0 = 0. Let y0 ∈ Γ (0) be fixed. Consider V ∈ VY and take W ∈ VY such that W + W + W ⊂ V . Since Γ is H-C-l.c. at 0, there exists U ∈ VX such that y0 ∈ Γ (u) + W + C for all u ∈ U . Since B is bounded, there exists µ ∈ ]1/2, 1[ ∩Λα with (µ−1 − 1)(B ∪ {y0 }) ⊂ W . Let u ∈ (1 − µ−1 )U ∈ VX . There exists u
∈ U such that 0 = µu + (1 − µ)u
. It follows that y0 = y
+ w + k with y
∈ Γ (u
), w ∈ W and k ∈ C. Therefore µΓ (u) + (1 − µ)y
⊂ Γ (0), whence µΓ (u) ⊂ Γ (0) + (1 − µ)(w + k − y0 ). Hence Γ (u) ⊂ Γ (0) + (µ−1 − 1)Γ (0) + (µ−1 − 1)W + C − (µ−1 − 1)y0 ⊂ Γ (0) + (µ−1 − 1)B + C + W + C + W ⊂ Γ (0) + W + W + W + C ⊂ Γ (0) + V + C. Therefore Γ is H-C-u.c. at x0 . The preceding results lead to the following important theorem.




Theorem 2.6.4. Let Γ be a C-nearly convex. Suppose that Γ is C-l.c. at (x0 , y0 ) ∈ gr Γ and Γ (x0 ) ⊂ B + C for some bounded set B ⊂ Y . Then Γ is C-H-continuous on int(dom Γ ). Proof. Of course, x0 ∈ int(dom Γ ) in our conditions. From Proposition 2.6.2, Γ (x) ⊂ Bx + C, with Bx bounded, for every x ∈ dom Γ . Using Theorem 2.6.1 we obtain that Γ is H-C-l.c. at every x ∈ int(dom Γ ). Then using Proposition 2.6.3 we obtain that Γ is H-C-u.c. at every x ∈ int(dom Γ ). Therefore Γ is C-H-continuous on int(dom Γ ).
From the preceding results we have the following characterizations of the continuity of nearly convex functions. Corollary 2.6.5. Let f : X → Y • be a nearly convex function and x0 ∈ int(dom f ). Suppose that C is normal. Then f is C-u.c. at x0 ⇔ Γf is l.c. at x0 ⇔ Γf is H-continuous at every x ∈ int(dom f ) ⇔ f is continuous on int(dom f ) ⇔ f is continuous at x0 .

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2 Functional analysis over cones

Proof. We have that f is C-u.c. at x0 ⇒ Γf is l.c. at x0 by Proposition 2.5.26 (ii) and Γf is l.c. at x0 ⇒ Γf is H-continuous at every x ∈ int(dom f ) by Theorem 2.6.1 and Proposition 2.6.3. If Γf is H-continuous at (any) x ∈ int(dom f ), by Proposition 2.5.26 (i) and (ii) we obtain that f is C-l.c. and C-u.c. at x; the continuity of f at x follows from the normality of C. Of course, if f is continuous at every x ∈ int(dom f ), then f is continuous at x0 , whence f is C-u.c. at x0 .
The next result corresponds to another well-known result of convex analysis. Theorem 2.6.6. Let Γ be a C-nearly convex. Suppose that there exist x0 ∈ int(dom Γ ), B ⊂ Y a bounded set, and U ∈ VX such that Γ (x0 ) ⊂ B + C and Γ (x0 + u) ∩ (B − C) = ∅ for every u ∈ U . Then Γ is C-H-continuous on int(dom Γ ). Proof. Let α ∈ ]0, 1[ be such that Γ is α-convex. Taking into account Theorem 2.6.1 and Propositions 2.6.2, 2.6.3, it is sufficient to show that Γ is H-C-l.c. at x0 . Consider V ∈ VY and take W ∈ VY such that W + W ⊂ V . There exists λ ∈ Λα \ {0, 1} such that λB ⊂ W . It follows that Γ (x0 ) ⊂ λΓ (x0 ) + (1 − λ)Γ (x0 ) ⊂ λ(B + C) + (1 − λ)Γ (x0 ) ⊂ (1 − λ)Γ (x0 ) + W + C. Let u ∈ U and y ∈ Γ (x0 + u) ∩ (B − C); hence y = b − k with b ∈ B and k ∈ C. Then λ(b − k) + (1 − λ)Γ (x0 ) ⊂ λΓ (x0 + u) + (1 − λ)Γ (x0 ) ⊂ Γ (x0 + λu) + C, whence (1 − λ)Γ (x0 ) ⊂ Γ (x0 + λu) + C − λb + λk ⊂ Γ (x0 + λu) + W + C. Therefore Γ (x0 ) ⊂ Γ (x0 + λu) + W + C + W + C ⊂ Γ (x0 + λu) + V + C. Hence Γ (x0 ) ⊂ Γ (x) + V + C for every x ∈ x0 + λU ∈ VX (x0 ). It follows that Γ is H-C-l.c. at x0 .
From Theorem 2.6.6, also taking into account Proposition 2.5.27, we obtain that if f : X → R is nearly convex, f (x0 ) ∈ R, and f is bounded above on a neighborhood of x0 , then f is (finite-valued and) continuous on int(dom f ). In fact, semicontinuous nearly convex multifunctions are very close to convex multifunctions. Proposition 2.6.7. Suppose that Γ : X ⇒ Y is closed-valued and nearly convex. If dom Γ is convex (for example, if dom Γ is closed or open) and Γ is H-u.c. at every point of dom Γ , then Γ is convex.

2.7 Tangent cones and differentiability of multifunctions

73

Proof. Since Γ is nearly convex, so is dom Γ ; so, if dom Γ is closed or open, from Corollary 2.4.4 we obtain that dom Γ is convex. Let α ∈ ]0, 1[ be such that Γ is α-convex. Consider x, y ∈ dom Γ and λ ∈ ]0, 1[. There exists a sequence (λn )n∈N ⊂ Λα converging to λ. Of course, λn Γ (x) + (1 − λn )Γ (y) ⊂ Γ (λn x + (1 − λn )y)

∀ n ∈ N.

Let u ∈ Γ (x) and v ∈ Γ (y) be fixed. Consider V ∈ VY , and take W ∈ VY such that W + W ⊂ V . Since Γ is H-u.c. at xλ := λx + (1 − λ)y ∈ dom Γ , there exists nV ∈ N such that Γ (λn x + (1 − λn )y) ⊂ Γ (xλ ) + W for all n ≥ nV . It follows that λn u + (1 − λn )v ∈ Γ (xλ ) + W for all n ≥ nV , whence λu + (1 − λ)v ∈"cl (Γ (xλ ) + W ) ⊂ Γ (xλ ) + W + W ⊂ Γ (xλ ) + V . Hence λu + (1 − λ)v ∈ V ∈VY (Γ (xλ ) + V ) = cl Γ (xλ ) = Γ (xλ ). Therefore λΓ (x) + (1 − λ)Γ (y) ⊂ Γ (λx + (1 − λ)y), which shows that Γ is convex.
The first part of Theorem 2.6.1 was obtained by Borwein [37] for Γ convex and C = {0}, while the second part was obtained by Nikodem [275] for Γ mid-convex, bounded-valued, and H-C-l.c. at x0 ; also, Nikodem obtained Proposition 2.6.3 and Theorem 2.6.6 for Γ mid-convex and bounded-valued in [275], and Proposition 2.6.7 for Γ mid-convex with bounded values and H-continuous at each point of its open convex domain in [276]. The equivalence “f is continuous at x0 ⇔ Γf is l.c. at x0 ” of Corollary 2.6.5 is stated by Borwein [36] for f convex.

2.7 Tangent Cones and Differentiability of Multifunctions Throughout this section X, Y are normed vector spaces. Consider A ⊂ X a nonempty set and a ∈ X. The Bouligand tangent cone or contingent cone of A at a is defined as the set TB (A; a) = lim sup t−1 (A − a) := lim sup RA,a (t), t↓0

t→0

where RA,a : [0, ∞[ ⇒ X, RA,a (t) = t−1 (A − a) for t > 0, RA,a (0) = ∅. Similarly, the Ursescu tangent cone or adjacent cone of A at a is defined by TU (A; a) = lim inf t−1 (A − a) := lim inf RA,a (t). t↓0

t→0

/ cl A. Taking this into Of course, TU (A; a) ⊂ TB (A; a) and TB (A; a) = ∅ if a ∈ account, in the sequel we consider only the case a ∈ cl A when considering tangent cones. It is known (and easy to see) that TB (A; a) = TU (A; a) = cl (]0, ∞[·(A − a)) if A is a convex set and a ∈ cl A. Using Proposition 2.5.3 we have the following result; the notation (tn ) → 0+ means that (tn )n∈N ⊂ ]0, ∞[ and (tn ) → 0.

74

2 Functional analysis over cones

Proposition 2.7.1. Let A ⊂ X and a ∈ cl A. Then #  $ #   TB (A; a) = u ∈ X ## lim inf dist u, t−1 (A − a) = 0 t↓0

= {u ∈ X | ∃ (tn ) → 0+ , ∃ (un ) → u, ∀ n ∈ N : a + tn un ∈ A} , and

#  $ #  −1  # TU (A; a) = u ∈ X # lim dist u, t (A − a) = 0 t↓0

= {u ∈ X | ∀ (tn ) → 0+ , ∃ (un ) → u, ∀ n ∈ N : a + tn un ∈ A} . Moreover, TB (A; a) and TU (A; a) are nonempty closed cones (in particular, they contain 0). Proof. The formulae above follow from Proposition 2.5.3. The sets TB (A; a) and TU (A; a) are closed, since they are limit superior and limit inferior of   multifunctions, respectively. Since a ∈ cl A, dist 0, t−1 (A − a) = 0 for every t > 0. So 0 ∈ TU (A; a). The fact that they are cones follows immediately from the sequential characterizations of their elements.
Let Γ : X ⇒ Y and (x, y) ∈ X × Y . Consider the multifunction  −1 t (Γ (x + tu) − y) if t > 0, Γx,y : R × X ⇒ Y, Γx,y (t, u) = ∅ if t ≤ 0. Of course, the domain of Γx,y is the set D := {(t, u) | t > 0, u ∈ X, x + tu ∈ dom Γ }. So (0, u) ∈ cl D if and only if u ∈ TB (dom Γ ; x). Definition 2.7.2. Let Γ : X ⇒ Y and (x, y) ∈ gr Γ . The Dini upper derivative of Γ at (x, y) in the direction u ∈ X is the set # DΓ (x, y)(u) := lim sup (Γx,y ) #D∪{(0,u)} (t, u
) = lim sup Γx,y (t, u
), (t,u )→(0,u)

(t,u )→(0,u)

while the Dini lower derivative of Γ at (x, y) in the direction u ∈ X is the set # % lim inf (Γx,y ) #D∪{(0,u)} (t, u
) if u ∈ TB (dom Γ ; x),  DΓ (x, y)(u) := (t,u )→(0,u) ∅ if u ∈ / TB (dom Γ ; x). Normally, the Dini upper derivative of Γ at (x, y) would be denoted by DΓ (x, y) (see Penot [289]); we prefer the notation DΓ (x, y) because the Dini lower derivative will be seldom used. It is obvious that DΓ (x, y)(u) = ∅ if u∈ / TB (dom Γ ; x), and so DΓ (x, y)(u) ⊂ DΓ (x, y)(u) ∀ u ∈ X, and these sets are closed. Moreover,

2.7 Tangent cones and differentiability of multifunctions

75

gr DΓ (x, y) = TB (gr Γ ; (x, y)) . These facts can be seen from the following characterizations for v ∈ DΓ (x, y)(u): ∃ (tn ) → 0+ , (un ) → u, (vn ) → v, ∀ n ∈ N : vn ∈ Γx,y (tn , un ), ∃ (tn ) → 0+ , (un , vn ) → (u, v), ∀ n ∈ N : (x, y) + tn (un , vn ) ∈ gr Γ, and for v ∈ DΓ (x, y)(u) when u ∈ TB (dom Γ ; x), ∀ D ⊃ ((tn , un )) → (0+ , u), ∃ (vn ) → v, ∃ n0 , ∀ n ≥ n0 : (x, y) + tn (un , vn ) ∈ gr Γ. Definition 2.7.3. Let Γ : X ⇒ Y and (x, y) ∈ gr Γ . We say that Γ is semidifferentiable at (x, y) in the direction u if DΓ (x, y)(u) = DΓ (x, y)(u), and Γ is semidifferentiable at (x, y) if Γ is semidifferentiable at every u ∈ X. The following theorem establishes sufficient conditions for semidifferentiability of convex multifunctions. By raint A we mean the algebraic relative interior of A. For the proof of the theorem we need the following lemma. Lemma 2.7.4. Let ∅ = A ⊂ X be a convex set and T : X → Y be a linear operator. (i) If int A = ∅ and T is open, then int(]0, ∞[·A) =]0, ∞[· int A and T (int A) = int T (A). (ii) If raint A = ∅ (e.g., if X is finite-dimensional), then raint(]0, ∞[·A) = ]0, ∞[· raint A and T (raint A) = raint T (A). Proof. (i) The inclusions ]0, ∞[· int A ⊂ int(]0, ∞[·A) and T (int A) ⊂ int T (A) are obvious. Let a ∈ int A be fixed. Consider x ∈ int(]0, ∞[·A). Then there exist x
∈ ]0, ∞[·A and λ ∈ ]0, 1[ such that x = (1 − λ)x
+ λa. Therefore there exist a
∈ A and t
> 0 such that x
= t
a
. Since a ∈ int A and A is convex, 1 (1 − λ)t
λ x= a
+ a ∈ int A,
(1 − λ)t + λ (1 − λ)t
+ λ (1 − λ)t
+ λ whence x ∈ ]0, ∞[· int A. Consider now y ∈ int T (A). Then there exist y
∈ T (A) and λ ∈ ]0, 1[ such that y = (1 − λ)y
+ λT (a). Take x
∈ A such that y
= T (x
) and consider x := (1 − λ)x
+ λa. Since a ∈ int A and A is convex, x ∈ int A. Therefore y = T (x) ∈ T (int A). (ii) The proof is similar; just note that a ∈ raint A if and only if for every x ∈ A there exists λ > 0 such that (1 + λ)a − λx ∈ A and if a ∈ raint A, x ∈ A and λ ∈]0, 1[, then (1 − λ)a + λx ∈ raint A.

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2 Functional analysis over cones

Lemma 2.7.5. Let A ⊂ X × Y be a nonempty convex set. Suppose that either (x, y) ∈ int A or (x, y) ∈ raint A and dim X < ∞. Then for every sequence (xn ) ⊂ PrX (A) with (xn ) → x there exists (yn ) → y such that (xn , yn ) ∈ A for every n ∈ N. Proof. Without loss of generality we may assume that (x, y) = (0, 0). Suppose first that (0, 0) ∈ int A. Since (xn , 0) → (0, 0) ∈ int A, (xn , 0) ∈ A for n ≥ n0 . Taking y0 = 0 for n ≥ n0 and yn arbitrary such that (xn , yn ) ∈ A for n < n0 , the conclusion follows. Suppose now that dim X < ∞ and (0, 0) ∈ raint A. From the preceding lemma with T = PrX , we have that 0 ∈ raint(PrX (A)). Let X0 := span(PrX (A)). Since dim X < ∞ and PrX (A) is convex, there exists r > 0 such that X0 ∩ {x ∈ X | x ≤ r} ⊂ PrX (A). For every n ∈ N∗ consider yn ∈ Y such that (xn , yn ) ∈ A and yn ≤ inf{ y | (xn , y) ∈ A} + n−1 . Suppose that (yn ) → 0. Taking eventually subsequences, we may suppose that

yn ≥ r1 > 0 for every n ∈ N and (xn / xn ) → x. There exists a base {u1 , u2 , . . . , up } of X0 such that ui = r for 1 ≤ i ≤ p and x ∈ int C, where p C = { i=1 λi ui | λi ≥ 0}. (For example, one takes a base {u
1 , u
2 , . . . , u
p } of  X0 , a bijective linear operator T : X0 → X0 such that x = p1 T (u1 + · · · + u
p ) and ui = T u
i for 1 ≤ i ≤ p.) Let (vi )pi=1 ⊂ Y be such that (ui , vi ) ∈ A for every i. Since (xn / xn ) → x ∈ int C, there exists n0 ∈ N such that xn ∈ C for every n ≥ n0 . Therefore, there exist the sequences (λni ) ⊂ [0, ∞[, 1 ≤ i ≤ p, p n ui for n ≥ n0 . Since (xn ) → 0, (λni ) → 0 for every such that xn = i=1 λi  p i. We may suppose that i=1 λni ≤ 1 for n ≥ n0 . Since (0, 0) ∈ A, it follows that   p p



n xn , λi vi = λni (ui , vi ) ∈ A, i=1

i=1

p p whence yn ≤ i=1 λni vi +n−1 ≤ i=1 λni vi +n−1 → 0, a contradiction.
Theorem 2.7.6. Let Γ : X ⇒ Y be a convex multifunction. Suppose that one of the following conditions holds: (i) int(gr Γ ) = ∅, x ∈ int(dom Γ ) and y ∈ Γ (x); (ii) X and Y are finite-dimensional, x ∈ raint(dom Γ ), and y ∈ Γ (x). Then Γ is semidifferentiable at (x, y). Proof. We give the proof of (ii), that of (i) being similar (in this case one takes into account that PrX is an open linear operator). We already know that DΓ (x, y)(u) ⊂ DΓ (x, y)(u) = {v ∈ Y | (u, v) ∈ TB (gr Γ, (x, y))}

∀ u ∈ X.

Let us show that v ∈ DΓ (x, y)(u) if (u, v) ∈ raint TB (gr Γ, (x, y)), even without asking that x ∈ raint(dom Γ ).

2.7 Tangent cones and differentiability of multifunctions

77

Without loss of generality we consider that (x, y) = (0, 0). Take (u, v) ∈ raint TB (gr Γ, (0, 0)) = raint(]0, ∞[· gr Γ ) = ]0, ∞[· raint(gr Γ ); we used the relation raint A = raint(cl A), valid if A is a convex subset of a finite-dimensional space, for the first equality and Lemma 2.7.4 for the last one. Then there exist λ
> 0 and (u
, v
) ∈ raint(gr Γ ) such that (u, v) = λ
(u
, v
). Let (tn ) → 0+ and (un ) → u such that tn un = x+tn un ∈ dom Γ for every n ∈ N. It follows that (un ) ⊂ aff(dom Γ ) = span(dom Γ ) and (un /λ
) → u
∈ raint(dom Γ ). So there exists n0 ∈ N such that un /λ
∈ dom Γ for n ≥ n0 . So, using Lemma 2.7.5, there exists a sequence (vn
) ⊂ Y such that (vn
) → v
and (un /λ
, vn
) ∈ gr Γ for n ≥ n0 . Let vn := λ
vn
for n ≥ n0 and arbitrary for n < n0 . So (vn ) → v. Since (tn ) → 0+ , there exists n1 ≥ n0 such that tn ≤ 1/λ
for n ≥ n1 . Since (0, 0) ∈ gr Γ , we obtain that (x, y) + tn (un , vn ) = tn λ
(un /λ
, vn
) ∈ gr Γ for n ≥ n1 . Therefore v ∈ DΓ (x, y)(u). Consider now u ∈ raint TB (dom Γ ; 0) = ]0, ∞[· raint(dom Γ ) and v ∈ DΓ (x, y)(u). Therefore u ∈ raint (]0, ∞[· dom Γ ) and (u, v) ∈ cl (]0, ∞[· gr Γ ). By Lemma 2.7.5, there exists v0 ∈ Y such that (u, v) ∈ raint (]0, ∞[· gr Γ ). It follows that for every λ ∈ ]0, 1[, (u, (1 − λ)v + λv0 ) = (1 − λ)(u, v) + λ(u, v0 ) ∈ raint (]0, ∞[· gr Γ ) . By what precedes, we have that (1 − λ)v + λv0 ∈ DΓ (x, y)(u) for every λ ∈ ]0, 1[. Taking into account that DΓ (x, y)(u) is a closed set, we obtain that v ∈ DΓ (x, y)(u) for λ → 0. Therefore for every u ∈ raint TB (dom Γ, x), DΓ (x, y)(u) = {v ∈ Y | (u, v) ∈ TB (gr Γ, (x, y))} = DΓ (x, y)(u); i.e., Γ is semidifferentiable at every u ∈ raint TB (dom Γ, x). If our initial condition x ∈ raint(dom Γ ) holds, then TB (dom Γ ; x) = span(dom Γ − x) = raint TB (dom Γ, x), and so Γ is semidifferentiable.
Remark 2.7.7. The notion of semidifferentiable multifunction is introduced by Penot in [289]. Theorem 2.7.6 (i) is established in [289, Prop. 3.5]. Consider now another normed vector space U, the multifunction Λ : U ⇒ X, and the function f : X × U → Y . With these we associate the multifunction f Λ : U ⇒ Y defined by f Λ(u) := f (Λ(u) × {u}). We are interested in evaluating Df Λ(u, y) for some (u, y) ∈ gr f Λ. Recall that a multifunction S : X ⇒ Y is upper Lipschitz at x ∈ dom S if there exist L, ρ > 0 such that S(x) ⊂ S(x) + L x − x BY for every x ∈ B(x, ρ). Theorem 2.7.8. Suppose that x ∈ Λ(u), y = f (x, u) and that f is Fr´echet differentiable at (x, u). Then ∇x f (x, u) (DΛ(u, x)(u)) + ∇u f (x, u)(u) ⊂ Df Λ(u, y)(u) and

∀u ∈ U

(2.48)

78

2 Functional analysis over cones

∇x f (x, u) (DΛ(u, x)(u)) + ∇u f (x, u)(u) ⊂ Df Λ(u, y)(u)

∀ u ∈ U. (2.49)

Consider Λ : U × Y ⇒ X,

 y) := {x ∈ Λ(u) | f (x, u) = y}. Λ(u,

Moreover, suppose that dim X < ∞ and either (a) Λ(u) = {x} and Λ is upper  y) = {x} and Λ is upper Lipschitz at (u, y). Then Lipschitz at u or (b) Λ(u, ∇x f (x, u) (DΛ(u, x)(u)) + ∇u f (x, u)(u) = Df Λ(u, y)(u)

∀ u ∈ U. (2.50)

Proof. Let x ∈ DΛ(u, x)(u) and take y = ∇x f (x, u)(x)+∇u f (x, u)(u). Then there exist (tn ) → 0+ , ((un , xn )) → (u, x) such that x + tn xn ∈ Λ(u + tn un ) for every n ∈ N. Since f is differentiable at (x, u), we have that for some Y ⊃ (vn ) → 0, f (x + tn xn , u + tn un ) = y + tn (∇x f (x, u)(xn ) + ∇u f (x, u)(un ) + vn ) for every n ∈ N. Taking yn := ∇x f (x, u)(xn ) + ∇u f (x, u)(un ) + vn , since ∇f (x, u) is continuous, we have that (yn ) → y. Since y + tn yn ∈ f Λ(u + tn un ) for every n ∈ N, y ∈ Df Λ(u, y)(u). Hence (2.48) holds. Consider x ∈ DΛ(u, x)(u) and y as above. Let ((tn , un )) → (0+ , u) with u + tn un ∈ dom f Λ (= dom Λ) for every n. Then there exists (xn ) → x such that x + tn xn ∈ Λ(u + tn un ) for n ≥ n0 . Continuing as above, we get (yn ) → y with y + tn yn ∈ f Λ(u + tn un ) for n ≥ n0 . Therefore y ∈ Df Λ(u, y)(u), whence (2.49). Suppose now that dim X < ∞ and (a) or (b) holds. Consider y ∈ Df Λ(u, y)(u); then there exist (tn ) → 0+ , ((un , yn )) → (u, y) such that y + tn yn ∈ f Λ(u + tn un ) for every n ∈ N. Let x n ∈ Λ(u + tn un ) be such  + tn un , y + tn yn ), for every n. that y + tn yn = f ( xn , u + tn un ); i.e., x n ∈ Λ(u Since Λ is upper Lipschitz at u and Λ(u) = {x} in case (a) and Λ is upper  y) = {x} in case (b), there exists a bounded seLipschitz at (u, y) and Λ(u, quence (xn ) ⊂ X such that x n = x + tn xn for every n ∈ N. Since dim X < ∞, there exists a subsequence (xnp )p∈N converging to some x ∈ X. It follows that x ∈ DΛ(u, x)(u), and so y ∈ ∇x f (x, u) (DΛ(u, x)(u)) + ∇u f (x, u)(u). Taking into account (2.48), we obtain that (2.50) holds.
In the following example one of the conditions in (a) and (b) of Theorem 2.7.8 is not satisfied, and (2.50) does not hold. Example 2.7.9. ([354]) Let U = X = Y = R, Λ, Λ
: U ⇒ X and f : X×U → Y be defined by Λ(u) := [0, max(1, 1 + u)],

Λ
(u) := [0, max(1, 1 + u)[,

f (x, u) := x2 − x,

u = 0, x = 0, y = f (0, 0) = 0. Then Λ, Λ
are upper Lipschitz at u, but Λ(u) = {x} = Λ
(u) (so (a) does not hold for Λ and Λ
) and

2.7 Tangent cones and differentiability of multifunctions

 f Λ(u) =

if u ≤ 0, [− 14 , 0] [− 14 , u(1 + u)] if u > 0,

f
Λ(u) =



79

if u ≤ 0, [− 14 , 0] [− 14 , u(1 + u)[ if u > 0.

Let {xy1 , xy2 } = {x ∈ R | x2 − x = y}, with xy1 ≤ xy2 , for y ≥ − 14 . Then ⎧ y y ⎨ {x1 , x2 } if u ∈ R, y ∈ [− 14 , 0],  if u > 0, y ∈ ]0, u(1 + u)], Λ(u, y) = {xy2 } ⎩ ∅ otherwise, ⎧ y y {x , x } if u ∈ R, y ∈ [− 14 , 0[, ⎪ ⎪ ⎨ 1 2 {0} if u ∈ R, y = 0, Λ
(u, y) = {xy2 } if u > 0, y ∈ ]0, u(1 + u)[, ⎪ ⎪ ⎩ ∅ otherwise.  y) = {0, 1} = {x}, while Λ
is Now, Λ is upper Lipschitz at (u, y) but Λ(u, not upper Lipschitz at (u, y) (so (b) does not hold for Λ and Λ
). One obtains that DΛ(u, x)(u) = DΛ
(u, x)(u) = R × R+ ,


∀ u ∈ R,

Df Λ(u, y)(u) = Df Λ (u, y)(u) = {(u, y) ∈ R2 | y ≤ max(0, u)},

∀ u ∈ R.

Since ∇x f (x, u) = −1 and ∇u f (x, u) = 0, ∇x f (x, u) (DΛ(u, x)(u))+∇u f (x, u)(u) = ]−∞, 0] = ]−∞, u] = Df Λ(u, x)(u) for every u > 0, and similarly for Λ
. For other types of tangent cones one can consult Aubin and Frankowska’s book [13]. Formulae (2.48) and (2.50) are stated by Tanino [354] under the supplementary conditions that U, X, and Y are finite-dimensional and f is of class C 1 and by Klose [212, Th. 4.1].

3 Optimization in Partially Ordered Spaces

3.1 Solution Concepts 3.1.1 Approximate Minimality In the last years several concepts for approximately efficient solutions of a vector optimization problem were published (see [110], [127], [93, 94], [241], [271], [329], [337, 338], [362]). The reason for introducing approximately efficient solutions is the fact that numerical algorithms usually generate only approximate solutions anyhow, and moreover, the set of efficient points may be empty, whereas approximately efficient points always exist under very weak assumptions. We begin with our concept given in [127]. Let M and B be subsets of the topological vector space Y with 0 ∈ bd B, let k 0 ∈ Y be such that cl B + (0, ∞)k 0 ⊂ int B, and let ε ≥ 0. Definition 3.1.1. (Approximate efficiency, εk 0 -efficiency) An element yε ∈ M is said to be εk 0 -efficient on M w.r.t. B if   M ∩ yε − εk 0 − (B \ {0}) = ∅. The set of εk 0 -efficient points of M w.r.t. B will be denoted by Eff(M, Bεk0 ). For the case ε = 0 the set Eff(M, Bεk0 ) coincides with the usual set Eff(M, B) of efficient points of M w.r.t. B (compare Remark 2.1.3). An illustration of the εk 0 -efficient points of M w.r.t. B is furnished by Figure 3.1.1; see also Figure 3.1.2 Because B + (0, ∞)k 0 ⊂ int B ⊂ B \ {0}, for 0 ≤ ε1 ≤ ε2 , we have that Eff(M, B) ⊆ Eff(M, Bε1 k0 ) ⊂ Eff(M, Bε2 k0 ). Approximate efficiency can also be defined by scalarization. Definition 3.1.2. Let C be a cone and ϕ : Y → R any C-monotone functional. An element yε ∈ M is said to be ε-efficient w.r.t. ϕ if

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3 Optimization in partially ordered spaces

6 b b b bbs y b   b b b bb 0 b y − εk − B b b b b

* k0  





M

B

-

Figure 3.1.1. The set of approximately efficient elements (where the distance between it and the set of efficient elements is unbounded).

∀ y ∈ M : y ∈ yε − C ⇒ ϕ(yε ) ≤ ϕ(y) + ε. The set of such ε-efficient points of M w.r.t. ϕ and C will be denoted by ε- Eff ϕ (M, C). Simple examples show that even if k 0 ∈ int C and cl B + int C ⊂ B, generally ε- Eff ϕ (M ) = Eff(M, Bεk0 ). So it seems to be very interesting to study relations between the Definitions 3.1.1 and 3.1.2. We use the functional ϕcl B,k0 introduced in Section 2.5: ϕcl B,k0 (y) := inf{t ∈ R | y ∈ tk 0 − cl B}. Theorem 3.1.3. Let M, B ⊂ Y be nonempty sets such that 0 ∈ bd B, let C ⊂ Y be a cone, k 0 ∈ int C, and ε ≥ 0. Assume that cl B + int C ⊂ B. Consider yε ∈ Eff(M, Bεk0 ) and ϕ : Y → R, ϕ(y) := ϕcl B,k0 (y − yε ). Then ϕ is a finite-valued, continuous, and strictly (int C)-monotone (even strictly C-monotone if cl B + (C \ {0}) ⊂ int B) functional such that ∀ y ∈ M : ϕ(yε ) ≤ ϕ(y) + ε. In particular, yε ∈ ε- Eff ϕ (M, C). Proof. By Proposition 2.3.4 we have that ϕcl B,k0 is finite-valued and continuous, while from Theorem 2.3.1(g) we obtain that ϕcl B,k0 is strictly (int C)monotone (strictly C-monotone if cl B + (C \ {0}) ⊂ int B). It is obvious that ϕ has the same properties.

3.1 Solution concepts

83

6 b

b

b b

b b

b bb b b bb bb bb bs y bb b y − εk0 − C b b bb bb bb b b bb b M b b b * k0     b

-

b

b bb C

Figure 3.1.2. The set of approximately efficient elements with a bigger cone C.

Assume that there exists y ∈ M such that ϕ(y) + ε < ϕ(yε ) = 0. It follows that there exists t < −ε with y − yε ∈ tk 0 − cl B, and so   y ∈ yε − εk 0 − cl B + (−ε − t)k 0 ⊂ yε − εk 0 − int B ⊂ yε − εk 0 − (B \ {0}), a contradiction with yε ∈ Eff(M, Bεk0 ).




Theorem 3.1.4. Let M, B ⊂ Y be proper sets, C ⊂ Y a cone, k 0 ∈ int C, and ε ≥ 0. Assume that cl B + (C \ {0}) ⊂ int B. If yε ∈ M is such that ∀ y ∈ M : ϕcl B,k0 (yε ) ≤ ϕcl B,k0 (y) + ε,

(3.1)

then there is an open set D ⊂ Y with 0 ∈ bd D and cl D + (C \ {0}) ⊂ D such that yε ∈ Eff(M, Dεk0 ). Proof. First of all, note that cl B is proper (otherwise, B = Y ) and 0 ∈ / int C (otherwise, C = Y , and so B = Y ). Taking into account Remark 2.3.3 we have that cl B + int C ⊂ cl B. Using Proposition 2.3.4 we obtain that ϕ := ϕcl B,k0 is a finite-valued continuous function, while from Theorem 2.3.1 we have that ϕ is strictly C-monotone. Consider the set D := {y ∈ Y | ϕ(yε − y) < ϕ(yε )} = yε − tε k 0 + int B, where tε := ϕ(yε ), the last equality being a consequence of (2.29). Hence D is open. Taking again into account Remark 2.3.3, we obtain that

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cl D = yε −tε k 0 +cl B, bd D = yε −tε k 0 +bd B = {y ∈ Y | ϕ(yε −y) = ϕ(yε )}. Therefore 0 ∈ bd D. From the inclusion cl B + (C \ {0}) ⊂ int B and the above formulae one obtains immediately that  cl D +0 (C \ {0})  ⊂ D. Using (2.29), condition (3.1) is equivalent to M ∩ (t − ε)k − int B = ∅, or equivalently, ε   M ∩ yε − εk 0 − D = ∅; i.e., yε ∈ Eff(M, Dεk0 ).
3.1.2 A General Scalarization Method Scalarization of a given vector optimization problem (VOP) means converting that problem into an optimization problem (or a family of such problems) with a (common) real-valued objective function to be minimized. If the solutions of the latter problems (often called scalarized problems) are also solutions of the given VOP, then the obtained scalarization is advantageous in solving the VOP, because methods of usual “scalar” optimization (nonlinear programming) can be used. For the finite-dimensional case see, for example, [259]. The most advantageous scalarization is obtained by using suitable monotone functionals; the following propositions serve as theoretical background. Proposition 3.1.5. Let Y be a linear space partially ordered by a nontrivial pointed convex cone C, let M be a nonempty subset of Y , and let ϕ : Y → R be a proper functional with M ∩ dom f = ∅. Assume that y 0 ∈ M satisfies ϕ(y) ≥ ϕ(y 0 )

∀ y ∈ M.

(3.2)

Then y 0 ∈ Eff(M, C) if one of the following conditions is satisfied: (i) ϕ is C-monotone on M and y 0 is uniquely determined, (ii) ϕ is strictly C-monotone on M ∩ dom f . Proof. Let y 1 ∈ M ∩ (y 0 − C) and assume that y 1 = y 0 . In case (i), because ϕ is C-monotone, we obtain that ϕ(y 1 ) ≤ ϕ(y 0 ), contradicting the uniqueness of y 0 . In case (ii), because ϕ is strictly C-monotone, we obtain that ϕ(y 1 ) < ϕ(y 0 ), contradicting (3.2). Therefore y 0 ∈ Eff(M, C).
Note that condition (3.2) may be replaced by an apparently weaker one: ϕ(y) ≥ ϕ(y 0 )

∀ y ∈ M ∩ (m − C),

where m ∈ M (or m ∈ Y with M ∩ (m − C) = ∅). Examples of C-monotone functionals are furnished by the elements of C + , while examples of strictly C-monotone functionals are given by the elements of C # . An easy application of the preceding result is the next corollary. Corollary 3.1.6. Let Y be an s.l.c.s., C ⊂ Y a convex cone with C # = ∅, and ∅ = M ⊂ Y . Then y 0 ∈ Eff(M, C) if and only if there exist m ∈ M and y ∗ ∈ C # such that y 0 is a solution of the following scalar minimization problem: y, y ∗  −→ min, s.t. y ∈ M, y ≤C m.

3.1 Solution concepts

85

Proof. If y 0 ∈ Eff(M, C), then M ∩ (y0 − C) = {y0 }, and so y 0 is a solution of the above problem for m = y 0 and arbitrary y ∗ ∈ C # . The converse implication follows immediately from case (ii) of the preceding proposition.
When M is C-convex; i.e., M + C is convex, the following result holds. Proposition 3.1.7. (Jahn) Let Y be an H.l.c.s., C ⊂ Y a proper pointed convex cone, and M ⊂ Y a nonempty C-convex set. If y 0 ∈ Eff(M, C) and M + C has nonempty interior, then there exists y ∗ ∈ C + \ {0} such that  0 ∗ y , y ≤ y, y ∗  ∀ y ∈ M. Proof. Since y 0 ∈ Eff(M, C), (M + C) ∩ (y 0 − C) = {y 0 }, and so y 0 ∈ / int(M + C). By Theorem 2.2.7 there exists y ∗ ∈ Y ∗ \ {0} such that  0 ∗ y , y ≤ y + c, y ∗  ∀ y ∈ M, ∀ c ∈ C. Replacing c ∈ C by tc with t ≥ 0 and c ∈ C, we obtain that y ∗ ∈ C + . Taking c = 0 in the above relation, we see that the conclusion holds.
An inspection of the proof of Proposition 3.1.5 shows that neither the convexity of C nor the fact that C is a cone was used. Also, additional properties of ϕ imply efficiency of y 0 w.r.t. larger sets having additional properties, as the next result shows. Theorem 3.1.8. (Gerth–Weidner [131]) Let B, M be nonempty subsets of the t.v.s. Y , and ϕ : Y → R a strictly B-monotone function. Assume that y 0 ∈ M is such that ϕ(y 0 ) ≤ ϕ(y) for all y ∈ M . Let H := {y ∈ Y | ϕ(y 0 − y) < ϕ(y 0 )}. Then B \{0} ⊂ H and y 0 ∈ Eff(M, H); in particular, y 0 ∈ Eff(M, B). Moreover, (a) if ϕ is convex, then H is convex; (b) if ϕ is continuous, then H is open and cl H + (B \ {0}) ⊂ H; (c) if ϕ is linear, then H ∪ {0} is a convex cone; (d) if 0 ∈ cl(B \ {0}), then 0 ∈ bd H. Proof. From the choice of H it is clear that M ∩ (y 0 − H) = ∅, and so y 0 ∈ Eff(M, H). Because ϕ is strictly B-monotone, it is also obvious that B \ {0} ⊂ H. Statements (a) and (c) are obvious. (b) By the continuity of ϕ we have that H is open and cl H ⊂ {y ∈ Y | ϕ(y 0 − y) ≤ ϕ(y 0 )}. Let y ∈ cl H and d ∈ B \ {0}. Since ϕ is strictly Bmonotone, ϕ(y 0 − y − d) < ϕ(y 0 − y) ≤ ϕ(y 0 ), and so y + d ∈ H. (d) Of course, 0 ∈ / H. Since B \ {0} ⊂ H, we have that 0 ∈ cl(B \ {0}) ⊂ cl H, and so 0 ∈ bd H.
Note that without asking 0 ∈ cl(B \ {0}) it is possible that 0 ∈ / cl H. Indeed, consider Y := R,

B := {0} ∪ [1, ∞),

M := [1, 4],

and the strictly B-monotone continuous function

y 0 := 1,

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3 Optimization in partially ordered spaces

ϕ : R → R,


 ϕ(y) := min y + 21 , max{1, y} .

/ cl H. Related to weakly efficient points the Then H = ( 21 , ∞), and so 0 ∈ following result holds; we make use of the function ϕD,k0 considered in Section 2.3. Theorem 3.1.9. Let Y be an s.t.v.s., B ⊂ Y a closed set with nonempty interior and 0 ∈ bd B, and M ⊂ Y a nonempty set. Assume that one of the following conditions holds: (i) there exists a cone C ⊂ Y with nonempty interior such that B +int C ⊂ B, (ii) B is convex and there exists k 0 ∈ Y such that B + R+ k 0 ⊂ B and B + Rk 0 = Y . Then y 0 ∈ w − Eff(M, B) if and only if y 0 ∈ M and there exists a continuous function ϕ : Y → R such that ϕ(y 0 − y) < 0 = ϕ(y 0 ) ≤ ϕ(x) ∀ y ∈ int B, ∀ x ∈ M.

(3.3)

Proof. If condition (3.3) we have that M ∩ (y 0 − int B) = ∅, and so y 0 ∈ w − Eff(M, B). For the necessity, take k 0 ∈ int C in case (i). In both cases consider D := B − y 0 . The conclusion follows from the nonconvex separation theorem (Theorem 2.3.6) taking ϕ := ϕD,k0 .
Remark 3.1.10. Knowing additional properties of the set B in the preceding theorem we can ask additional properties for the function ϕ (see Theorem 2.3.1). For example, in case (ii) of Theorem 3.1.9 we can ask that ϕ be convex (and surjective in all the cases). Theorems 3.1.8 and 3.1.9 can be found in a more detailed form in Gerth and Weidner’s paper [131]. Special cases of those theorems had been proved earlier; for example, scalarization results for weakly efficient points in the case Y = Rn and B = y 0 − Rn+ , k 0 ∈ int Rn+ with the functional ϕ(y) = inf{t ∈ R | y ∈ y 0 − Rn+ + tk 0 }, had been obtained by Bernau [27] and for k 0 = (1, . . . , 1)T by Brosowski and Conci [49]. Furthermore, compare Jahn [191, 192, 196, 197]. In the nineties, results of such types were useful to get general variants of Ekeland’s variational principle in the case in which the performance space Y has dimension greater than 1.

3.2 Existence Results for Efficient Points In this section we establish several existence results for maximal points with respect to transitive relations; then we apply them in topological vector spaces for preorders generated by convex cones. A comparison of existence results for efficient points is made at the end of the section.

3.2 Existence results for efficient points

87

3.2.1 Preliminary Notions and Results Concerning Transitive Relations In the sequel X is a nonempty set and t ⊂ X × X; i.e., t is a relation on X. If ∅ = Y ⊂ X and t ⊂ X × X, the restriction of t to Y is denoted by tY ; i.e., tY := t ∩ (Y × Y ). With the relation t on X we associate the following relations: tR := t ∪ ∆X ,

tN := t \ t−1 = t \ (t ∩ t−1 ),

tN R = (tN )R .

Hazen and Morin [160] call tN the asymmetric part of t. Some properties of these relations are given in the following lemma; the first three properties mentioned in (ii) are stated by Dolecki and Malivert in [98]. Lemma 3.2.1. Let t be a transitive relation on X and ∅ = Y ⊂ X. (i) tR is reflexive and transitive; tN ∩ ∆X = ∅; (ii) t◦tN ⊂ tN , tN ◦t ⊂ tN , tN ◦tN ⊂ tN , (tR )N = tN , tR ◦tN = tN ◦tR = tN ; (tN )N = tN ; (iii) tN R is reflexive, antisymmetric, and transitive; (tN )N = (tN R )N = tN ;    Y  Y (iv) tY R = (tR ) , tY N = (tN ) . Proof. The proof is easy; one uses only the definitions.
Y Y Y , t , and t the Taking into account the above lemma, we denote by t R N NR      relations tY R , tY N , and tY N R , respectively. The preceding lemma shows that with every transitive relation one can associate a partial order. It is useful to know whether they determine the same maximal and minimal points. As noted in Section 2.1, Max(Y ; t) = Min(Y, t−1 ), where t ⊂ X × X is a (transitive) relation and ∅ = Y ⊂ X; so it is sufficient to study only the problems related to maximal points. Corollary 3.2.2. Let ∅ = Y ⊂ X and let t be a transitive relation on X. Then Max(Y ; t) = Max(Y ; tR ) = Max(Y ; tN ) = Max(Y ; tN R ). / tN for every Proof. Let y¯ ∈ Y . Note first that y¯ ∈ Max(Y, tN ) iff (y, y) ∈ y ∈ Y . Using this remark we have immediately that Max(Y ; t) = Max(Y ; tN ). Applying the preceding lemma, we get tN = (tR )N = (tN )N = (tN R )N . The other equalities are now obvious.
The above corollary shows that the problem of existence (for example) of maximal points w.r.t. a transitive relation t reduces, theoretically, to the same problem for the partial order tN R . Another way to reduce this problem to one for a partial order is given by the following known result. −1

Proposition 3.2.3. Let t be a transitive relation on X and ρ = tR ∩ (tR ) . Then ρ is an equivalence relation, and & t = {(& x, y&) | (x, y) ∈ t} is a partial & = X/ρ, where x order on X & is the class of x ∈ X with respect to ρ. Moreover, & & if x ∈ X, x ∈ Max(X; t) if and only if x & ∈ Max(X; t).

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In the sequel we shall also use the notation Yt+ (x) and Yt− (x) for the upper and lower sections of Y ⊂ X with respect to t and x ∈ X. So Yt+ (x) := {y ∈ Y | (x, y) ∈ tR },

Yt− (x) := {y ∈ Y | (y, x) ∈ tR },

respectively; the most common case is for x ∈ Y . Similarly, for Z ⊂ X, we consider  Yt+ (Z) := {y ∈ Y | ∃ z ∈ Z : (z, y) ∈ tR } = Yt+ (z), z∈Z

Yt− (Z) Note that

Yt+



:= {y ∈ Y | ∃ z ∈ Z : (y, z) ∈ tR } =

Yt+ (Z)





Yt− (z).

z∈Z

=

Yt+ (Z)

(so one may suppose that Z ⊂ Y ) and

Max(Yt+ (Z); t) = Yt+ (Z) ∩ Max(Y ; t),

Min(Yt− (Z); t) = Yt− (Z) ∩ Min(Y ; t). (3.4) We say that ∅ = Y ⊂ X has the domination property (w.r.t. t) (DP) if Max(Yt+ (y); t) = ∅, for every y ∈ Y (i.e., every element of Y is dominated by a maximal element of Y ). A quite important problem is how to extend other notions related to partially ordered sets, like chain or increasing net, to sets endowed with transitive relations. Related to this problem we have the next result. Proposition 3.2.4. Let t be a transitive relation on X and ∅ = Y ⊂ X. (i) If tYR is a partial order on Y , then tYR ⊂ tN R . (ii) If µ is a total order on Y such that µ ⊂ tN R , then µ = tYN R . Proof. (i) Let µ = tYR be a partial order on Y and consider (x, y) ∈ µ. If x = y, we have that (x, y) ∈ tN R . Suppose now that x = y. It follows that (x, y) ∈ t. Assuming that (x, y) ∈ / tN R , we have that (x, y) ∈ / t \ t−1 , whence −1 (x, y) ∈ t . Therefore (y, x) ∈ t, and so (y, x) ∈ µ. Since µ is antisymmetric, it follows that x = y, a contradiction. Hence (x, y) ∈ tN R . (ii) Suppose that (Y, µ) is totally ordered and µ ⊂ tN R . Of course, µ ⊂ tYN R . Consider (x, y) ∈ tYN R (⊂ t) and suppose that (x, y) ∈ / µ. Of course, x = y. Since µ is a total order on Y , we have that (y, x) ∈ µ, and so (y, x) ∈ tN = / t, a contradiction. Therefore µ = tYN R .
t \ t−1 . It follows that (x, y) ∈ Note that in (ii) we cannot replace tN R by t or tR (take X = R and t = R×R). Let X be endowed with the transitive relation t and let (xi )i∈I ⊂ X be a net. We say that (xi ) is t-increasing [t-decreasing] if (xi , xj ) ∈ t [(xj , xi ) ∈ t] for all i, j ∈ I with i  j and i = j; (xi ) is strictly t-increasing [strictly t-decreasing] if (xi ) is tN -increasing [tN -decreasing]. In the sequel we say that ∅ = A ⊂ (X, t), with t a transitive relation on X, is a chain (w.r.t. to t) if tA R is a total order on A, while A is well-ordered (w.r.t. to t) if A is well-ordered by tA R. The following interesting result is due to Gajek and Zagrodny (see [123]).

3.2 Existence results for efficient points

89

Proposition 3.2.5. Let t be a transitive relation on X. Then there exists a nonempty and well-ordered subset W ⊂ X such that for every x ∈ X \ W there exists w ∈ W with (x, w) ∈ t or (w, x) ∈ / t. Proof. Let L = {U ⊂ (X, t) | U well-ordered}. Then L is nonempty because {x} ∈ L for every x ∈ X. Consider the relation η = {(U, V ) ∈ L × L | U ⊂ V and ∀ v ∈ V, ∀ u ∈ U : (v, u) ∈ t ⇒ v ∈ U }. It is easy to see that (L, η) is a partially ordered set. Let us show that (L, η) has maximal elements. For this let U ⊂ L be a (nonempty) chain, and show that U is upper bounded in L. Consider U = ∪U. We have that U ∈ L; for this note first that t is antisymmetric on U . Indeed, let x, y ∈ U be such that (x, y) ∈ t and (y, x) ∈ t. Because U is a chain, there exists U ∈ U ⊂ L such that x, y ∈ U . Since t is antisymmetric on U , we obtain that x = y. Similarly, we obtain that U is totally ordered. Now let ∅ = A ⊂ U . There exists U0 ∈ U such that A0 = A ∩ U0 = ∅. Let a0 ∈ A0 be the least element of A0 as a subset of U0 . Let us show that a0 is the least element of A. Let a ∈ A \ {a0 }; there exists U ∈ U such that a ∈ U . If U η U0 , then a ∈ A ∩ U0 , and so (a0 , a) ∈ t. In the contrary case, U0 η U . Of course, a and a0 are comparable. Suppose that (a, a0 ) ∈ t; from the definition of η, we obtain that a ∈ U0 , whence a ∈ A0 . It follows that (a0 , a) ∈ t. Therefore a0 is the least element of A. Hence U ∈ L. It is easy to see that U η U for every U ∈ U. Indeed, let u ∈ U and u ∈ U ∈ U such that (u, u) ∈ t. There exists U
∈ U such that u ∈ U
. If U
η U , there is nothing to prove. If U η U
, by the very definition of η, we have that u ∈ U . By Zorn’s lemma, L has a maximal element W . Let x ∈ X \ W . Suppose that the conclusion is not true. Then for every w ∈ W we have that (w, x) ∈ t and (x, w) ∈ / t. Consider W
= W ∪ {x}. It is obvious that W
is antisymmetric and totally ordered w.r.t. tR . Let ∅ = A
⊂ W
; if A
= {x}, of course, x is the least element of A
. In the contrary case we have that ∅ = A = A
\ {x} ⊂ W . Taking a the least element of A, it is obvious that a is also the least element of A
. Therefore W
∈ L. But W η W
, contradicting the maximality of W .
3.2.2 Existence of Maximal Elements with Respect to Transitive Relations We begin with the following result. Proposition 3.2.6. Let t be a transitive relation on X and ∅ = Y ⊂ X. Suppose that one of the following conditions holds: (i) every nonempty set A ⊂ Y with A × A ⊂ t ∪ t−1 ∪ ∆X is upper bounded in Y ; (ii) every chain in Y is upper bounded in Y ; (iii) every well-ordered subset of Y is upper bounded in Y . Then Y has the domination property. In particular, Max(Y ; t) = ∅.

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3 Optimization in partially ordered spaces

Proof. Without loss of generality we may suppose that Y = X. It is obvious that (i) ⇒ (ii) ⇒ (iii). Assume that (iii) holds. By Gajek and Zagrodny’s lemma (Proposition 3.2.5) there exists a well-ordered nonempty set W ⊂ X such that for every x ∈ X \ W , there exists w ∈ W with (x, w) ∈ t or (w, x) ∈ / t. By hypothesis there exists x ∈ X such that (w, x) ∈ tR for every w ∈ W . Assume that for some u ∈ X we have (x, u) ∈ tN . Of course, u∈ / W and (w, u) ∈ tR ◦ tN ⊂ tN for every w ∈ W . By the choice of W , there exists wu ∈ W such that (u, wu ) ∈ t. It follows that (u, u) ∈ t ◦ tN ⊂ tN , a contradiction. Therefore x ∈ Max(X; t). The fact that X has (DP) follows from the fact that if one of the conditions (i), (ii), (iii) holds for X, it holds also for Xt+ (x) for every x ∈ X.
Corollary 3.2.7. Let t be a transitive relation on X and ∅ = Y ⊂ X. Suppose that one of the following conditions holds: (i) every t-increasing net of Y is upper bounded in Y ; (ii) every strictly t-increasing net of Y is upper bounded in Y ; (iii) every t-increasing net (yi )i∈I ⊂ Y with I totally ordered is upper bounded in Y ; (iv) every strictly t-increasing net (yi )i∈I ⊂ Y with I totally ordered is upper bounded in Y ; (v) every t-increasing net (yi )i∈I ⊂ Y with I well-ordered is upper bounded in Y ; (vi) every strictly t-increasing net (yi )i∈I ⊂ Y with I well-ordered is upper bounded in Y . Then Y has the domination property. In particular, Max(Y ; t) = ∅. Proof. Of course, (i) ⇒ (ii), (iii) ⇒ (iv), (v) ⇒ (vi), (i) ⇒ (iii) ⇒ (v), and (ii) ⇒ (iv) ⇒ (vi). On the other hand, if (vi) holds, then condition (iii) of the preceding proposition is satisfied. Therefore the conclusions hold.
The following result is due, essentially, to Gajek and Zagrodny [123]. Proposition 3.2.8. Let t be a transitive relation on X and ∅ = Y ⊂ X. Suppose that the following two conditions hold: (i) every nonempty well-ordered subset W of Y is at most countable; (ii) every strictly t-increasing sequence of Y is upper bounded in Y . Then Y has the domination property. In particular, Max(Y ; t) = ∅. Proof. Without loss of generality we may suppose that Y = X. In order to apply Proposition 3.2.6 we show that every well-ordered subset W of X is upper bounded. So let W ⊂ X be a well-ordered set. By hypothesis W is at most countable. If W has a greatest element, this is also an upper bound for W . Suppose that W does not have a greatest element. It follows that W is not finite (otherwise, since W is also totally ordered, it has a

3.2 Existence results for efficient points

91

greatest element). Therefore there exists p : N → W bijective. Observe first that for every k ∈ N the set {n ∈ N | n > k, (p(k), p(n)) ∈ t} is nonempty. In the contrary case we have that (p(n), p(k)) ∈ t for every n > k. Taking p(i) = max{p(0), . . . , p(k)}, p(i) is the greatest element of W , a contradiction. Let n0 = 0, n1 = min{n > n0 | (p(n0 ), p(n)) ∈ t}, and so on; therefore nk+1 = min{n > nk | (p(nk ), p(n)) ∈ t} for every k ∈ N. Define xk = p(nk ) ∈ X. It is clear that (xk ) is a strictly t-increasing sequence. Therefore there exists x such that (xk , x) ∈ tR for every k. We have that (p(n), x) ∈ tR for every n. In the contrary case for some n ∈ N, (p(n),x) ∈ / tR . It follows that (p(n),p(nk )) ∈ / tR , and so, by Proposition 3.2.4, (p(nk ), p(n)) ∈ tN , for every k; in particular, n = nk for every k. Since n > 0 = n0 , it follows that n ≥ n1 , and so n > n1 . Continuing in this way we get that n ≥ nk for every k, which, of course, is not possible. It follows that x is an upper bound of W . Therefore condition (iii) of Proposition 3.2.6 is satisfied, and so the conclusions hold.
Note that condition (i) of Corollary 3.2.7 is equivalent to    ∀ (yi )i∈I ⊂ Y, (yi ) t-increasing : Y ∩ Xt+ (yi ) = ∅, (3.5) i∈I

while (ii) is equivalent to  ∀ (yi )i∈I ⊂ Y strictly t-increasing : Y ∩



 Xt+N R (yi )

= ∅.

i∈I

A sufficient condition for (3.5) is the following one: ∀A ⊂ Y : Y ⊂



n     X \ Xt+ (a) ⇒ ∃ a1 , . . . , an ∈ A : Y ⊂ X \ Xt+ (ai ) . i=1

a∈A

(3.6) If τ is a topology on X and the upper sections of X are closed (for example, if t is closed in X × X), then the sets X \ Xt+ (a) are open; in this situation condition (3.6) is a kind of compactness of Y . The following result is related to this kind of condition. Proposition 3.2.9. Let t be a transitive relation on X and ∅ = Y ⊂ X. Assume that there exists a relation s on X such that s ◦ tN ⊂ t R and

(3.7) 

∀ (yi )i∈I ⊂ Y strictly t-increasing : Y ∩



 Xs+ (yi )

= ∅.

i∈I

Then Y has the domination property. In particular, Max(Y ; t) = ∅.

(3.8)

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3 Optimization in partially ordered spaces

Proof. Once again, without loss of generality we may suppose that Y = X. In order to apply Proposition 3.2.6 (ii) we show that every chain C of X is upper " bounded. So let ∅ = C ⊂ X be a chain. By hypothesis there exists x ∈ c∈C {x ∈ X | (c, x) ∈ s}. If C has a greatest element, there is nothing to prove. Let the contrary case hold and c ∈ C. Then there exists c
∈ C such that (c, c
) ∈ tN . Of course, (c
, x) ∈ s; using (3.7) we obtain that (c, x) ∈ tR . Hence x is an upper bound for C. Using Proposition 3.2.6 the conclusions follow.
The above proof shows that the conclusions of Proposition 3.2.9 hold if in (3.8) I is totally ordered [even well-ordered, applying in this case Proposition 3.2.6 (ii)]. In the sequel, in this section, we suppose that (X, τ ) is a topological space. In this situation one can formulate other conditions for the existence of maximal points. Having a net (xi )i∈I , by (xϕ(j) )j∈J , or simply (xϕ(j) ), we denote a subnet of (xi ). Corollary 3.2.10. Let t be a transitive relation on the topological space (X, τ ) and ∅ = Y ⊂ X. Assume that one of the following conditions holds: ∀ (yi )i∈I ⊂ Y t-increasing, ∃ (yϕ(j) )j∈J → y ∈ Y, ∀ j ∈ J : (yϕ(j) , y) ∈ tR , (3.9) ∀ (yi )i∈I ⊂ Y strictly t-increasing, ∃ (yϕ(j) )j∈J → y ∈ Y, ∀ j ∈ J : (yϕ(j) , y) ∈ tR .

(3.10)

Then Y has the domination property. In particular, Max(Y ; t) = ∅. Proof. If (3.9) or (3.10) holds, then condition (i) or (ii) of the preceding corollary is satisfied, respectively. Indeed, suppose that (3.9) holds and let (yi )i∈I ⊂ Y be t-increasing. There exists a subnet (yϕ(j) )j∈J → y ∈ Y such that (yϕ(j) , y) ∈ tR for every j ∈ J. Fix k ∈ I. For every i ∈ I there is some ji ∈ J with i  ϕ(j) for all j ∈ J, ji  j. Since (yi ) is t-increasing, (yk , yϕ(jk ) ) ∈ tR ; since (yϕ(jk ) , y) ∈ tR , we get (yk , y) ∈ t. Therefore y is an upper bound for (yi ) in Y . The conclusion holds.
When the upper or lower sections of X are closed, we may consider other conditions, too. Proposition 3.2.11. Let t be a transitive relation on the topological space (X, τ ) and ∅ = Y ⊂ X. Consider the following conditions: ∀ (yi )i∈I ⊂ Y t-increasing : ∃ (yϕ(j) )j∈J → y ∈ Y, ∀ (yi )i∈I ⊂ Y strictly t-increasing : ∃ (yϕ(j) )j∈J → y ∈ Y, ∀ (yi )i∈I ⊂ Y t-increasing, ∃ y ∈ Y : yi → y, and

(3.11) (3.12) (3.13)

3.2 Existence results for efficient points

∀ (yi )i∈I ⊂ Y strictly t -increasing, ∃ y ∈ Y : yi → y.

93

(3.14)

If the upper sections of X are closed, then (3.11) ⇔ (3.9) and (3.12) ⇔ (3.10), while if t is a partial order and the upper and lower sections of X (w.r.t. t) are closed, then (3.11) ⇔ (3.13) and (3.12) ⇔ (3.14). Proof. The implications (3.9) ⇒ (3.11), (3.13) ⇒ (3.11), (3.10) ⇒ (3.12), and (3.14) ⇒( 3.12) are always true. Let us show that (3.11) ⇒ (3.9) and (3.13) ⇒ (3.9) under the mentioned supplementary conditions, the other two being proved similarly. Suppose that the upper sections of X are closed and (3.11) holds. Consider (yi )i∈I ⊂ Y a t-increasing net. There exists the subnet (yϕ(j) )j∈J converging to y ∈ Y . With the notation from the proof of the preceding proposition, we have that (yϕ(j) )ji j ⊂ Xt+ (yi ). Taking the limit, we get that lim yϕ(j) = y ∈ cl Xt+ (yi ) = Xt+ (yi ) for all i ∈ I. Therefore y is an upper bound for (yi )i∈I in Y , whence y is an upper bound for (yϕ(j) ), too. Suppose now that t is a partial order and the upper and lower sections of X are closed. Let us show that (3.11)⇒(3.13). So, suppose that (3.11) holds and (yi )i∈I ⊂ Y is t-increasing. By hypothesis there exists a subnet (yϕ(j) )j∈J converging to y ∈ Y . As in the proof of (3.11)⇒(3.9), we have that yi ∈ Xt− (y) for every i ∈ I. Suppose that yi →y. It follows that there exists a neighborhood V0 of y in X such that I0 := {i ∈ I | yi ∈ / V0 } is cofinal in I. Of course, (yi )i∈I0 is a t-increasing net of Y . Therefore, by (3.11), there exists (yψ(k) )k∈K a subnet of (yi )i∈I0 such that yψ(k) → y0 ∈ Y ; of course, (yψ(k) )k∈K is also a subnet of (yi )i∈I . The same argument as above gives yi ∈ Xt− (y0 ) for every i ∈ I. From (yϕ(j) )j∈J ⊂ Xt− (y0 ) we get y ∈ Xt− (y0 ), while from (yψ(k) )k∈K ⊂ Xt− (y0 ) we get y0 ∈ Xt− (y). Since t is antisymmetric, it follows that y = y0 , which contradicts the choice of I0 (because V0 is a neighborhood of y0 in this case).
Corollary 3.2.12. Let t be a transitive relation on the separated topological space (X, τ ) and ∅ = Y ⊂ X. Assume that the upper sections of X are closed and Y is compact. Then Y has the domination property. In particular, Max(Y ; t) = ∅. Proof. In the hypotheses of the corollary condition (3.11) holds, and because the upper sections of X are closed, (3.9) holds, too. The conclusions follow from Corollary 3.2.10.
Hazen and Morin proved the result of Corollary 3.2.12 in [160, Cor. 2.8]; note that Theorem 2.2 of [160] does not follow from any of the preceding results. One can ask naturally whether a kind of converse of Proposition 3.2.6 (Corollary 3.2.7) is true, in the sense that if (X; t) has the domination property, is it true that every chain (every t-increasing net) of X is upper bounded? The answer is negative, as the following example shows.

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3 Optimization in partially ordered spaces

Example 3.2.13. Let X = {fn | n ∈ N∗ } ∪ {gn | n ∈ N∗ }, where fn , gn : [0, 1] → R, fn (x) = xn and % xn if x ∈ [0, 2−1/n ], 1/n gn (x) = 2 (1 − x) if x ∈ (2−1/n , 1]. 2(21/n −1) For f, g ∈ X we put f  g iff g(x) ≤ f (x) for every x ∈ [0, 1]. Note that gn is strictly increasing on [0, 2−1/n ] and strictly decreasing on [2−1/n , 1], and so attains its greatest value, 21 , for x = 2−1/n . It follows that for n, m ∈ N∗ , n = m, gn , and gm are not comparable. Moreover, fn  fn+1  gn+1 for every n ∈ N∗ , but fn gm for n, m ∈ N∗ , n > m. It follows that the set of maximal points of X is {gn | n ∈ N∗ }. So (X, ) has the domination property, but the chain {fn | n ∈ N∗ } is not upper bounded. 3.2.3 Existence of Efficient Points with Respect to Cones To begin with, let X be a real vector space and ∅ = C ⊂ X a convex cone. As usual, with C we associate (see Theorem 2.1.13) the reflexive and transitive relation (3.15) ≤C := t := {(x, y) ∈ X × X | y − x ∈ C}. Taking L := C ∩ (−C), the lineality space of C, the equivalence relation ρ := t ∩ t−1 is {(x, y) ∈ X × X | y − x ∈ L}. So we get tN = {(x, y) ∈ X × X | y − x ∈ C \ L}.

(3.16)

Using Lemma 3.2.1 we obtain that C + (C \ L) = C \ L,

(C \ L) + (C \ L) ⊂ C \ L;

the above formulae were obtained by Luc in [243, 244]. So, tN R = {(x, y) ∈ X × X | y − x ∈ (C \ L) ∪ {0}}. It follows that (C \ L) ∪ {0} is a pointed convex cone. Note that for x ∈ X and Y ⊂ X one has YC+ (x) := Y≤+C (x) = Y ∩(x+C) and YC− (x) = Y ∩(x−C). Therefore the upper and lower sections of X (w.r.t. C) are closed if and only if C is closed. Similarly, for Z ⊂ X, YC+ (Z) := Y ∩ (Z + C) and YC− (Z) := Y ∩ (Z − C). In accordance with the notions introduced in Section 3.2.1, the net (xi )i∈I ⊂ X is [strictly] C-increasing if xj − xi ∈ C [xj − xi ∈ (C \ L) ∪ {0}] for all i, j ∈ I, i  j, i = j; (xi )i∈I is C-decreasing if (−xi )i∈I is Cincreasing. Of course, the set Y ⊂ X is C-upper (lower) bounded if Y ⊂ x0 −C (Y ⊂ x0 +C) for some x0 ∈ X. Moreover, the set Max(Y, ≤C ) will

3.2 Existence results for efficient points

95

be denoted by Max(Y ; C). An element y ∈ Max(Y ; C) is called an efficient point of Y (w.r.t. C). It is obvious that for ∅ = Y ⊂ X and C = X or C = {0} we have that Max(Y ; C) = Y . Taking into account this fact, in the sequel we shall suppose that C is proper, i.e., {0} = C = X. To Proposition 3.2.6 and Corollary 3.2.7 corresponds the following result. Proposition 3.2.14. Let ∅ = Y ⊂ X. Suppose that one of the following conditions holds: (i) every nonempty set Z ⊂ Y such that Z − Z ⊂ C ∪ (−C) is C-upper bounded in Y ; (ii) every chain in Y w.r.t. ≤C is C-upper bounded in Y ; (iii) every well-ordered subset of Y (w.r.t. ≤C ) is C-upper bounded in Y ; (iv) every C-increasing net of Y is C-upper bounded in Y . Then Y has the domination property. In particular, Max(Y ; C) = ∅. In the following proposition we gather some properties of efficient sets, most of them appearing in the literature in particular cases. Proposition 3.2.15. Let C, K ⊂ X be convex cones such that C ⊂ K, x ∈ X, and ∅ = Y, Z ⊂ X. (i) Max(YK+ (Z); C) = YK+ (Z) ∩ Max(Y ; C). In particular, we have that Max(YC+ (x); C) = YC+ (x) ∩ Max(Y ; C). (ii) If Y ⊂ Z ⊂ Y − K and K ∩ (−C) ⊂ C, (3.17) then Max(Z; K) ⊂ Max(Y ; C) + (K ∩ (−K)) . In particular, if Max(Z; K) = ∅, then Max(Y ; C) = ∅. (iii) If (3.17) holds, then Max(Y ; K) ⊂ Max(Y ; C). (iv) If Y ⊂ Z ⊂ Y − C, then Max(Y ; K) ⊂ Max(Z; K); moreover, if K ∩ (−C) = {0}, then Max(Y ; K) = Max(Z; K). (v) Suppose that (3.17) and K + (C \ L) ⊂ C

(3.18)

hold. If Y ⊂ Max(Y ; K) − K, then Y ⊂ Max(Y ; C) − C (i.e., if Y has (DP) w.r.t. K, then Y has (DP) with respect to C). (vi) Suppose that K ∩ (−C) = {0} and Y ⊂ Max(Y ; K) − C. Then Max(Y ; C) = Max(Y ; K).   Proof. (i) Let y ∈ Max YK+ (Z); C . Then y ∈ YK+ (Z) and YK+ (Z)∩(y +C) ⊂ y − C. Let y
∈ Y ∩ (y + C). It follows that y
∈ y + C ⊂ z + K + C ⊂ z + K for some z ∈ Z, and so y
∈ YK+(Z). Therefore y
∈ y − C, which shows  + that y ∈ Max (Y ; C). Hence Max YK (Z); C ⊂ YK+ (Z) ∩ Max (Y ; C); since

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3 Optimization in partially ordered spaces

the converse inclusion is immediate, we have the first relation. The other one follows immediately taking K = C and Z = {x}. (ii) Let z ∈ Max(Z; K) ⊂ Z ⊂ Y − K. Then z = y − k with y ∈ Y and k ∈ K. Since y ∈ Z, we obtain that k = y − z ∈ (Z − z) ∩ K ⊂ −K, and so k ∈ K ∩(−K). Let us show that y ∈ Max(Y ; C). For this let c ∈ (Y −y)∩C. It follows that y+c ∈ Y ⊂ Z, whence y+c−z = y+c−y+k = c+k ⊂ C+K ⊂ K. Therefore c + k ∈ −K, and so −c ∈ K + k ⊂ K. Using (3.17) we get c ∈ −C. Hence y ∈ Max(Y ; C). (iii) Although (iii) is not a particular case of (ii) (considering Z = Y ), the proof is the same taking k = 0. (iv) Let y ∈ Max(Y ; K) ⊂ Y ⊂ Z. If k ∈ (Z − y) ∩ K, then y + k ∈ Z, whence y+k = y
−c
with y
∈ Y and c
∈ C; so y
−y = k+c
∈ (Y −y)∩K ⊂ −K. It follows that k ∈ −c
− K ⊂ −K. Therefore y ∈ Max(Z; K). Suppose that C ∩ (−K) = {0} and consider z ∈ Max(Z; K). It follows that z = y − c for some y ∈ Y and c ∈ C. Hence y ∈ Z ∩ (z + K) ⊂ z − K. Therefore y = z − k for some k ∈ K. We obtain that c + k = 0, whence c = −k ∈ C ∩(−K) = {0}. Thus z = y ∈ Y and Y ∩(z −K) ⊂ Z ∩(z +K) ⊂ z −K, which shows that z ∈ Max(Y ; K). Therefore Max(Y ; K) = Max(Z; K). (v) Suppose that (3.17), (3.18) hold and Y ⊂ Max(Y ; K) − K. Let y ∈ Y . If y ∈ Max(Y ; C), there is nothing to prove. Assume that y ∈ / Max(Y ; C). It follows that y = y
− c
with y
∈ Y and c
∈ C \ L. Since Y has (DP) w.r.t. K, y
= y − k with y ∈ Max(Y ; K) and k ∈ K. By (iii) we have that y ∈ Max(Y ; C). Moreover, y = y − k − c
= y − (k + c
) ∈ Max(Y ; C) − (K + (C \ L)) ⊂ Max(Y ; C) − C. Therefore Y ⊂ Max(Y ; C) − C. (vi) Since K ∩ (−C) ⊂ C, by (iii), one has that Max(Y ; K) ⊂ Max(Y ; C). Since Max(Y ; K) ⊂ Y ⊂ Max(Y ; K) − C and C ∩ (−C) = {0}, by (iv) one has Max(Y ; C) = Max(Max(Y ; K); C) ⊂ Max(Y ; K). The proof is complete.
In the sequel we suppose that X is a real Hausdorff topological vector space (H.t.v.s. for short), its topology being denoted by τ . Applying the results from Section 3.2.2 we obtain several existence theorems for efficient points w.r.t. cones. Before stating them, let us recall or introduce some notions or, more exactly, some possible properties of the cone C. So, C is (sequentially Daniell) Daniell if every C-upper bounded and C-increasing (sequence) net in X has a least upper bound (or supremum) and converges to it. Because we suppose that X is Hausdorff, every (sequentially) Daniel cone is pointed. Other similar conditions are: (P1) Every C-increasing and C-upper bounded net in C is convergent to an element of C.

3.2 Existence results for efficient points

97

(P2) Every C-increasing and C-upper bounded net in X is fundamental. (P3) Every C-increasing and τ -bounded net in C is convergent to an element of C. (P4) Every C-increasing and τ -bounded net in X is fundamental. (P5) Every C-increasing and τ -bounded net in X that is contained in a complete set is convergent. The sequential variants of (P1)–(P4) are: (SP1) Every C-increasing and C-upper bounded sequence in C is convergent to an element of C. (SP2) Every C-increasing and C-upper bounded sequence in X is fundamental. (SP3) Every C-increasing and τ -bounded sequence in C is convergent to an element of C. (SP4) Every C-increasing and τ -bounded sequence in X is fundamental. If one of the above conditions holds, then, necessarily, C is pointed. Note that an equivalent formulation for (P1) [and similarly for (P3), (SP1), and (SP3)] is: (P1
) Every C-increasing and C-upper bounded net in X is convergent to an element of X that is a C-upper bound for the net. Of course, (P1
)⇒(P1). For the converse implication consider a C-increasing and C-upper bounded net (xi )i∈I ⊂ X. Fix k ∈ I. Then the net (yi )k i , with yi := xi − xk ∈ C, is C-increasing and C-upper bounded. By (P1) there exists y ∈ C such that yi → y. It follows that xi → x = xk + y, and so xk ≤C x. Since k ∈ I is arbitrary, (P1
) holds. Note that when X is a normed space, Krasnosel’skij [217] says that a convex cone C satisfying (SP2) is regular, while a convex cone C satisfying (SP4) is completely regular. Isac [175] says that C is completely regular if every C-increasing and τ -bounded net of C is convergent. Of course, if X is quasicomplete (i.e. every closed and bounded subset of X is complete), then C is completely regular in Isac’s sense if and only if C has property (P4). Nemeth [270] says that a convex cone C satisfying (P1) [(SP1)] is regular [sequentially regular] ; moreover, he says that C is fully regular if C satisfies (P3). Condition (P5) was introduced by Ha [148] under the name of property (∗∗ ). We have the following scheme of implications, where (D) means Daniell, while (sD) means sequentially Daniell.

←

←

(D) ⇒ (P1) ⇒ (P2) (P3) ⇒ (P4) ⇒ (P5) ⇓↑ ⇓↑  ⇓↑  ↓ ← ← (sD) ⇒ (SP1) ⇒ (SP2) (SP3) ⇒ (SP4) (P3). The implications marked by ⇒, ⇓, and  always hold; those marked by ↑ are valid when C is complete or when 0 ∈ X has a countable basis of

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3 Optimization in partially ordered spaces

neighborhoods; the implications ↓ and ← are valid if C is complete; those marked  are valid for C closed. The nontrivial implications are stated in the next proposition. Moreover, the implications → below hold if C is normal, while the implications ← hold if C is boundedly order complete, i.e., every C-increasing and τ -bounded net of C has a supremum: (P3) ← → (P1), (P4) ← → (P2), (SP3) ← → (SP1) and (SP4) ← → (SP2).

(3.19)

The proof of the following result uses an idea from [270]. Note that Borwein (see Proposition 2.7 (ii) of [40]) showed that C is Daniell iff C is sequentially Daniell when “C admits a strict monotone functional.” Proposition 3.2.16. Let C ⊂ X be a convex cone. Then (i) (P 2) ⇔ (SP 2) and (P 4) ⇔ (SP 4). (ii) If C is complete, or 0 ∈ X has a countable basis of neighborhoods, then (P 1) ⇔ (SP 1), (P 3) ⇔ (SP 3), and C is Daniell if and only if C is sequentially Daniell. Proof. We need to show only the implications “⇐” because the others are trivial. (i) Let us show that (SP2)⇒(P2). Consider (xi )i∈I ⊂ C a C-increasing and C-upper bounded net, and suppose that (xi ) is not fundamental. Therefore there exists V0 a neighborhood of 0 ∈ X such that for every i ∈ I there exist j, k ∈ I with i  j, k and xj − xk ∈ / V0 . There exists a symmetric neighborhood U0 of 0 such that U0 + U0 ⊂ V0 . For j, k found above, there / U0 or xk − xl ∈ / U0 exists l ∈ I such that j, k  l. We have that xj − xl ∈ (otherwise, xj − xk ∈ U0 − U0 ⊂ V0 ). Taking this into account, we have that ∀ i ∈ I, ∃ j, k ∈ I : i  j  k, xj − xk ∈ / U0 .

(3.20)

Let i0 ∈ I be fixed. Then taking i = i0 in (3.20), there exist i1 , i2 ∈ I / U0 . Taking i = i2 in (3.20), there exist such that i0  i1  i2 and xi1 − xi2 ∈ i3 , i4 ∈ I such that i2  i3  i4 and xi3 − xi4 ∈ / U0 . Continuing in this way we obtain an increasing sequence (in )n∈N ⊂ I such that xi2n+1 − xi2n+2 ∈ / U0 for every n ∈ N. Of course, the sequence (xin )n∈N ⊂ C is C-increasing and C-upper bounded, and so (xin )n∈N is fundamental by (SP2). This contradicts the choice of this sequence. The proof of (SP4)⇒(P4) is practically the same. (ii) I. Suppose first that C is complete. The equivalence of (P1) and (SP1) is obtained from the statement (3.1) of [270] taking K0 = K = C, while for the other equivalences the proofs are similar. II. Suppose now that 0 has a countable basis of neighborhoods, say (Vn )n∈N . Without loss of generality we may suppose that Vn ⊃ Vn+1 + Vn+1 for every n ∈ N.

3.2 Existence results for efficient points

99

Let us show that (SP1) ⇒ (P1). So, suppose that (SP1) holds. Since (SP1) ⇒ (SP2), by (i) we have that (P2) holds. Consider (xi )i∈I ⊂ C to be C-increasing and C-upper bounded. It follows that (xi ) is Cauchy. Hence for every n ∈ N there exists in ∈ I such that xj − xk ∈ Vn for all j, k ∈ I, in  j, k. Taking into account that I is upward directed, we may consider that (in )n∈N is increasing. Therefore the sequence (xin )n∈N ⊂ C is C-increasing and C-upper bounded. By hypothesis xin → x ∈ C. For p ∈ N there exists np ∈ N, np ≥ p + 1, such that xin − x ∈ Vp+1 for every n ≥ np . But for i ∈ I, (ip+1 ) inp  i, we have that xi − xinp ∈ Vp+1 . It follows that xi − x = xi − xinp + xinp − x ∈ Vp+1 + Vp+1 ⊂ Vp for every i  inp , and so xi → x. Therefore (P1) holds. The proof for (SP3)⇒(P3) is the same. Suppose now that C is sequentially Daniell and show that C is Daniell. As in the proof of (SP1)⇒(P1), consider the C-increasing and C-upper bounded net (xi )i∈I . Without loss of generality we may suppose that (xi )i∈I ⊂ C. With the notation of that proof, there exists x ∈ X such that xin → x and x is the least upper bound of (xin )n∈N . With the same proof, we have that xi → x. Using an argument similar to that in the proof of (P1)⇒(P1
) above, we have that x is a C-upper bound for (xi ). The proof is complete.
Before stating the next result we introduce other two notions. We say that the subset Y of X is C-complete (resp. sequentially C-complete) if every Cauchy C-increasing net (resp. Cauchy C-increasing sequence) is convergent to an element of C. Proposition 3.2.17. Let Y ⊂ X be a nonempty set. Assume that one of the following conditions holds: (i) C satisfies (P1), while Y is closed and C-upper bounded; (ii) C is closed and satisfies (SP2), while Y is C-complete and C-upper bounded; (iii) C satisfies (P3), while Y is closed and τ -bounded; (iv) C is closed and satisfies (SP4), while Y is C-complete and τ -bounded. Then Y has the domination property. In particular, Max(Y ; C) = ∅. Proof. Let (yi )i∈I ⊂ Y be a C-increasing net. In case (i) or (iii), since C satisfies (P1) or (P3), and Y is C-bounded from above or τ -bounded, respectively, (yi ) is convergent to some y ∈ X that is a C-upper bound for (yi ). Since Y is closed, y ∈ Y . Therefore the condition (3.9) holds. In case (ii) and (iv), since C satisfies (SP2) or (SP4), by the preceding proposition, C satisfies (P2) or (P4), respectively. Since Y is bounded from above or τ -bounded, respectively, (yi ) is Cauchy. Since Y is C-complete, yi → y ∈ Y . Therefore in both cases condition (3.13) holds. Since C is closed, by

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3 Optimization in partially ordered spaces

Proposition 3.2.11, the condition (3.9) holds, too. Applying Corollary 3.2.10 the conclusions follow.
When every well-ordered subset W of X w.r.t. ≤C (see page 88) is at most countable (i.e., (X, ≤C ) is countable orderable in the sense of Gajek and Zagrodny [123]) the closedness and completeness in the preceding result can be taken in the weaker sequential sense. Proposition 3.2.18. Let Y ⊂ X be a nonempty set. Assume that every wellordered subset W of X w.r.t. ≤C is at most countable and one of the following conditions holds: (i) C satisfies (SP1), while Y is sequentially closed and C-upper bounded; (ii) C is closed and satisfies (SP2), while Y is sequentially C-complete and C-upper bounded; (iii) C satisfies (SP3), while Y is sequentially closed and τ -bounded; (iv) C is closed and satisfies (SP4), while Y is sequentially C-complete and τ -bounded. Then Y has the domination property. In particular, Max(Y ; C) = ∅. Proof. Let W ⊂ Y be well-ordered with respect to ≤C . Therefore, by our hypothesis, W is at most countable. If W is finite, then W is bounded from above by its greatest element (which is in Y ). In the contrary case W = {yn | n ∈ N}. Consider zn := max{yk | k ≤ n}. It is obvious that (zn ) ⊂ Y is a C-increasing sequence. As in the proof of the preceding proposition we obtain that (zn ) converges to some z ∈ Y with the property that zn ≤C z for every n. Of course, yn ≤C z for every n; i.e., W is C-upper bounded in Y . The conclusion follows by applying Proposition 3.2.15 (iii).
Note that using Proposition 3.2.15, in the above two propositions one can only ask that the hypotheses on Y be satisfied by a nonempty upper section YC+ (x), or even a nonempty set of the form YC+ (Z), to have Max(Y ; C) = ∅. The next result is due to Ha [149]. Proposition 3.2.19. Assume that cl C satisfies (P5) and Y ⊂ X is nonempty, complete, and τ -bounded. Then Max(Y ; C) = ∅. Proof. The proof is the same as that of assertion (iii) of the preceding proposition (but obtaining directly that (yi ) is convergent).
Proposition 3.2.20. Assume that C is closed and Y ⊂ X is nonempty and compact. Then Y has the domination property. In particular, Max(Y ; C) = ∅. Proof. The relation t =≤C and Y satisfy the conditions of Corollary 3.2.12. The conclusions follow.


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Corollary 3.2.21. Let ∅ = Y ⊂ X and assume that cl C ∩ (−C) ⊂ C.

(3.21)

If there exists Z ⊂ X such that YC+ (Z) or Ycl+C (Z) is nonempty and compact (even weakly compact if X is locally convex), then Max(Y ; C) = ∅. Proof. Suppose first that YC+ (Z) is nonempty and compact (or weakly compact). By the preceding proposition, Max(YC+ (Z); cl C) = ∅. Using Proposition 3.2.15 (iii) and (i) for K = cl C, we have that Max(YC+ (Z); cl C) ⊂ Max(YC+ (Z); C) ⊂ Max(Y ; C), whence Max(Y ; C) = ∅. Similarly, if Ycl+C (Z) is nonempty and compact (or weakly compact), then ∅ = Max(Ycl+C (Z); cl C) ⊂ Max(Y ; cl C) ⊂ Max(Y ; C).


Therefore the conclusion holds in both cases.

Proposition 3.2.22. Let C, K ⊂ X be convex cones and ∅ = Y ⊂ X. Assume that (3.18) and    ∀ (yi )i∈I ⊂ Y strictly C-increasing : Y ∩ (yi + K) = ∅ (3.22) i∈I

hold. Then Y has the domination property. In particular, Max(Y ; C) = ∅. Proof. Consider t defined by (3.15) and s := {(x, y) ∈ X × X | y − x ∈ K}. Taking into account (3.15) and (3.16), (3.18) is equivalent to (3.7), while from (3.22) we obtain that (3.8) holds. The conclusions follow by applying Proposition 3.2.9.
Note that in locally convex spaces, every result stated above has a weak version when the topology τ is replaced by the weak topology on X. 3.2.4 Types of Convex Cones and Compactness with Respect to Cones In Sections 2.1 and 2.2 we introduced several types of cones, some of them being useful for existence of efficient points. The following result is stated in Jameson [199, Th. 3.8.7] (there the set B being also a base for C in the sense of Definition 2.1.14). Theorem 3.2.23. Let X be an H.t.v.s. and C be a nontrivial well-based convex cone. Then condition (P4) holds.

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Proof. By Proposition 3.2.16 (i) it is sufficient to show that condition (SP4) is satisfied. Let C = R+ B, where B is a bounded convex set with 0 ∈ / cl B, and (xn )n∈N a τ -bounded C-increasing sequence. Without loss of generality we assume that x0 = 0. Let yn := xn − xn−1 ∈ C for n > 0; then yn = tn bn with tn ≥ 0 and bn ∈ B for n ≥ 1. Since xn = y1 + · · · + yn = t1 b1 + · · · + tn bn , from the convexity of B we have that xn = (t1 + · · · + tn )b
n = t
n b
n with t
n := t1 + · · · + tn and b
n ∈ B for every n ≥ 1. Since 0 ∈ / cl B and  (xn )n≥1 is bounded we have that (t
n )n≥1 is bounded, and so the series n≥1 tn is convergent. Fix V ∈ NX . Since B is bounded, n there exists ε > 0 such that [0, ε]B ⊂ V . There exists n0 ≥ 0 such that k=m+1 tk < ε for n > m ≥ n0 . n Then xn − xm = yn + · · · + ym+1 ∈ ( k=m+1 tk )B ⊂ [0, ε]B ⊂ V for n > m ≥ n0 . This shows that the sequence (xn ) is Cauchy.
When X is an l.c.s. the preceding result may be reinforced (take into account Proposition 2.2.15). Proposition 3.2.24. Let (X, P) be an H.l.c.s. If C is supernormal, then C satisfies condition (P4), and therefore (P2). Proof. Suppose that C is supernormal and consider (xi )i∈I ⊂ X a Cincreasing and τP -bounded net. Consider p ∈ P. Because C is supernormal, there exists x∗ ∈ C + such that p(x) ≤ x, x∗  for every x ∈ C. Because (xi )i∈I is C-increasing and τP -bounded, the net (xi , x∗ )i∈I ⊂ R is bounded and nondecreasing. It follows that (xi , x∗ )i∈I is convergent, and so fundamental. Since p(xj − xi ) ≤ xj − xi , x∗  for all i, j ∈ I, i  j, it follows immediately that (xi )i∈I is fundamental. Therefore (P4) holds. Now (P2) follows from the normality of C.
The next result is useful in the context of Proposition 3.2.18. Proposition 3.2.25. Assume that either (i) C has a base in the sense of Definition 2.1.14 or (ii) X is an H.t.v.s. and C is based. Then every wellordered subset of X w.r.t. ≤C is at most countable. Proof. Let W ⊂ X be well-ordered with respect to ≤C . Replacing W by W + c0 − w0 with c0 ∈ C0 := C \ {0} and w0 the least element of W , we may assume that W ⊂ C0 . (i) Assume that C has a base. By Theorem 2.1.15, there exists a linear functional ϕ : X → R such that ϕ(x) > 0 for x ∈ C \ {0}. Since ϕ(x) < ϕ(x + x
), provided that x, x
∈ C0 , we obtain that ϕ(W ) is a well-ordered subset of R endowed with the usual order, and so ϕ(W ) is at most countable (see Remark 2.1.6). Hence W is at most countable. (ii) By Definition 2.2.14, there exists a convex set B such that 0 ∈ / cl B and C = R+ B. Consider C0 := C \ {0} and νB : C0 → R defined by νB (x) := sup{t > 0 | x ∈ tB}. Of course, νB (x) > 0 for every x ∈ C0 ; moreover, νB (x) < ∞, because otherwise, 0 ∈ cl B. If x ∈ tB and x
∈ t
B with t, t
> 0, then x + x
∈ (t + t
)B, whence νB (x + x
) ≥ νB (x) + νB (x
). So, if x, x
∈ C0 , then νB (x) < νB (x + x
). As in (i) we obtain that W is at most countable.


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Another type of cone is the (π)-cone. Suppose that X is an l.c.s.; the cone C is a (π)-cone if there exists x∗ ∈ C + such that {x ∈ C | x, x∗  ≤ 1} is relatively weakly compact (see [332, Def. 2.1] and [60, Def. 4.1]). SternaKarwat showed in [332, Prop. 2.1] that C is (π)-cone ⇔ cl C has a weakly compact base. As seen in Proposition 2.2.15, if the cone C has a bounded base B, then C is supernormal; furthermore, if B is complete, it follows easily that C is complete, too. This discussion shows that a cone with compact base is Daniell, supernormal, and complete. In particular, a closed (π)-cone is Daniell and complete w.r.t. the weak topology. Luc in [243, Def. 2.3] and [244, Def. I.1.1] says that C is correct if (3.18) holds for K = cl C. Every domination cone in the sense of Henig [162, p. 112] is correct. Concerning cone compactness, recall the following notions. Hartley [158, p. 214] says that Y is C-compact if Ycl+C (y) is compact for every y ∈ Y . Corley in Definition 2.5 of [82] says that Y ⊂ X is C-semicompact if (3.6) holds for t equal to ≤cl C . As a generalization of this notion, Luc ([243, Def. 2.1] and [244, Def. II.3.2]) says that Y is C-complete if (3.22) holds for K = cl C. Postolic˘ a [303] and Isac [178] say that Y is C-bounded if there exists a τ -bounded set Y0 ⊂ Y such that Y ⊂ Y0 − C, Y is C-closed if Y − C is closed, and Y is C-semicompact if Y is C-bounded and C-closed; in fact, these notions are also used in [175, Def. 3], but with Y0 a singleton. Dedieu [92] says that Y is asymptotically compact (a.c. for short) if there exist γ > 0 and U a neighborhood of 0 ∈ X such that ([0, γ]Y ) ∩ U is relatively compact; note that Y is a.c. iff cl Y is a.c. (see [389, Prop. 2.2(i)]). Of course, every subset of Rm is asymptotically compact. Several properties of a.c. sets can be found in [389, Prop. 2.2]. The asymptotic cone of the nonempty set A ⊂ X is A∞ := {x ∈ X | ∃ (ti )i∈I ⊂ (0, ∞), ti → 0, ∃ (ai )i∈I ⊂ A : ti ai → x}; if X is a normed space, in particular, for X =Rm , one can use sequences instead of nets. Note that A∞ = (cl A)∞ . If A is closed and " convex, then A∞ is given by the known formula from convex analysis A∞ = t>0 t(A − a) for some fixed a ∈ A. The importance of this notion in our context is shown by the following result. Proposition 3.2.26. Let C ⊂ X be a closed convex cone and ∅ = Y ⊂ X closed. (i) If Y is asymptotically compact and Y∞ ∩ C = {0}, then Y ∩ (x + C) is compact for every x ∈ X (hence Y is C-compact).

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(ii) Suppose that there exists a compact set Q ⊂ X such that 0 ∈ / Q and C = [0, ∞) · Q and Y∞ ∩ C = {0}. Then Y ∩ (x + C) is compact for every x ∈ X. Moreover, if C is pointed, then Y −C is closed and (Y −C)∩(x+C) is compact for every x ∈ X. Conversely, if Y is convex and Y ∩ (x + C) is nonempty and compact for some x ∈ X, then Y∞ ∩ C = {0}. Proof. Suppose that Y ∩ (x + C) = ∅. Both in (i) and (ii) we have that {0} ⊂ [Y ∩ (x + C)]∞ ⊂ Y∞ ∩ (x + C)∞ = Y∞ ∩ C = {0}. If C = [0, ∞) · Q with Q compact and 0 ∈ / Q, one obtains easily that C is locally compact. Using [389, Prop. 2.2(ii)] we get that C is an a.c. set. So, in both cases Y ∩ (x + C) is a.c. as a subset of such a set. Applying [389, Prop. 2.3] we obtain that Y ∩ (x + C) is relatively compact; since the set is closed, it is compact, too. Consider now case (ii) and suppose that C is pointed. Applying [389, Cor. 3.12] we have that Y − C is closed and (Y − C)∞ = Y∞ − C. Let z ∈ (Y − C)∞ ∩ C. It follows that there are y ∈ Y∞ and c ∈ C such that z = y − c. Since z ∈ C, we obtain that y = z + c ∈ C ∩ Y∞ = {0}. Since C is pointed, we get z = 0. By what was obtained above we have that (Y − C) ∩ (x + C) is compact for every x ∈ X. Now let Y be convex and assume that Y∞ ∩ C = {0}. Since Y is also closed, fixing y ∈ Y and 0 = u ∈ Y∞ ∩ C, we have that y + tu ∈ Y for every t ≥ 0. Therefore {y + tu | t ≥ 0} ⊂ Y ∩ (y + C), whence Y ∩ (y + C) is not compact.
Corollary 3.2.27. Let C ⊂ X be a closed convex cone and ∅ = Y ⊂ X closed. Assume that Y∞ ∩ C = {0}. If Y is asymptotically compact or C = [0, ∞) · Q for some compact set Q ⊂ X with 0 ∈ / Q, then Y has the domination property. + Proof. By the preceding proposition  YC (y) is compact for every y ∈ Y ; + by Proposition 3.2.20 Max YC (y); C = ∅. Therefore Y has the domination property.


3.2.5 Classification of Existence Results for Efficient Points In the sequel we mention several existence results for efficient points that can be found in the literature in equivalent formulations (some of them were stated for minimization problems, or using other terminology). In all the results below C ⊂ X is a convex cone and Y ⊂ X is a nonempty set. Results Corresponding to Proposition 3.2.14 (The Nontopological Case) Theorem 3.2.28. (Corley [82, Th. 3.1]) Let X be a separated t.v.s. and cl C pointed. Suppose that Y is C-semicompact in Corley’s sense. Then Max(Y ; C) = ∅.

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Proof. By hypothesis, Y satisfies (3.6), and therefore (3.5), for t =≤cl C . It follows that (iv) of Proposition 3.2.14 holds for cl C and Y , and so Max(Y ; cl C) = ∅. Using Proposition 3.2.15 (iii) for K = cl C, the conclusion follows.
We put this result here because applied to K = cl C, the result is not topological. Theorem 3.2.29. (Chew [77]) Let X be real vector space. Suppose that the intersection of every nonempty chain (w.r.t. inclusion) of C-upper sections of Y is nonempty. Then Max(Y ; C) = ∅.   Proof. Taking into account Proposition 3.2.4, YC+ (yi ) i∈I , with (yi )i∈I ⊂ Y , is a chain if and only if (yi )i∈I is a chain in (Y, ≤C ). So, using Proposition 3.2.14 (ii), the conclusion follows.
Note that the above formulation covers Proposition 4.6 (where it is supposed that C is pointed), Corollary 4.4, and Proposition 4.7 from [77] (in Proposition 4.7 it is assumed that X is a Hilbert space and C ∩ (−C) is closed). Results Corresponding to Propositions 3.2.17 and 3.2.18 Theorem 3.2.30. (Jameson [199, Cor. 3.8.10]) Let X be an H.t.v.s. and C well-based and closed. If Y is complete and bounded, then Y has the domination property. Proof. By Theorem 3.2.23 condition (P4) is satisfied; so, apply Proposition 3.2.17 (iv).
Theorem 3.2.31. (Penot [288, Th. 3.3]) Let X be an H.t.v.s. and let C be Daniell and closed. If Y is C-upper bounded and closed, then Max(Y ; C) = ∅. Proof. Apply Proposition 3.2.17 (i).




Theorem 3.2.32. (Cesari and Suryanarayana [60, Lemma 4.1]) Let X be a Banach space and C a closed (π)-cone. Suppose that Y is C-upper bounded. Then Max(w − cl Y ; C) = ∅. Proof. We noted in the preceding section that C is Daniell and complete w.r.t. w. It follows that C and w − cl Y satisfy condition (i) of Proposition 3.2.17 for τ = w.
The above result also follows from Corollary 3.2.27 because Y ⊂ y0 − C implies Y∞ ∩ C = {0}. Theorem 3.2.33. (Borwein [39, Th. 1]) Let X be an l.c.s. and let C be Daniell and closed. If (i) there exists y ∈ Y such that YC+ (y) is C-upper bounded and closed, or (ii) Y is τ -bounded and closed, while C is boundedly order complete, then Max(Y ; C) = ∅.

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Proof. In the first case apply Proposition 3.2.17 (i) for YC+ (y) and then Proposition 3.2.15 (i). In the second case (P3) holds [see also (3.19)], so that the conclusion follows by applying Proposition 3.2.17 (iii).
Theorem 3.2.34. (Isac [175, Th. 2]) Let X be an l.c.s. and let C be supernormal. Suppose that there exists ∅ = Z ⊂ Y such that YC+ (Z) is complete and τ -bounded. Then Max(Y ; C) = ∅. Proof. By Proposition 3.2.24 cl C satisfies (P4). Using also Proposition 3.2.15 (iii) for K = cl C and (i) we have    
∅ = Max YC+ (Z); cl C ⊂ Max YC+ (Z); C ⊂ Max (Z; C) . Note that Postolic˘ a [304, Th. 3.2] obtained a slightly more general variant: X, C, and Y being as in Isac’s theorem, he supposes that there exists Z such + (Y0 ) is complete and that Y ⊂ Z ⊂ Y − C and ∅ = Y0 ⊂ Y , so that ZC τ -bounded; then Max(Y ; C) = ∅. The conclusion follows from Isac’s theorem applied to Z and Proposition 3.2.15 (iv). Theorem 3.2.35. (Postolic˘ a [304, Cor. 3.2.1]) Let X be an l.c.s. and C have a complete bounded base. Suppose that Y is bounded and closed. Then Y has the domination property. Proof. As observed in the previous section, C is complete and supernormal; therefore C satisfies (P3). Applying Proposition 3.2.17 (iii), the conclusion follows.
Theorem 3.2.36. (Attouch and Riahi [10, Th. 2.5]) Let X be a Banach space and C a closed convex cone satisfying (2.21) for some x∗ ∈ X ∗ . Suppose that Y is closed and x∗ (Y ) is bounded above (in R). Then Y has the domination property. Proof. To obtain the conclusion we use Proposition 3.2.14, but the ideas are those from Proposition 3.2.17; in fact, when Y is bounded the result follows from either of the preceding two theorems. Let (yi )i∈I ⊂ Y be a C-increasing net. It follows that (yi , x∗ )i∈I ⊂ [0, ∞) is increasing and bounded, and therefore is a Cauchy net. Taking ε > 0, there exists iε ∈ I such that |yi , x∗  − yj , x∗ | < ε/2 for iε  i, j. From (2.21) we obtain that for i, j ∈ I, iε  i, j,

yi − yj ≤ yi − yiε + yj − yiε ≤ |yi − yiε , x∗ | + |yj − yiε , x∗ | < ε. Therefore (yi ) is a Cauchy net, and so yi → y ∈ X. Since Y is closed, y ∈ Y . From Proposition 3.2.11 we have that y is a C-upper bound for (yi ). Applying Proposition 3.2.14 (iv), the conclusion follows.
Theorem 3.2.37. (Ha [148, 149]) Let X be a quasi-complete H.t.v.s. Assume that cl C satisfies (P 5) and (A+cl C)∩(A−cl C) is bounded for every bounded set A ⊂ X. If Y − cl C is closed and Y ⊂ Y0 − C for some bounded Y0 ⊂ X, then Max(Y ; C) = ∅.

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Proof. The set cl C satisfies (P 3) by what was observed after the introduction of conditions (P1)–(P5). Let y0 ∈ Y and consider Y1 = (Y − cl C)+ cl C (y0 ); Y1 is closed and bounded, being included in ((Y0 ∪ {y0 }) + cl C)∩ ((Y0 ∪ {y0 }) − cl C). By Proposition 3.2.17 (iii), Max(Y1 ; cl C) = ∅, whence, by Proposition 3.2.15 (i) and (ii), Max(Y ; C) = ∅.
Theorem 3.2.38. (Ng and Zheng [272, Th. 3.1]) Let X be an H.t.v.s. and let C be well-based and closed. If there exists y0 ∈ Y such that YC+ (y0 ) is sequentially C-complete and bounded, then Max(Y ; C) = ∅. Proof. By Theorem 3.2.23 condition (SP4) holds. The conclusion follows using Propositions 3.2.25 and 3.2.18 (iv).
Even if the next result is a special case of the preceding theorem, we state it mainly because we are in a position to furnish a direct and constructive proof of it. Theorem 3.2.39. Let X be an H.l.c.s. and let C be well-based and closed. If there exists y0 ∈ Y such that YC+ (y0 ) is sequentially C-complete and bounded, then Max(Y ; C) = ∅. Proof. Because C is well-based, there exists y ∗ ∈ Y ∗ such that B := {y ∈ C | y, y ∗  = 1} is a bounded base of C. Because A := YC+ (y0 ) is bounded, α0 := sup{y, y ∗  | y ∈ A} = sup{y, y ∗  | y ∈ A ∩ (y0 + C)} ∈ R. Take y1 ∈ A ∩ (y0 + C) such that y1 , y ∗  ≥ α0 − 2−1 . Then α1 := sup{y, y ∗  | y ∈ A ∩ (y1 + C)} ≤ α0 ; take y2 ∈ A ∩ (y1 + C) such that y2 , y ∗  ≥ α1 − 2−2 . Continuing in this way we obtain the C-increasing sequence (yn )n≥0 ⊂ A and the decreasing sequence (αn ) ⊂ R such that yn+1 ∈ A ∩ (yn + C), αn = sup{y, y ∗  | y ∈ A ∩ (yn + C)} and yn+1 , y ∗  ≥ αn − 2−n−1 for n ≥ 0. It follows that αn+1 ≥ yn+1 , y ∗  ≥ αn − 2−n−1 ,

yn+1 , y ∗  ≥ yn , y ∗ 

∀ n ≥ 0. (3.23)

Because yn+1 − yn ∈ C, we have that yn+1 − yn = λn bn with bn ∈ B and λn = yn+1 − yn , y ∗  ≥ 0. From (3.23) we obtain that λn = yn+1 , y ∗  − yn , y ∗  ≤ αn+1 +2−n −αn−1 ≤ 2−n for n ≥ 1. Let p : X → R be a continuous seminorm. Because (bn ) is bounded, there exists Mp > 0 such that p(bn ) ≤ Mp for every n ≥ 0. Then p(yn+m − yn ) ≤

n+m−1

k=n

p(yk+1 − yk ) =

n+m−1

λk p(bk ) ≤ 2−n+1 Mp

k=n

for all n, m ≥ 1. Hence the sequence (yn ) ⊂ A is Cauchy. Because A is sequentially C-complete, (yn ) converges to some y ∈ A. Since yn+m ∈ A ∩ (yn + C) for m ≥ 0 and C is closed, we have that y ∈ yn + C for every n ≥ 0. In particular, y ≥ y0 . Take z ∈ A ∩ (y + C). Then αn ≥ z, y ∗  ≥ y, y ∗ . Because, by (3.23), lim αn = y, y ∗ , we obtain that z, y ∗  = y, y ∗ . Since
y ∗ ∈ C # , we obtain that z = y, and so y ∈ Max(A, C) ⊂ Max(Y, C).

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Remark 3.2.40. When C is well-based and Y is C-bounded from above, or more generally, when y ∗ is bounded above on Y for some y ∗ ∈ C # with B := {y ∈ C | y, y ∗  = 1} bounded, YC+ (y) is bounded for every y ∈ Y . Indeed, if z ∈ YC+ (y), then z − y = λb for some λ ≥ 0 and b ∈ B. Thus, 0 ≤ λ = z − y, y ∗  ≤ sup y ∗ (Y ) − y, y ∗  =: γ. Hence YC+ (y) ⊂ [0, γ] · B. Results Corresponding to Proposition 3.2.20 and Its Corollary Theorem 3.2.41. (Bitran and Magnanti [32, Prop. 3.1]) Let X = Rm , let C be closed with C # = ∅, and Y closed and convex. Then Max(Y ; C) = ∅ if and only if Y∞ ∩ C = {0}. Proof. The necessity follows from the last part of Proposition 3.2.26, while the sufficiency follows from both parts of Corollary 3.2.27.
Note that the convexity of Y is used only for the sufficiency part. Theorem 3.2.42. (Nieuwenhuis [273, Th. I-14]) Let X be a Banach space and C have nonempty interior. If Y is compact, then Max(Y ; C0 ) = ∅, where C0 = int C ∪ {0}. Proof. By Proposition 3.2.20, Max(Y ; cl C) = ∅. The conclusion follows from Proposition 3.2.15 (iii) because cl C ∩ (−C0 ) = {0} ⊂ C0 .
Theorem 3.2.43. (Henig [162, Th. 2.1]) Let X =Rm and let C # be nonempty. Suppose that there exists Z such that Y ⊂ Z ⊂ Y − cl C, Z is closed, and Z∞ ∩ cl C = {0}. Then Max(Y ; C) ∩ (y0 + cl C) = ∅ for every y0 ∈ Y . + Proof. As in (or from) Bitran–Magnanti’s theorem, Max(Zcl C (y0 ); cl C) = ∅. The conclusion follows from Proposition 3.2.15 (iii) taking K = cl C.


Theorem 3.2.44. (Borwein [39, Th. 1]) Let X be an l.c.s. and C closed. If YC+ (y) is compact for some y ∈ Y , then Max(Y ; C) = ∅. Proof. Apply Corollary 3.2.21 for Z = {y}.




Theorem 3.2.45. (Penot and Sterna-Karwat [291, Rem. 3.3]) Let X be an H.t.v.s., Y a closed convex set, and C ⊂ X have a compact base. If Max(Y ; C) = ∅, then Y has the domination property. Proof. Taking y ∈ Max(Y ; C), YC+ (y) is nonempty and compact, being a singleton. Using Proposition 3.2.26, Y∞ ∩ C = {0}, whence, by Corollary 3.2.27, Y has the domination property.
Theorem 3.2.46. (Jahn [194, Th. 2.3(b), Th. 2.6(b)]) Let (X, · ) be a reflexive Banach space and Y ⊂ X have a nonempty C-upper section YC+ (y) that is weakly closed and C-upper bounded. (i) Suppose that x < x + y for all x, y ∈ C \ {0}. Then Max(Y ; C) = ∅.

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(ii) Suppose that aint C = ∅ and x < x + y for all x ∈ C, y ∈ aint C. Then Max(Y ; C) = ∅. Proof. Note first that when A (subset of an H.t.v.s.) is convex and cor A = ∅, then cl A = cl(aint A). Indeed, let u ∈ aint A and x ∈ cl A. There exists (xi )i∈I ⊂ A with xi → x. For every λ ∈ (0, 1) and i ∈ I we have that (1 − λ)u + λxi ∈ cor A, and so (1 − λ)u + λx ∈ cl(aint A). Taking λ → 1, we get x ∈ cl(aint A). Therefore cl A = cl(aint A). Let x, y ∈ cl C \ {0}. There exist (xn )n∈N , (yn )n∈N sequences from C \ {0} in case (i) and from aint C in case (ii) such that xn → x, yn → y. Since

xn < xn + yn in both cases, it follows that x ≤ x + y . Therefore

x ≤ x + y for all x, y ∈ cl C in both cases. In particular, C is an acute (normal) cone. From the normality of C (or directly from the monotonicity conditions), YC+ (y) is also bounded. It follows that YC+ (y) is w-compact. Since (3.21) is satisfied, C being acute, the conclusion follows by applying Corollary 3.2.21.
Note that Jahn stated a similar result when X is the dual of a normed space and YC+ (y) is w∗ -closed. Theorem 3.2.47. (Tanaka [350, Lemma 2.4]) Let X =Rm and let C # be nonempty. Suppose that Y is C-compact. Then Max(Y ; C) = ∅. Moreover, if C is correct, then Y has the domination property. Proof. Note that by taking K = cl C, condition (3.17) is satisfied, while when C is correct, (3.18) holds, too. Since Y is C-compact, Ycl+C (y) is compact for  every y ∈ Y . From Corollary 3.2.21 we have that Max Ycl+C (y); cl C = ∅ for every y ∈ Y , and so Y has the domination property w.r.t. cl C. In particular, by Proposition 3.2.15 (iii), Max(Y ; C) = ∅, while when C is correct we obtain that Y has (DP) using assertion (v) of the same proposition.
Results Corresponding to Proposition 3.2.22 Theorem 3.2.48. (Luc [243, Th. 2.6, Cor. 2.12], [244, Th. II.3.3]) Assume that C is correct and Y satisfies (3.22) for K = cl C. Then Y has the domination property. Proof. The conditions of Proposition 3.2.22 are satisfied in this case.




Theorem 3.2.49. (Malivert [248, Th. 3.5]) Let C be a Daniell cone with int C = ∅. Suppose that Y is closed and for every y ∈ Y \ Max(Y ; C0 ) there exists y0 ∈ (y + int C) ∩ Y such that (y0 + int C) ∩ Y is C-upper bounded. Then Y has the domination property w.r.t. C0 , where C0 = {0} ∪ int C.

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Proof. Let Y0 = (y0 + int C) ∩ Y , and consider (zi )i∈I ⊂ Y0 be strictly C0 increasing. It follows that (zi ) is C-increasing and C-upper bounded. Since C is Daniell, zi → z ∈ X and zi ≤C z for every i ∈ I. Since Y is closed, z ∈ Y . Since (zi ) ⊂ Y0"is strictly C0-increasing, it follows that z ∈ Y0 . So, we obtain that z ∈ Y0 ∩ i∈I (zi + C) . Therefore the hypotheses of Proposition 3.2.22
are satisfied by C, K, Y replaced by C0 , C, Y0 , respectively. Note that Propositions 3.2.17–3.2.22 have different fields of application, although these fields are not disjoint. For example, even in finite-dimensional spaces there exist convex cones whose closure is pointed that are not correct; for such a cone and a compact set one can apply Corollary 3.2.21 but not Proposition 3.2.22 in Luc’s version. Also note that there are other results concerning existence of efficient points w.r.t. cones that do not enter into our classification. To our knowledge, among these results are Theorem 4.1 of Hartley [158], Proposition 4.10 and Proposition 4.11 of Chew [77], and Theorem 2.2 of Sterna-Karwat [330] (see also the paper of Turinici [359] for a discussion related to the use of ordinals). The presentation of this section follows that in the paper by Sonntag and Z˘ alinescu [327]; however, we modified slightly Proposition 3.2.17 and introduced Propositions 3.2.18 and 3.2.25 in order to cover Theorem 3.2.30 by Jameson [199] and Theorem 3.2.38 by Ng and Zheng [272]. 3.2.6 Some Density and Connectedness Results Let ∅ = Y ⊂ X, and C ⊂ X a convex cone. We say that y0 ∈ Y is properly maximal w.r.t. C if there exists x∗ ∈ C # such that y0 , x∗  ≥ y, x∗  for every y ∈ Y ; we denote the set of properly maximal points of Y w.r.t. C by PrMax(Y ; C). Another notion of proper maximality is due to Henig [161]; namely, y0 ∈ Y is Henig proper maximal w.r.t. C if there exists a proper convex cone K ⊂ X such that C \ {0} ⊂ int K and y0 ∈ Max(Y ; K). We denote by HMax(Y ; C) the set of Henig proper maximal points of Y w.r.t. C. We have that PrMax(Y ; C) ⊂ HMax(Y ; C) ⊂ Max(Y ; C).

(3.24)

For the first inclusion in (3.24) take y0 ∈ PrMax(Y ; C) and K := {x ∈ X | x, x∗  ≥ 0}, where x∗ ∈ C # is the element in the definition of proper maximality, while the second inclusion in (3.24) is obvious. Note that when one of the first two sets in (3.24) is nonempty, then C is a pointed cone. Lemma 3.2.50. If the set Y is convex, then PrMax(Y ; C) = HMax(Y ; C). Proof. If y0 ∈ HMax(Y ; C), then there exists a proper convex cone K ⊂ X such that C \ {0} ⊂ int K and (Y − y0 ) ∩ K = {0}; hence (Y − y0 ) ∩ int K = ∅. Applying a separation theorem, we get x∗ ∈ X ∗ such that y − y0 , x∗  ≤ 0 < x, x∗ 

∀ y ∈ Y, ∀ x ∈ int K.

3.2 Existence results for efficient points

111

It follows that x∗ ∈ C # and y, x∗  ≤ y0 , x∗  for every y ∈ Y . Therefore y0 ∈ PrMax(Y ; C).
The elements of PrMax(Y ; C) are obtained by linear scalarization (compare with Section 3.1.1), and so they can be obtained (quite) easily. So, for practical purposes, it is important to have the possibility to approximate each element in Max(Y ; C) by elements of PrMax(Y ; C). The first result in this direction was obtained by Arrow, Barankin, and Blackwell [9] for X = Rn , C = Rn+ , and Y a compact convex set. This result was later extended in several directions. One of these extensions was realized by Henig [161, Th. 5.1],   who proved that Max(Y ; C) ⊂ cl HMax(Y ; C) when X = Rn , C is a closed pointed convex cone, and there exists A ⊂ C such that 0 ∈ A, Y − A is closed, and (Y − A)∞ ∩ C = {0}. In the sequel we extend this result to the case in which (X, · ) is a normed vector space. When no mention of the topology is made for a topological notion, this is considered w.r.t. the norm topology. First we prove an auxiliary result; this is due essentially to Gong [136, Lem. 2.1] (see also Sterna-Karwat [334, Th. 3.1]). Lemma 3.2.51. Let C ⊂ (X, · ) be a based closed convex cone with base B and take δ := d(0, B) > 0. For ε ∈ ]0, δ[ consider Bε := {x ∈ X | d(x, B) ≤ ε} and Cε := [0, ∞[ ·Bε the cone generated by Bε . Then (i) Cε is a closed convex cone for every ε ∈ ]0, δ[; (ii) if 0 < γ < ε < δ, then C \ {0} ⊂ Cγ \ {0} ⊂ int Cε ; (iii) C = ∩ε∈ ]0,δ[ Cε = ∩n∈N Cεn , where (εn ) ⊂ ]0, δ[ converges to 0. Proof. (i) Fix ε ∈ ]0, δ[. Because B is convex, d(·, B) is a convex (continuous) function, and so Bε is a closed convex set with 0 ∈ / Bε . We have to show that Cε is closed. So, let y ∈ cl Cε . Fix (εn ) ⊂ ]ε, ∞[ converging to ε. Because Bε ⊂ B + ε
UX for ε
> ε, there exist (λn ) ⊂ [0, ∞[, (bn ) ⊂ B and (un ) ⊂ UX such that yn := λn (bn + εn un ) → y. The sequence (λn ) is bounded above. b
n := b'n + Otherwise, there exists a subsequence (λnk ) with limit ∞. Taking '
' εn un ,'it follows that bnk → 0. Since δ = d(0, B) ≤ bnk = εnk unk − b
nk ' ≤ '
' ' εnk + bnk , we get the contradiction δ ≤ ε. So, we may assume that λn → λ ∈ [0, ∞[. If λ = 0, then y = lim λn bn ∈ C' ⊂ Cε , because ' 'C is closed. Let λ > ' 0. Then'b
n → λ−1 y' and d(λ−1 y, B) ≤ 'λ−1 y − bn ' = 'λ−1 y − b
n + εn un ' ≤ εn + 'λ−1 y − b
n '. It follows that d(λ−1 y, B) ≤ ε, and so λ−1 y ∈ Bε , whence y ∈ Cε . (ii) Let 0 < γ < ε < δ. The first inclusion being obvious, let us prove the second. Let y ∈ Cγ \ {0}; i.e., y = λx with λ > 0 and x ∈ Bγ . Because Bγ ⊂ B + ε int UX ⊂ int Cε , the conclusion follows. (iii) The second equality being obvious, taking into account (ii), it is sufficient to prove that ∩n∈N Cεn ⊂ C for a fixed (εn ) ⊂ ]0, δ[ convergent to 0. So let y ∈ ∩n∈N Cεn . There exist (λn ) ⊂ [0, ∞[, (bn ) ⊂ B, and (un ) ⊂ UX such that yn := λn (bn + 2εn un ) → y. As in (i) we obtain that (λn ) is bounded above. It follows that y = lim λn bn ∈ C.
The cone Cε introduced in Lemma 3.2.51 is called a Henig dilating cone.

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3 Optimization in partially ordered spaces

Remark 3.2.52. Borwein and Zhuang [42] say that y0 is a super efficient point of Y w.r.t. C, written y0 ∈ SE(Y ; C), if (UX + C) ∩ R+ (Y − y0 ) ⊂ µUX for some µ > 0. Under the hypothesis that C has a bounded base B, in [42, Prop. 2.5] one proves that y0 ∈ SE(Y ; C) if and only if y0 ∈ Max(Y ; Cε ) for some 0 < ε < d(0, B). Theorem 3.2.53. Let X be a normed vector space, C ⊂ X a based closed convex cone, and τ a locally convex topology on X compatible with the dual system (X, X ∗ ). Assume that Y is a nonempty τ -closed and τ -asymptotically compact subset of X with Y∞ ∩ C = {0}, the asymptotic cone being taken w.r.t. τ . Then   Max(Y ; C) ⊂ τ - cl HMax(Y ; C) . (3.25)   Proof. Taking y ∈ Max(Y ; C), we must show that y ∈ τ - cl HMax(Y ; C) . Without loss of generality, we take y = 0; hence Y ∩ C = {0}. Let B be a base for C; because C is closed, we (may) assume that B is closed; of course, B is convex, too. Because Y is τ -a.c., by [389, Prop. 2.2 (i)], so is Y∞ . Using [389, Cor. 3.12] we obtain that B − Y∞ is τ -closed, and so B − Y∞ is (norm) closed. Because / B − Y∞ . It follows that Y∞ ∩ C = {0}, we have that Y∞ ∩ B = ∅, whence 0 ∈ γ := d(B, Y∞ ) := inf{ b − y | b ∈ B, y ∈ Y∞ } = d(0, B − Y∞ ) > 0. (3.26) γ Let Bn = {x ∈ X | d(x, B) ≤ 2n+1 } for n ∈ N∗ ; Bn is a closed convex set with 0 ∈ / Bn . From Lemma 3.2.51 we have that Cn := [0, ∞) · Bn is a based closed convex cone such that C \ {0} ⊂ int Cn . Consider Yn := Y ∩ Cn . Since Y∞ ∩ Bn = ∅, we have that Y∞ ∩ Cn = {0}, and so, by Proposition 3.2.26, Yn is τ -compact for every n ≥ 1. Now using Proposition 3.2.20 we get the existence of yn ∈ Max(Yn ; Cn ). From Proposition 3.2.15 (i) we have that yn ∈ Max(Y ; Cn ), and so yn ∈ HMax(Y ; Cn ). Of course, (yn ) ⊂ Y1 ; since Y1 is τ -compact, (yn ) is bounded. Because yn ∈ Cn , there exist (λn ) ⊂ [0, ∞[, (bn ) ⊂ B, and (xn ) ⊂ X such that xn ≤ γ/2n and yn = λn (bn + xn ) for every n ≥ 1. Let us prove that λn → 0. In the contrary case there exists a subnet (λθ(j) )j∈J with 0 < λθ(j) → λ ∈ ]0, ∞]. Because Y1 is τ -compact, we may assume that (yθ(j) )j∈J τ -converges to some y ∈ Y1 . If λ = ∞, from bθ(j) = λ−1 θ(j) yθ(j) − xθ(j) we obtain the contradiction 0 ∈ B. Hence λ < ∞. If λ ∈ ]0, ∞[, then λ−1 y ∈ B, whence y ∈ C ∩ Y . It follows that y = 0, and so we get the contradiction 0 ∈ B. Therefore λn → 0. Let us prove that yn →τ 0. In the contrary case there exists a τ neighborhood U of 0 such that the set P := {n ∈ N∗ | yn ∈ / U } is infinite. As above, there exists a subnet (yθ(j) )j∈J of (yn )n∈P τ -convergent to some y ∈ Y1 ⊂ Y . Then y = τ − lim λθ(j) bθ(j) ∈ C ∩ Y , and so y = 0. It follows that there exists some j0 ∈ J such that yθ(j) ∈ U for j ≥ j0 ; this is a contradiction
because θ(j0 ) ∈ P . Hence 0 = τ − lim yn ∈ τ - cl HMax(Y ; C) .

3.2 Existence results for efficient points

113

The convergence in the preceding theorem is strong if the classes of weak (or even τ ) and norm neighborhoods of y in Y coincide (that is, y is a continuity point of Y ); this happens, for example, if Y − y is contained in a Phelps cone. The next result furnishes other conditions for having strong convergence at every y ∈ Max(Y ; C). Theorem 3.2.54. Let X be a normed vector space, τ a locally convex topology on X compatible with the dual system (X, X ∗ ), C ⊂ X a closed convex cone, and ∅ = Y ⊂ X. Then   Max(Y ; C) ⊂ cl HMax(Y ; C) , (3.27) provided one of the following conditions is satisfied: (i) C has a bounded base, Y is τ -closed, τ -asymptotically compact, and Y∞ ∩ C = {0}, the asymptotic cone being taken w.r.t. τ ; (ii) C has a w-compact base, Y is τ -complete, and Y ⊂ A − C for some bounded set A ⊂ X. (iii) Y is a closed convex set and either (a) C is based and Y is locally compact, (b) C has a bounded base and Y is τ -locally compact, or (c) C has a w-compact base and the τ -closed bounded subsets of Y are τ -complete. Proof. We follow the lines of the proof of the preceding theorem, and we point out only the respective differences. Let y = 0 ∈ Max(Y ; C). (i) The base B of C is, moreover, bounded. With the same notation as in the proof of Theorem 3.2.53, we have that λn → 0. Because (bn ) is bounded, we obtain that (yn ) strongly converges to 0. (ii) The base B of C is w-compact, and so bounded. This time Y∞ ⊂ (A − C)∞ = −C, because A is bounded, and so B − Y∞ ⊂ B + C = [1, ∞[ ·B (the asymptotic cone being taken w.r.t. w). The last set being closed, γ > 0. With the notation of Theorem 3.2.53 let us show that Yn is bounded. Indeed, Yn ⊂ Cn ∩ (A − C) ⊂ (A0 + Cn ) ∩ (A0 − Cn ), where A0 = A ∪ {0}. Since Cn is well-based, Cn is normal, whence the boundedness of Yn follows. Moreover, Yn is τ -complete as a τ -closed subset of a τ -complete set. Applying Propositions 3.2.24 and 3.2.17(iv), we find that (yn ) ⊂ Y1 ∩ HMax(Y ; C), as in Theorem 3.2.53. Exploiting this time the w-compactness of B we obtain that λn → 0, and so yn → 0 for the strong topology. (iii) (a) Let U be a convex neighborhood of 0 such that Y0 := Y ∩ U is compact. Of course, 0 ∈ Max(Y0 ; C). By Theorem 3.2.53 for the norm topology, there exists a sequence (yn ) ⊂ HMax(Y0 ; C) strongly converging to 0. There exists n0 such that yn ∈ int U for n ≥ n0 . Because Y0 is convex, by Lemma 3.2.50, yn ∈ PrMax(Y0 ; C), and so yn maximizes some x∗n ∈ C # on Y0 . Let y ∈ Y . Then there exists t ∈ ]0, 1[ such that (1 − t)yn + ty ∈ Y0 ,

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and so (1 − t)yn + ty, x∗n  ≤ yn , x∗n , whence y, x∗n  ≤ yn , x∗n . Hence yn ∈ PrMax(Y ; C). In case (b) we take U to be a τ -closed and convex neighborhood of 0 such that Y0 := Y ∩ U is τ -compact, while in case (c) one takes UX instead of U (then Y0 is a τ -complete convex and bounded set). The proof continues as for (a) but using assertion (i) or (ii) above instead of Theorem 3.2.53.
Note  that for a fixed  y ∈ Max(Y ; C), when Y is a convex set, the fact that y ∈ cl PrMax(Y ; C) can be obtained as above, for (a) C has a bounded base and Y ∩ (y + rUX ) is τ -compact, or (b) C has a w-compact base and Y ∩ (y + rUX ) is τ -complete, where r is some positive number. Remark 3.2.55. In the statements of Theorems 3.2.53 and 3.2.54 besides Y we can consider another set Z satisfying Y ⊂ Z ⊂ Y − C and ask that this set Z satisfy the conditions imposed on Y , the conclusion remaining the same for the set Y . Moreover, when Y (or Z) is closed and convex, there is no need for the hypothesis Y∞ ∩ C = {0} in Theorems 3.2.53 and 3.2.54(i), because in the contrary case, Max(Y ; C) = ∅. Density results like those proved in Theorems 3.2.53 and 3.2.54 can be used to establish the connectedness of the set Max(Y ; C). Connectedness of an efficient point set provides a possibility of moving continuously from one optimal solution to any other along optimal alternatives only. Theorem 3.2.56. Let X be a normed vector space, C ⊂ X a closed convex cone, and ∅ = Y, Z ⊂ X such that Y ⊂ Z ⊂ Y − C and Y − C is convex. (i) If Z is w-compact and C is based, then PrMax(Y ; C) and Max(Y ; C) are connected w.r.t. the weak topology w. (ii) If Z is compact and C is well-based, then PrMax(Y ; C) and Max(Y ; C) are connected. Proof. (i) Because PrMax(Y ; C) = PrMax(Y − C; C) = PrMax(Z; C) and Max(Y ; C) = Max(Y − C; C) = Max(Z; C), we (may) assume that Y is wcompact. Consider the multifunction Γ : C # ⇒ Y,

Γ (x∗ ) := {y ∈ Y | y, x∗  ≥ y
, x∗  ∀ y
∈ Y }.

It is clear that PrMax(Y ; C) = Im Γ = Γ (C # ). Because Y is w-compact, Γ (x∗ ) is a nonempty w-compact set for every x∗ ∈ C # , and so dom Γ = C # . Observe that sup{y
, x∗  | y
∈ Y } = sup{y
, x∗  | y
∈ Y − C}, and so Γ (x∗ ) is convex for every x∗ ∈ C # . In particular, Γ (x∗ ) is w-connected for every x∗ ∈ C # . Because C # is a convex set (in fact, C # ∪ {0} is a convex cone), and so a (norm) connected set, in order to obtain that Γ (C # ) is wconnected, by Proposition 2.5.8(ii), it is sufficient to show that Γ is (norm–w)

3.3 Continuity properties with respect to a scalarization parameter

115

u.c. Assume that Γ is not · -w u.c. at x∗ ∈ C # . Then there exist a weak open set U ⊂ X and a sequence (x∗n ) ⊂ C # converging to x∗ such that Γ (x∗ ) ⊂ U and Γ (x∗n ) ⊂ U for every n ∈ N. Take yn ∈ Γ (x∗n ) \ U ⊂ Y for n ∈ N. Because ˇ Y is weakly compact, by Eberlein–Smulian theorem (see [168]), there exists a subsequence (ynp )p∈N converging weakly to y ∈ Y . But ynp , x∗np  ≥ y, x∗np 

∀ y ∈ Y,

and so, taking the limit (for y ∈ Y fixed), we obtain that y ∈ Γ (x∗ ) ⊂ U . Since U is w-open, we get the contradiction that ynp ∈ U for p ≥ p0 , for some p0 ∈ N. Now 3.2.53 we have that PrMax(Y ; C) ⊂ Max(Y ; C) ⊂ w using Theorem  cl PrMax(Y ; C) . We obtain that Max(Y ; C) is connected w.r.t. w by a wellknown result in topology. (ii) The proof is similar.
Note that in general, Max(Y ; C) is not connected, as the following example shows. Example 3.2.57. Let Y = R3 , C = R3+ , C0 = {0} ∪ int C, and 
 (0, 0, k), (0, 1, k), (1, 0, k), (1 − 2−k , 1 − 2−k , k) . Y = conv n∈N

Then Max(Y ; C) = ∅, but Max(Y ; C0 ) = {(1, 0, z) | z ≥ 0} ∪ {(0, 1, z) | z ≥ 0} is obviously not connected. The most recent density results in the literature, for normed vector spaces, are those of Ferro [116] and Daniilidis [89] (stated for Y convex). So, in [116, Th. 3.1] one obtains Theorem 3.2.53 for Y weakly compact, Theorem 3.2.54(ii) for C well-based and Y w-compact, and Theorem 3.2.54(iii) for X a Banach space, C with w-compact base, and Y closed, while in [89, Th. 2] one obtains Theorem 3.2.53 for Y τ -compact. Theorems 3.2.53 and 3.2.54 also subsume the density results obtained by Jahn [197, Th. 3.1], Petchke [295, Th. 4.1], Borwein and Zhuang [42, Th. 4.2] (see also Remark 3.2.52), Zhuang [393, Th. 3.3], and Gong [136, Th. 3.1]. There are several density results established for topological vector spaces; we mention only the paper by Makarov and Rachkovski [247]. Theorem 3.2.56(i) was established by Song in [325, Th. 3.1], while Theorem 3.2.56(ii ) was essentially established by Gong [136, Th. 4.1].

3.3 Continuity Properties with Respect to a Scalarization Parameter Let Y be a topological vector space and C ⊂ Y a proper convex cone with nonempty interior and ∅ = A ⊂ Y . Let k0 ∈ int C. For every p ∈ Y consider the minimization problem

116

P (p)

3 Optimization in partially ordered spaces

minimize t

subject to t ∈ R, y ∈ A, p ∈ tk0 − y − C.

Consider also the marginal function m : Y → R associated with problems P (p) defined by m(p) := inf{t ∈ R | y ∈ A, p ∈ tk0 − y − C} = inf{t ∈ R | p ∈ tk0 − (A + C)}. We have the following relations between the weak efficient points of A w.r.t. C and the solutions of P (p). Let us denote the set B \ int B by ∂B; note that ∂B differs from the usual boundary bd B of B. Proposition 3.3.1. (i) If (t, y) ∈ R × A is a solution of P (p), then y ∈ w Eff(A; C) and tk0 − p ∈ ∂(A + C). (ii) If y ∈ w Eff(A; C), then (0, y) is a solution of P (−y). Proof. (i) Let (t, y) ∈ R × A be a solution of P (p) and suppose that y ∈ / w Eff(A; C). Then there exists k ∈ int C such that y := y − k ∈ A. There exists δ > 0 such that k − δk0 ∈ C. So p − (t − δ)k0 ∈ −y + δk0 − C = −y − (k − δk0 ) − C ⊂ −y − C, contradicting the optimality of (t, y). Therefore y ∈ w Eff(A; C). If tk0 − p ∈ / ∂(A + C), then tk0 − p ∈ int(A + C). There exists δ > 0 such that (t − δ)k0 − p ∈ A + C, which shows that (t, y) is not a solution of P (p), a contradiction. Therefore tk0 − p ∈ ∂(A + C). (ii) Let y ∈ w Eff(A; C); of course, (0, y) is an admissible solution of P (−y). Suppose that (0, y) is not a solution of P (−y). Then there exist t < 0 and y ∈ A such that −y ∈ tk0 − y − C, whence y ∈ y − int C, a contradiction.
We say that the segment [y, z] ⊂ Y with y = z is parallel to C if y ≤C z or z ≤C y. Lemma 3.3.2. Let ∅ = A, V ⊂ Y be such that ∂(A + C) ∩ V contains no segments parallel to C. Then ∂(A + C) ∩ aint V ⊂ Eff(A; C). Proof. Let y ∈ [(A + C) \ int(A + C)] ∩ aint V . Suppose that y − k ∈ A for some k ∈ C \ {0}. Since y ∈ aint V , there exists δ ∈ ]0, 1[ such that y + tk ∈ V for every t ∈ [−δ, δ]. Of course, for t ∈ [0, δ] we have that y − tk = y − k + (1 − t)k ⊂ A + C. Suppose that y − tk ∈ int(A + C) for some t ∈ ]0, δ]; i.e. y − tk + W ⊂ A + C for some neighborhood W of 0 in Y . Then y + W ⊂ A + C + tk ⊂ A + C, which shows that y ∈ int(A + C), a contradiction. Therefore y − tk ∈ ∂(A + C) ∩ V for every t ∈ [0, δ]. Since the segment [y, y − δk] is parallel to C contained in ∂(A + C) ∩ V , we get a contradiction. It follows that A ∩ (y − C) ⊂ {y}. Since y ∈ A + C, we obtain
that y ∈ A and, furthermore, y ∈ Eff(A; C). As application of the preceding lemma we have the following result.

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117

Proposition 3.3.3. Let p ∈ Y be such that the problem P (p) has at least a solution. If there exists an open set V ⊂ Y such that m(p)k0 − p ∈ V and ∂(A + C) ∩ V contains no segments parallel to C, then y ∈ A obtained by solving P (p) coincides with m(p)k0 − p and belongs to Eff(A; C). Proof. The conclusion follows immediately by using Proposition 3.3.1 (i) and Lemma 3.3.2.
Concerning the continuity properties of the marginal function m we have the following results. Proposition 3.3.4. The marginal function m is upper continuous at every p ∈ dom m. Proof. Indeed, let p0 ∈ dom m and λ ∈ R such that m(p0 ) < λ. Then there exists t < λ such that p0 ∈ tk0 − (A + C). It follows that p0 ∈ λk0 − [(λ − t)k0 + A + C] ⊂ λk0 − (A + int C) ⊂ λk0 − int(A + C). Therefore U := λk0 − (A + C) is a neighborhood of p0 . Of course, for p ∈ U we have that m(p) ≤ λ. Hence m is upper continuous at p0 .
Let M := {p ∈ Y | m(p) ∈ R} and consider the function g : M → Y defined by g(p) := m(p)k0 − p. Corollary 3.3.5. The function g is C-u.c. on M . Proof. Let p0 ∈ M be fixed and V ∈ VY . Consider V1 ∈ VY such that V1 + V1 ⊂ V . There exists ε > 0 such that εk0 ∈ V1 . Since m is upper continuous at p0 , there exists U1 ∈ VY such that m(p) < m(p0 ) + ε for every p ∈ p0 + U1 . Consider U := U1 ∩ V1 and p ∈ M ∩ (p0 + U ). Then m(p) − m(p0 ) = ε − γ with γ > 0, whence, for p ∈ M ∩ (p0 + U ), (m(p) − m(p0 )) k0 − (p − p0 ) ⊂ (ε − γ)k0 + V1 ⊂ V1 + V1 − γk0 ⊂ V − C. It follows that g(p) ∈ g(p0 ) + V − C, which shows that g is C-u.c. at p0 .




Proposition 3.3.6. Suppose that A+C is closed. Then m is lower continuous on Y and continuous on dom m. Proof. Taking T : Y × R → Y defined by T (p, t) := tk0 − p, T is a continuous linear operator. Therefore the set T −1 (A + C) is closed. It is also a set of epigraph type. It follows that epi m = T −1 (A + C), and so m is lower continuous. The continuity of m on dom m follows now from the preceding proposition.
Note that in the case in which A + C is closed, M is the set of those p ∈ Y for which problem P (p) has solutions; in this case, by Proposition 3.3.1 (i), we have that g(p) ∈ w Eff(A; C) for every p ∈ M .

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Corollary 3.3.7. Suppose that A + C is closed. Then g is C-continuous on M . Moreover, if C is normal, then g is continuous on M . Proof. The proof of the C-lower continuity of g is similar to that of Corollary 3.3.5, so we omit it. Therefore g is C-continuous on M . When C is normal, C-continuity and usual continuity are equivalent.
Combining Proposition 3.3.3 and Corollary 3.3.7 we obtain the following result. Corollary 3.3.8. Suppose that A + C is closed and that there exists an open set V ⊂ Y such that g(p0 ) ∈ V for some p0 ∈ M , and ∂(A + C) ∩ (V − C) contains no segments parallel to C. Then there exists a neighborhood U of p0 in M such that g is C-continuous on U with values in Eff(A; C). Proof. By the preceding corollary we have that g(p) ∈ V − C for p in a neighborhood U of p0 in M . Applying now Proposition 3.3.3 to each g(p) with p ∈ U , we obtain that g(p) ∈ Eff(A; C). The C-continuity of g is stated in the preceding corollary.
Consider now f : X → Y , where (X, τ ) is a topological space, and consider A := f (X). We associate with f the multifunction Σ : Y ⇒ X defined by  {x ∈ X | f (x) ∈ m(p)k0 − p − C} if p ∈ M, Σ(p) := ∅ if p ∈ Y \ M. Of course, Σ(p) is the set of weak efficient solutions of the problem to minimize f (x) w.r.t. C, by solving P (p). The next result holds. Proposition 3.3.9. Suppose that f is continuous, C is closed, and m is continuous at p0 ∈ M . Then Σ is closed at p0 . Proof. Let ((pi , xi ))i∈I ⊂ gr Σ converging to (p0 , x). Then m(pi )k0 − f (xi ) − pi ∈ C for every i ∈ I. Taking the limit, we obtain that m(p0 )k0 − f (x) − p0 ∈ C, which shows that x ∈ Σ(p0 ).
Of course, if Σ has some compactness properties at p0 , then Σ is even u.c. at p0 . Note that Pascoletti and Serafini [286] considered Y to be a finitedimensional space and a supplementary parameter q in span C; also, SternaKarwat considered this supplementary parameter q in the case in which the intrinsic core of C is nonempty, Y being an arbitrary topological vector space. Our motivation to consider a fixed parameter q (= k0 ) is that with the help of it one can find all the weak efficient elements of the set A. The results stated above correspond to similar ones in [331].

3.4 Well-posedness of vector optimization problems

119

3.4 Well-Posedness of Vector Optimization Problems Throughout this section X is a separated topological space, Y is a separated topological vector space, and C ⊂ Y is a convex cone with C = Y . Recall that for A ⊂ Y a nonempty set, Eff(A; C) := {a ∈ A | A ∩ (a − C) ⊂ a + C}. Of course, Eff(∅; C) = ∅. We say that A has the domination property (DP) (w.r.t. C) if A ⊂ Eff(A; C) + C. When int C = ∅ we denote by w Eff(A; C) the set Eff(A; {0} ∪ int C). Remark 3.4.1. If A ⊂ Y and int C = ∅, then A ∩ cl (Eff(A; C)) ⊂ w Eff(A; C) = A ∩ cl (w Eff(A; C)) . In particular, if A is closed, then w Eff(A; C) is closed, too. Indeed, let w Eff(A; C) ⊃ (yi )i∈I → y ∈ A. Suppose that y ∈ / w Eff(A; C). Then there exists y
∈ A ∩ (y − int C). It follows that yi − y
→ y − y
∈ int C, whence yi −y
∈ int C for i  i0 , contradicting the fact that yi0 ∈ w Eff(A; C). The other assertions are obvious. Let f : X → Y be a function and A ⊂ X a nonempty set. The optimal value of the vector optimization problem (P) C-minimize f (x) subject to x ∈ A is Ω = Ω(f, A; C) := Eff(f (A); C), while its solution set is Σ(f, A; C) := A∩f −1 (Ω); an element of Σ(f, A; C) is called an optimal solution, or simply solution, of problem (P). We associate with (P) and η ∈ Ω the multifunction Π η : Y ⇒ X defined by Π η (ε) := {x ∈ A | f (x) ≤C η + ε} for ε ∈ C and We also consider the multifunction Π : Y ⇒ X defined Π η (ε) := ∅ otherwise. ! by Π(ε) := η∈Ω Π η (ε). In the case X = Y and f = IdX we denote Π η and Π by Π0η and Π0 , respectively. An element of Π(ε) is called an ε-optimal solution of (P). Definition 3.4.2. Let f : X → Y be a function and A ⊂ X a nonempty set. We say that (i) (P) is η-well-posed if Ω = ∅ and Π η is u.c. at 0 for every η ∈ Ω; (ii) (P) is well-posed if Ω = ∅ and Π is u.c. at 0; (iii) (P) is weakly well-posed if X is a topological vector space, Ω = ∅, and Π is H-u.c. at 0. Before stating the next result, let us introduce another notion. We say that the nonempty set A ⊂ Y has the containment property (CP) if ∀ W ∈ VY , ∃ V ∈ VY

: [A \ (Eff(A; C) + W )] + V ⊂ Eff(A; C) + C.

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If int C = ∅, then (CP) is equivalent to ∀ W ∈ VY , ∃ V ∈ VY , ∀ y ∈ A \ (Eff(A; C) + W ) , ∃ y ∈ Eff(A; C), ∃ k ∈ C : y = y + k, k + V ⊂ C. (3.28) It is obvious that (3.28) is sufficient for (CP) (and implies that int C = ∅ if A ⊂ cl (Eff(A; C))). Suppose that (CP) holds and int C = ∅. For W ∈ VY let V0 ∈ VY be such that [A \ (Eff(A; C) + W )] + V0 ⊂ Eff(A; C) + C. Consider k ∈ V0 ∩ int C; there exists V ∈ VY such that k + V ⊂ C. Let y ∈ A\(Eff(A; C) + W ). It follows that y −k = y0 +k
for some y0 ∈ Eff(A; C) and k
∈ C. Hence y = y0 + k0 with k0 = k + k
. Since k0 + V = k + k
+ V ⊂ C + C = C, (3.28) holds. Generally the condition (DP) does not imply and is not implied by (CP). However, if (CP) holds and int C = ∅, then w Eff(A; C) = A ∩ cl (Eff(A; C)) and A ⊂ w Eff(A; C) + C. Indeed, suppose that (CP) holds and y ∈ A \ cl (Eff(A; C)). Then there exists W ∈ VY such that y ∈ A \ (Eff(A; C) + W ). By (3.28), y = y + k with y ∈ Eff(A; C) and k ∈ int C. Therefore y ∈ / w Eff(A; C). Hence w Eff(A; C) = A∩cl (Eff(A; C)). The preceding argument shows also that A ⊂ w Eff(A; C)+ C. We have the following result. Proposition 3.4.3. Let A ⊂ Y be nonempty and compact and let C be closed and with nonempty interior. If Eff(A; C) = w Eff(A; C), then (CP) holds for A. Proof. It is known that A has (DP) in this case. Since C ∩ (−C0 ) ⊂ C0 and C + (C0 \ {0}) ⊂ C0 , where C0 := {0} ∪ int C, we have that (DP) holds w.r.t. C0 , too. Therefore, taking into account that Eff(A; C) = w Eff(A; C), A ⊂ Eff(A; C0 ) + C0 = Eff(A; C) ∪ (Eff(A; C) + int C) . Let W ∈ VY . From the above inclusion we have that every y ∈ A \ Eff(A; C) has a representation y = ey + ky with ey ∈ Eff(A; C) and ky ∈ int C; let Vy ∈ VY be such that ky + Vy ⊂ C. There exists Vy1 ∈ VY open such that
 Vy1 +Vy1 ⊂ Vy . It follows that the family y+Vy1 | y ∈ A\(Eff(A; C) + int W ) represents an open cover of the compact set A\(Eff(A; C) + int W ). Therefore there exist y1 , . . . , yn ∈ A \ (Eff(A; C) + int W ) such that A \ (Eff(A; C) + int W ) ⊂ "n

n  

 yi + Vy1i .

i=1

Let V := i=1 Vy1i ∈ VY and consider y ∈ A \ (Eff(A; C) + int W ). Then there exists 1 ≤ i ≤ n such that y ∈ yi + Vy1i . So, using the above notation, y = eyi + kyi + vi , with vi ∈ Vy1i . Taking ky
:= kyi + vi , we have that ky
+ V = kyi + vi + V ⊂ kyi + Vy1i + V ⊂ kyi + Vyi ⊂ C. Therefore condition (3.28) holds, and so (CP) holds, too.


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121

Proposition 3.4.4. Let A ⊂ Y be a nonempty set. Then (i) Π0 is (−C)-H-u.c. at 0. (ii) If (DP) holds for A, then Π0 is C-u.c. at 0. (iii) If (CP) holds for A, then Π0 is H-u.c. at 0. (iv) If C is normal and closed, Eff(A; C) is compact, and (DP) holds for A, then Π0 is u.c. at 0. Proof. (i) Let V ∈ VY . Consider ε ∈ V ∩ C and y ∈ Π0 (ε). By definition, there exists η ∈ Π0 (0) with y ≤C η + ε, whence y ∈ Π0 (0) + V − C. Therefore Π0 (V ) ⊂ Π0 (0) + V − C, which shows that Π0 is (−C)-H-u.c. at 0. (ii) Since (DP) holds, Π0 (ε) ⊂ A ⊂ A + C = Π0 (0) + C ⊂ D + C for every ε ∈ X and every open set D ⊂ Y with Π0 (0) ⊂ D. Therefore Π0 is C-u.c. at 0. (iii) Let W ∈ VY . Since (CP) holds for A, there exists V ∈ VY such that [A \ (Π0 (0) + W )] + V ⊂ Π0 (0) + C. Consider ε ∈ C ∩ (V ∩ W ) and y ∈ Π0 (ε). Suppose that y ∈ / Π0 (0) + W ; then y + V ⊂ Π0 (0) + C. Since y ∈ Π0 (ε), y = η + ε − k for some η ∈ Π0 (0) and k ∈ C. But y − ε ∈ y + V , whence y − ε = η − k = η
+ k
with η
∈ Π0 (0) and k
∈ C. Therefore η = η
+ (k + k
). Since η, η
∈ Eff(A; C) and C is pointed, it follows that k = k
= 0, and so y ∈ Π0 (0) + W , a contradiction. Hence Π0 (V ∩ W ) ⊂ Π0 (0) + W , which shows that Π0 is H-u.c. at 0. (iv) Suppose that Π0 is not u.c. at 0; then there exist D0 ⊂ Y an open set with Π0 (0) ⊂ D0 and a net (εi )i∈I ⊂ C with (εi ) → 0 and Π0 (εi ) ⊂ D0 for every i ∈ I. Therefore for every i ∈ I there exists yi ∈ Π0 (εi ) \ D0 . It follows that for every i there exist ηi ∈ Π0 (0) and ki ∈ C such that yi = ηi + εi − ki . Since (DP) holds for A, yi = ηi
+ ki
with ηi
∈ Π0 (0) and ki
∈ C for every i. It follows that ηi + εi = ηi
+ (ki
+ ki ) for i ∈ I. Since Π0 (0) is compact, passing to subnets if necessary, we may suppose that (ηi ) and (ηi
) converge to η and η
from Π0 (0), respectively. We obtain that (ki
+ ki ) → η − η
∈ C. Since η, η
∈ Eff(A; C), we obtain that η = η
, and so (ki
+ ki ) → 0. Since C is normal, it follows that (ki ) → 0, and so (yi ) → η ∈ Π0 (0) ⊂ D0 . We get that yi ∈ D0 for i0  i (for some i0 ∈ I), a contradiction. Therefore Π0 is u.c. at 0.
In the general case we have the following result concerning the relations between η-well-posedness and well-posedness. Proposition 3.4.5. Let f : X → Y , and A ⊂ X a nonempty set. Suppose that Eff (f (A); C) is compact, int C = ∅, and (P) is η-well-posed. Then (P ) is well-posed. Proof. Consider D ⊂ X an open set such that Π(0) ⊂ D. Since Π η is u.c. at 0 and Π η (0) ⊂ Π(0) for every η ∈ Eff (f (A); C), for each η ∈ Eff (f (A); C) there exists Vη ∈ VY such that Π η (Vη ) = A ∩ f −1 (η + Vη ∩ C − C) ⊂ D. Of course, for every η ∈ Eff (f (A); C) there exists an open neighborhood Vη
of 0 ∈ Y such that Vη
+ Vη
⊂ Vη . Since the family

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3 Optimization in partially ordered spaces




η + (Vη
∩ int C) − (Vη
∩ int C) | η ∈ Eff(f (A); C)



represents an open cover of the compact set Eff (f (A); C), there exist η1 , . . . , ηn in the set Eff (f (A); C) such that Eff (f (A); C) ⊂

n  

 ηi + (Vη
i ∩ int C) − (Vη
i ∩ int C) .

(3.29)

i=1

"n Consider V := i=1 Vη
i ∈ VY . Let ε ∈ V ∩ C and x ∈ Π(ε). There exist η ∈ Eff (f (A); C) and k ∈ C such that f (x) = η + ε − k. From (3.29) we have that η = ηi +ki −ki
for some ki , ki
∈ Vη
i ∩int C. It follows that ε+ki ∈ Vηi ∩C, whence f (x) ∈ ηi + Vηi − C, and so x ∈ Π ηi (Vηi ) ⊂ D. Therefore Π(V ) ⊂ D. The proof is complete.
The notion of well-posedness for vector optimization problems, as well as that of the containment property, was introduced by Bednarczuk (see [17], [18]) in the framework of topological vector spaces ordered by closed and pointed convex cones with nonempty interior. In this framework, assertions (ii) and (iii) of Theorem 3.4.4 are proved in [17], [18], while assertion (iv) is proved in [18] for dim Y < ∞. Proposition 3.4.5 is stated by Bednarczuk [18], too. Proposition 3.4.3 is proved in [19].

3.5 Continuity Properties 3.5.1 Continuity Properties of Optimal-Value Multifunctions As in the preceding section we consider X a topological space, Y a topological vector space, C ⊂ Y a convex cone with C = Y , and the multifunction Γ :X ⇒Y. With Γ we associate the optimal-value multifunctions Ω, Ω : X ⇒ Y defined by   Ω(x) := Eff (Γ (x); C) , Ω(x) := Eff Γ (x); C . When int C = ∅ we also consider the weak optimal-value multifunctions Ωw , Ω w obtained from Ω and Ω, respectively, by replacing C with {0} ∪ int C. We say that Γ has the domination property (around x0 ∈ X) if Γ (x) has the domination property at every x ∈ X (at every x ∈ U0 , where U0 is some neighborhood of x0 ). Note that when Γ has the domination property, then ΓC = ΩC , and so Ω is C-(nearly) convex if X is a topological vector space and Γ is C-(nearly) convex. An important special case occurs when U is another topological (vector) space and Γ is replaced by f Λ, where f : X × U → Y and Λ : U ⇒ X, i.e., Γ (u) := f (Λ(u) × {u}) and Ω(u) := Eff (f (Λ(u) × {u}) ; C) (and similarly for Ωw (u)).

3.5 Continuity properties

123

In the first two results we give continuity properties of Ω under convexity assumptions for Γ . Proposition 3.5.1. Let X be a topological vector space and let Γ be C-nearly convex and have the domination property. If Γ is C-l.c. at (x0 , y0 ) ∈ gr Γ and Γ (x0 ) ⊂ B + C for some bounded set B ⊂ Y , then Ω is C-H-continuous on int(dom Γ ). Proof. Since Γ has the domination property, we have that ΓC = ΩC . Because Γ is C-l.c. at (x0 , y0 ) ∈ gr Γ ⊂ gr ΓC = gr ΩC , ΩC is C-l.c. at (x0 , y0 ). Of course, ΩC (x0 ) ⊂ B + C. Applying Theorem 2.6.4 we obtain that ΩC is CH-continuous on dom ΩC , whence Ω is C-H-continuous on dom Γ = dom Ω.
Proposition 3.5.2. Let X be a topological vector space and let Γ be Cnearly convex and have the domination property. Suppose that there exist x0 ∈ int(dom Γ ), B ⊂ Y a bounded set, and U ∈ VX such that Γ (x0 ) ⊂ B + C and Γ (x0 + u) ∩ (B − C) = ∅ for every u ∈ U . Then Ω is C-H-continuous on int(dom Γ ). Proof. Since Γ has the domination property, it follows that Ω(x0 ) ⊂ B + C and Ω(x0 + u) ∩ (B − C) = ∅ for every u ∈ U . Since Ω is C-nearly convex, we obtain that Ω is C-H-continuous on int(dom Γ ) = int(dom Ω), by applying Theorem 2.6.6.
In the sequel we study the continuity properties of the optimal multifunction Ω without convexity assumptions. We begin with a result concerning the lower semicontinuity of the optimal multifunction. Proposition 3.5.3. Suppose that Γ is C-l.s.c. and (DP) holds for Γ (x) for every x ∈ X. Then Ω is C-l.s.c., too. Proof. Let y ∈ Y and (xi )i∈I ⊂ levΩ (y), (xi )i∈I → x ∈ X. Then (xi )i∈I ⊂ levΓ (y). Since Γ is C-lsc, x ∈ levΓ (y); therefore Γ (x) ∩ (y − C) = ∅. Since Γ (x) ⊂ Ω(x) + C, it follows that Ω(x) ∩ (y − C) = ∅, whence x ∈ levΩ (y).
We say that the set A ⊂ Y is C-complete if every C-decreasing net of A is C-lower bounded by an element of A. Corollary 3.5.4. Let G : X ⇒ Y be C-l.s.c. and Λ : U ⇒ X be compact at every u ∈ U. If C is closed and G(x) + C is C-complete for every x ∈ X, then Γ = GΛ has (DP) at every u ∈ U and Ω is C-l.s.c. Proof. From Proposition 2.5.28 (v) we have that Γ is C-l.s.c.; in order to apply the preceding result we must show that Γ (u) has (DP) for every u ∈ U. In order to apply Proposition 3.2.14 (iv), let (yi )i∈I ⊂ Γ (u) be a C-decreasing net. For every i ∈ I there exists xi ∈ Λ(u) such that yi ∈ G(xi ). Since Λ(u) is compact, there exists the subnet (xϕ(j) )j∈J converging to x ∈ Λ(u). Let i ∈ I; there exists ji ∈ J such that ϕ(j)  i for every j  ji . It follows that

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3 Optimization in partially ordered spaces

yϕ(j) ≤C yi for all j  ji . Hence (xϕ(j) )j ji ⊂ levG (yi ). Since G is C-l.s.c., x ∈ levG (yi ); i.e., (yi )i∈I ⊂ G(x) + C. By hypothesis, there exist y0 ∈ G(x) and k0 ∈ C such that y0 + k0 ≤C yi for every i ∈ I. Therefore y0 ∈ Γ (u), and it is a minorant for (yi )i∈I . By the above-mentioned result we have that Γ (u) has (DP).
We continue with a closedness result for the weak optimal multifunction. Theorem 3.5.5. Suppose that int C = ∅ and Γ is C-l.c. at x0 . If Γ or Γ is closed at x0 , then Ωw or Ω w is closed at x0 , respectively. Proof. Suppose first that Γ is closed at x0 . Let y ∈ / Ωw (x0 ). If y ∈ / Γ (x0 ), since Γ is closed at x0 , there exist U ∈ VX (x0 ) and V ∈ VY (y) such that Γ (U ) ∩ V = ∅, whence Ωw (U ) ∩ V = ∅. Suppose now that y ∈ Γ (x0 ). Then there exists z ∈ Γ (x0 ) ∩ (y − int C). Consider W ∈ VY such that W + W ⊂ z − y + int C. Since Γ is C-l.c. at (x0 , z), there exists U ∈ VX (x0 ) such that Γ (x) ∩ (z + W − C) = ∅; i.e., z ∈ Γ (x) + W + C, for every x ∈ U . Let z
∈ y + W . It follows that z
∈ y + W − z + Γ (x) + W + C ⊂ Γ (x) + int C + C ⊂ Γ (x) + int C, which means that z
∈ / Ωw (x). Therefore Ωw (U ) ∩ W = ∅. Hence Ωw is closed at x. If Γ is closed at x0 , since in our conditions Γ is C-l.c. at x0 , by applying the first part we get that Ω is closed at x0 .
Corollary 3.5.6. Suppose that int C = ∅, Λ : U ⇒ X is l.c. at u0 , and f : X × U → Y is C-u.c. on Λ(u0 ) × {u0 }. If f Λ is closed at u0 , then Ωw is closed at x0 . Proof. Using Proposition 2.5.28 (iii) we have that f Λ is C-l.c. at x0 . The conclusion follows from the preceding theorem.
Using the preceding theorem we get also an upper continuity result for Ωw . Corollary 3.5.7. Suppose that int C = ∅, Γ is u.c., and C-l.c. at x0 and Γ (x0 ) or cl Γ (x0 ) is compact. Then Ωw or Ω w is u.c. at x0 , respectively. Proof. Suppose that Γ (x0 ) is compact. Using Proposition 2.5.10 (iii) we have that Γ is closed at x0 . Then using Theorem 3.5.5 we have that Ωw is closed at x0 . Since Ωw (x) = Ωw (x) ∩ Γ (x) for every x ∈ X, using Proposition 2.5.11 (i), we obtain that Ωw is u.c. at x0 . Suppose now that cl Γ (x0 ) is compact. Using Proposition 2.5.20 (iii) and (iv) we obtain that Γ is C-l.c. and u.c. at x0 . The conclusion follows from the first part.
The following examples show that the hypotheses on Γ in Theorem 3.5.5 and Corollary 3.5.7 are essential.

3.5 Continuity properties

125

Example 3.5.8. ([356]) Let X = ] − ∞, 0[, Y = R2 , C = R2+ , and let the multifunction Γ be defined by  if x = −1, {(y1 , y2 ) ∈ R2 | y2 ≥ xy1 , y1 ≤ 1, y2 ≤ 1} Γ (x) = {(y1 , y2 ) ∈ R2 | y2 ≥ −y1 + 1, y1 ≤ 1, y2 ≤ 1} if x = −1. Then Γ is l.c. (in particular, C-l.c.), but neither closed nor u.c., at x0 = −1; moreover, Γ (x0 ) is compact. We have that  {(y1 , y2 ) ∈ R2 | y2 = xy1 , x−1 ≤ y1 ≤ 1} if x = −1, Ω(x) = Ωw (x) = {(y1 , y2 ) ∈ R2 | y2 = −y1 + 1, 0 ≤ y1 ≤ 1} if x = −1; Ω is not closed at x0 . Example 3.5.9. ([356]) Let X = ] − ∞, 0[, Y = R2 , C = R2+ , and let the multifunction Γ be defined by  if x = −1, {(y1 , y2 ) ∈ R2 | y2 ≥ xy1 , y1 ≤ 1, y2 ≤ 1} Γ (x) = {(y1 , y2 ) ∈ R2 | y2 ≥ −y1 − 1, y1 ≤ 1, y2 ≤ 1} if x = −1. Then Γ is is closed and u.c., but not C-l.c., at x0 = 1; moreover, Γ (x0 ) is compact. We have that  if x = −1, {(y1 , y2 ) ∈ R2 | y2 = xy1 , x−1 ≤ y1 ≤ 1} Ω(x) = Ωw (x) = {(y1 , y2 ) ∈ R2 | y2 = −y1 − 1, −2 ≤ y1 ≤ 1} if x = −1; Ω is not closed at x0 . Proposition 3.5.10. Suppose that Γ is C-u.c. at x0 and (DP) holds for Γ (x0 ). Then Ω is C-u.c. at x0 . Proof. Let D ⊂ Y be an open set such that Ω(x0 ) ⊂ D. Then Γ (x0 ) ⊂ Ω(x0 ) + C ⊂ D + C. Since Γ is C-u.c. at x0 and D + C is open, there exists U ∈ VX (x0 ) such that Γ (U ) ⊂ D + C + C, whence Ω(U ) ⊂ D + C. Therefore Ω is C-u.c. at x0 .
In Example 3.5.8 (DP) holds for Γ (x0 ), but Ω is not C-u.c. at x0 , showing that the C-upper continuity of Γ at x0 is essential. Corollary 3.5.11. Let Λ : U ⇒ X be u.c. at u0 ∈ U. (i) If g : X → Y is C-l.c. on Λ(u0 ) and (DP) holds for gΛ(u0 ), then Ω is C-u.c. at u0 . (ii) Assume that f : X × U → Y is C-l.c. on Λ(u0 ) × {u0 }, (DP) holds for f Λ(u0 ), and either the set Λ(u0 ) is compact or the multifunction Λ : U ⇒  X × U, Λ(u) = Λ(u) × {u}, is u.c. at u0 ; then Ω is C-u.c. at u0 .

126

3 Optimization in partially ordered spaces

Proof. (i) Using Proposition 2.5.28 (i), we have that gΛ is C-u.c. at u0 . The conclusion follows from the preceding proposition. (ii) If Λ(u0 ) is compact, then as seen in the proof of Proposition 2.5.28 (ii), Λ is u.c. at u0 . So, in both cases the conclusion follows from (i) applied  respectively. with g and Λ replaced by f and Λ,
In order to obtain the P -H-upper continuity of Ω for a convex cone P ⊂ Y we introduce the multifunction  if x = x0 , Γ (x0 ) Γ P : X ⇒ Y, Γ P (x) := Γ (x) \ (Ω(x0 ) + P ) if x = x0 . For P = {0} the next result establishes another sufficient condition for Ω to be H-u.c. at u0 . Theorem 3.5.12. Suppose that int C = ∅ and (CP) holds for Γ (x0 ). If Γ is uniformly C-l.c. at x0 on Ω(x0 ) and Γ P is H-u.c. at x0 , then Ω is H-P -u.c. at x0 . Proof. When x0 ∈ / dom Γ , the conclusion is obvious. Let x0 ∈ dom Γ and suppose that Ω is not H-P -u.c. at x0 . Then there exists W ∈ VY such that ∀ U ∈ VX (x0 ), ∃ xU ∈ U, ∃ yU ∈ Ω(xU ) \ (Ω(x0 ) + W + P ) .

(3.30)

Consider W1 ∈ VY such that W1 + W1 ⊂ W . Since (CP) holds for Γ (x0 ), there exists V ∈ VY open such that ∀ y ∈ Γ (x0 )\(Ω(x0 ) + W1 ) , ∃ ey ∈ Ω(x0 ), ky ∈ C : y = ey +ky , ky +V ⊂ C. Let V1 ∈ VY be such that V1 + V1 ⊂ V . Since Γ P is H-u.c. at x0 , there exists U1 ∈ VX (x0 ) such that Γ (x) \ (Ω(x0 ) + P ) ⊂ Γ (x0 ) + V1 ∩ W1

∀ x ∈ U1 .

(3.31)

Since Γ is uniformly C-l.c. at x0 on Ω(x0 ), there exists U2 ∈ VX (x0 ) such that Ω(x0 ) ⊂ Γ (x) + V1 + C ∀ x ∈ U2 . (3.32) From (3.30) we obtain x
∈ U1 ∩ U2 and y
∈ Ω(x
) \ (Ω(x0 ) + W + P ) ⊂ Γ (x
) \ (Ω(x0 ) + P ). By (3.31), y
= y0 + w
for some y0 ∈ Γ (x0 ) and w
∈ V1 ∩ W1 . If y0 ∈ Ω(x0 ) + W1 , then y
∈ Ω(x0 ) + W + P , a contradiction. / Ω(x0 ) + W1 , whence y0 = e0 + k0 with e0 ∈ Ω(x0 ) and Therefore y0 ∈ k0 + V ⊂ int C. Since x
∈ U2 , by (3.32), there exist y

∈ Γ (x
), v
∈ V1 , and k
∈ C such that e0 = y

+ v
+ k
. Hence y
= y

+ v
+ k
+ k0 + w
∈ Γ (x
) + k0 + V + C ⊂ Γ (x
) + int C, contradicting the fact that y
∈ Ω(x
). The proof is complete.
P Note that for P = {0}, Γ is H-u.c. at x0 if and only if Γ is H-u.c. at x0 .

3.5 Continuity properties

127

Corollary 3.5.13. Suppose that Γ is H-u.c. at x0 ∈ X, and C is proper, closed, with nonempty interior. If Γ (x0 ) is compact, Eff(Γ (x0 ); C) = w Eff(Γ (x0 ); C), and Γ is C-l.c. at (x0 , y) for all y ∈ Ω(x0 ), then Ω is u.c. at x0 . Proof. We may suppose that x0 ∈ dom Γ . Using Proposition 3.4.3, we have that (CP) holds for Γ (x0 ), while from Remark 2.5.19 we have that Γ is uniformly C-l.c. at x0 on Ω(x0 ). Then, from the preceding theorem for P = {0}, we obtain that Ω is H-u.c. at x0 . Since Eff(A; C) = w Eff(A; C), Ω(x0 ) is closed, and so compact. Hence Ω is actually u.c. at x0 .
In Examples 3.5.8 and 3.5.9 all the conditions in Theorem 3.5.12 (for P = {0}) and Corollary 3.5.13 are satisfied, except the fact that Γ is H-u.c. at x0 in Example 3.5.8 and Γ is uniformly C-l.c. at x0 on Ω(x0 ) in Example 3.5.9, showing that these two conditions are essential in those results. The next example shows that the condition (CP) for Γ (x0 ) is also essential in Theorem 3.5.12. Example 3.5.14. Let X = [0, 1], Y = R2 , C = R2+ , and let Γ be defined by  if x = 0, {(y1 , y2 ) | y1 ≥ 0, y2 = min(y12 , e−y1 +1 )} Γ (x) = {(y1 , y2 ) | x ≤ y1 ≤ 2 − 2 ln x, y2 = min(y12 , e−y1 +1 )} if x = 0. Then Γ is H-u.c. and l.c. at 0, whence Γ P is H-u.c. at x0 for every convex cone P ⊂ Y , Γ (0) has the domination property, but (CP) does not hold for Γ (0). Moreover,  {(0, 0)} if x = 0, Ω(x) = {(x, x2 )} ∪ {(y, e1−y ) | 1 − 2 ln x < y ≤ 2 − 2 ln x} if x = 0. So Γ is uniformly C-l.c. at x0 on Ω(x0 ), but Ω is not H-P -u.c. at x0 for P ⊂ −C. Note that asking Γ to be C-l.c. at x0 makes the preceding corollary a particular