Variational methods in partially ordered spaces

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.

© 2003 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1

SPIN 10911884

Typesetting: Pages created by the authors using the Springer www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH

2e macro package (w/svmono.cls).

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)

28

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 + ).

34

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.

46

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,

50

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

52

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

54

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.

56

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).

58

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:

60

2 Functional analysis over cones

(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)

2.5 Continuity notions for multifunctions

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.

62

2 Functional analysis over cones

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,

2.5 Continuity notions for multifunctions

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

64

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.

66

2 Functional analysis over cones

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 .

68

2 Functional analysis over cones

(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].

70

2 Functional analysis over cones

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 .

72

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.

76

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

82

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

84

3 Optimization in partially ordered spaces

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,

86

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).

88

3 Optimization in partially ordered spaces

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) = ∅.

90

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)

92

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.

94

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

96

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

98

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

100

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.


3.2 Existence results for efficient points

101

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.

102

3 Optimization in partially ordered spaces

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.


3.2 Existence results for efficient points

103

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).

104

3 Optimization in partially ordered spaces

(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) = ∅.

3.2 Existence results for efficient points

105

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) = ∅.

106

3 Optimization in partially ordered spaces

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) = ∅.

3.2 Existence results for efficient points

107

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).

108

3 Optimization in partially ordered spaces

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) = ∅.

3.2 Existence results for efficient points

109

(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.

110

3 Optimization in partially ordered spaces

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.

112

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 ,

114

3 Optimization in partially ordered spaces

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.

3.3 Continuity properties with respect to a scalarization parameter

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 .

118

3 Optimization in partially ordered spaces

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.

120

3 Optimization in partially ordered spaces

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.


3.4 Well-posedness of vector optimization problems

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

122

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

124

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 case of Corollary 3.5.7 because Ω(x0 ) = Ωw (x0 ) and Ω(x) ⊂ Ωw (x) for x ∈ X. Corollary 3.5.15. Suppose that C is closed, proper, with nonempty interior, Λ : U ⇒ X is u.c. and l.c. at u0 , Λ(u0 ) is compact, and f : X × U → Y is continuous on Λ(u0 ) × {u0 }. If w Eff (f Λ(u0 ); C) = Eff (f Λ(u0 ); C), then Ω is u.c. at u0 . Proof. Since Λ(u0 ) is compact and f is continuous on Λ(u0 ) × {u0 }, using Proposition 2.5.28 (ii) and (vi), f Λ is u.c. and l.c. at u0 ; moreover, f Λ(u0 ) is compact. Using the preceding corollary we obtain the conclusion.
In the following results we establish sufficient conditions for the lower continuity of the optimal multifunction Ω.

128

3 Optimization in partially ordered spaces

Proposition 3.5.16. Suppose that Γ is C-l.c. at (x0 , y0 ) with y0 ∈ Ω(x0 ). If (DP) holds for Γ around x0 , then Ω is C-l.c. at (x0 , y0 ). Proof. Take U0 ∈ VX (x0 ) such that Γ (x) ⊂ Ω(x) + C for x ∈ U . Let W ∈ VY . Since Γ is C-l.c. at (x0 , y0 ), there exists U1 ∈ VX (x0 ) such that Γ (x) ∩ (W − C) = ∅ for every x ∈ U1 . Let x ∈ U0 ∩ U1 . Then there exists y ∈ Γ (x) ∩ (W − C); it follows that y = y
+ k with y
∈ Ω(x) and k ∈ C. Therefore y
∈ W − C − C = W − C, whence Ω(x) ∩ (W − C) = ∅. Hence Ω is C-l.c. at (x0 , y0 ).
In Example 3.5.8 (DP) holds for Γ (x) for every x ∈ X, but Ω is not C-l.c. at (x0 , y) for every y ∈ Ω(x0 ); Γ is not C-l.c. at (x0 , y) for every y ∈ Ω(x0 ). In the next example (DP) does not hold for Γ around x0 . Example 3.5.17. ([356]) Let X = [0, ∞[, Y = R2 , C = R2+ , and let Γ be defined by  {(y1 , y2 ) ∈ R2 | y12 + y22 < x2 } if x = 1, Γ (x) = {(y1 , y2 ) ∈ R2 | y12 + y22 ≤ 1} if x = 1. Then Γ is l.c. (and so C-l.c.) at every x ∈ X and  ∅ if x = 1, Ω(x) = {(y1 , y2 ) ∈ R2 | y12 + y22 = 1, y1 ≤ 0, y2 ≤ 0} if x = 1. So (DP) does not hold for Γ around x0 = 1, and Ω is not C-l.c. at (x0 , y) for every y ∈ Ω(x0 ). Corollary 3.5.18. Suppose that Λ : U ⇒ X is l.c. at u0 ∈ U, and f : X ×U → Y is C-u.c. on Λ(u0 ) × {u0 }. If (DP) holds for f Λ around u0 , then Ω is C-l.c. at u0 . Proof. Using Proposition 2.5.28 (iii), f Λ is C-l.c. at u0 . The conclusion follows from the preceding proposition.
In order the obtain the next result we need a uniform (CP) condition. So, we say that (CP) holds for Γ uniformly around x0 ∈ X if there exists U ∈ VX (x0 ) such that ∀ W ∈ VY , ∃ V ∈ VY , ∀ x ∈ U : [Γ (x) \ (Ω(x) + W )] + V ⊂ Ω(x) + C. When int C = ∅, one can write a similar condition to (3.28). Theorem 3.5.19. Suppose that int C = ∅, (CP) holds for Γ uniformly around x0 ∈ X, and Γ is H-C-u.c. at x0 . If Γ is l.c. at (x0 , y0 ), where y0 ∈ Ω(x0 ), then Ω is l.c. at (x0 , y0 ). Moreover, if Γ is uniformly {0}-l.c. at x0 on Ω(x0 ), then Ω is H-l.c. at x0 .

3.5 Continuity properties

129

Proof. Let U0 ∈ VX (x0 ) be such that (CP) holds uniformly for Γ (x) with x ∈ U . Let us fix W ∈ VY and consider W1 ∈ VY such that W1 + W1 ⊂ W . There exists V ∈ VY open such that ∀x ∈ U0 , ∀y ∈ Γ (x)\(Ω(x) + W1 ) , ∃e ∈ Ω(x), k ∈ C : y = e+k, k +V ⊂ C. (3.33) There exists V1 ∈ VY such that V1 + V1 ⊂ V . Since Γ is l.c. at (x0 , y0 ), there exists U1 ∈ VX (x0 ) such that y0 ∈ Γ (x) + V1 ∩ W1

∀ x ∈ U1 .

(3.34)

Now, since Γ is H-C-u.c. at x0 , there exists U2 ∈ VX (x0 ) such that Γ (x) ⊂ Γ (x0 ) + V1 + C

∀ x ∈ U2 .

(3.35)

Let U := U0 ∩ U1 ∩ U2 and take x ∈ U . From (3.34) we have that y0 = yx + wx with yx ∈ Γ (x) and wx ∈ V1 ∩ W1 . Suppose that yx ∈ / Ω(x) + W1 . From (3.33) we get ex ∈ Ω(x) and kx ∈ C such that yx = ex + kx and kx + V ⊂ int C. From (3.35) we have that ex = yx0 + vx
+ kx
with yx0 ∈ Γ (x0 ), vx
∈ V1 , and kx
∈ C. So, y0 = yx0 + vx
+ kx
+ kx + wx , whence y0 ∈ yx0 + kx + V1 + V1 + C ⊂ yx0 + int C + C ⊂ yx0 + int C, contradicting the fact that y0 ∈ Ω(x0 ). Therefore yx ∈ Ω(x) + W1 , whence y0 ∈ Ω(x) + W . Hence Ω is l.c. at (x0 , y0 ). If Γ is uniformly {0}-l.c. at x0 on Ω(x0 ), relation (3.34) holds for every y0 ∈ Ω(x0 ), and so Ω(x0 ) ⊂ Ω(x) + W for all x ∈ U .
In Examples 3.5.8, 3.5.9, and 3.5.17 all the hypotheses of the preceding theorem are satisfied at x0 and y0 ∈ Ω(x0 ), except, respectively, the fact that Γ is H-C-u.c. at x0 , Γ is l.c. at (x0 , y0 ), and (CP) holds for Γ , uniformly around x0 ; Ω is not l.c. at (x0 , y0 ). The assumption that (CP) holds uniformly for Γ around x0 is sufficiently strong. This condition may be relaxed if Γ (x0 ) or Ω(x0 ) is compact. Theorem 3.5.20. Suppose that C is closed and normal, Γ satisfies (DP) around x0 ∈ X, Γ is H-C-u.c. at x0 , and either Γ (x0 ) is closed and Ω(x0 ) is relatively compact (for example, if Γ (x0 ) is compact) or Ω(x0 ) is compact. Then, if Γ is l.c. at (x0 , y0 ) ∈ gr Ω, Ω is l.c. at (x0 , y0 ), too. Moreover, if Γ is l.c. at (x0 , y) for every y ∈ Ω(x0 ), then Ω is l.c. at x0 (even H-l.c. at x0 if Ω(x0 ) is compact). Proof. We give the proof for X, Y first-countable spaces; in the general case one uses the equivalence of (i) and (ii) of Proposition 2.5.6 and assertion (ii) of Proposition 2.5.14 instead of the equivalence of (i) and (iii), and assertion (iv), respectively. Let us note first that C is necessarily pointed because C is normal and Y is separated. Let X ⊃ (xn ) → x0 . Since Γ is l.c. at (x0 , y0 ), using Proposition

130

3 Optimization in partially ordered spaces

2.5.6, there exists Y ⊃ (yn ) → y0 such that yn ∈ Γ (xn ) for every n ≥ n0 . We may suppose that (DP) holds for Γ (xn ) for n ≥ n0 . Hence yn = y n + kn with y n ∈ Ω(xn ) ⊂ Γ (xn ) and kn ∈ C for n ≥ n0 . Since ΓC is H-u.c. at x0 , using Proposition 2.5.14, there exist (yn0 ) ⊂ Γ (x0 ) and (kn0 ) ⊂ C such that 0 wn := y n − (yn0 + kn0 ) → 0. Since Γ (x0 ) has (DP), we have that yn0 = y 0n + k n 0 with y 0n ∈ Ω(x0 ) and k n ∈ C. So, for n ≥ n0 , 0

yn = y n + kn = yn0 + wn + (kn0 + kn ) = y 0n + wn + (k n + kn0 + kn ), 0

where (yn0 ) ⊂ Γ (x0 ), (y 0n ) ⊂ Ω(x0 ), (k n ), (kn0 ), (kn ) ⊂ C and (wn ) → 0. Since Ω(x0 ) is relatively compact, there exists a subsequence (y 0np ) → y 0 ; 0

in both cases y 0 ∈ Γ (x0 ), and so (k n + kn0 p + knp ) → y0 − y 0 ∈ C. Since 0

y0 ∈ Ω(x0 ) and C is pointed, we have that y0 = y 0 , and so (k n +kn0 p +knp ) → 0. Since C is normal, we obtain that (knp ) → 0, whence (y np ) → y0 . Arguing by contradiction we obtain, in fact, that (y n ) → y0 . Using again Proposition 2.5.6 we obtain that Ω is l.c. at (x0 , y0 ). The second part is an immediate consequence of the first one.
In Examples 3.5.8, 3.5.9, and 3.5.17 all the hypotheses of the preceding theorem are satisfied at x0 and y0 ∈ Ω(x0 ), except, respectively, the fact that Γ is H-C-u.c. at x0 , Γ is l.c. at (x0 , y0 ), and (DP) holds for Γ around x0 ; Ω is not l.c. at (x0 , y0 ). In the next result we suppose that Y is a normed vector space. Theorem 3.5.21. Suppose that Y is a normed vector space, C is closed and normal, and Γ is l.c. and closed at x0 . If there exists a neighborhood U0 of x0 such that Γ (x) satisfies (DP) for x ∈ U0• := U0 \ {x0 }, Γ (U0• ) is C-bounded from below (i.e., Γ (U0• ) ⊂ B0 + C for some bounded set B0 ⊂ Y ), and the bounded subsets of Ω(U0• ) are relatively compact, then Ω is l.c. at x0 . Proof. Consider y0 ∈ Ω(x0 ) and X ⊃ (xi )i∈I → x0 . We may suppose that (xi )i∈I ⊂ U0• . Since Γ is l.c. at x0 , by Proposition 2.5.6, there exist a subnet (xϕ(j) )j∈J of (xi ) and a net (yj )j∈J → y0 with yj ∈ Γ (xϕ(j) ) for every j ∈ J. Because Γ (xi ) satisfies (DP) for i ∈ I, there exist y j ∈ Ω(xϕ(j) ) and kj ∈ C such that yj = y j + kj for all j ∈ J. Since (yj ) → y0 , there exists a bounded set B1 ⊂ Y such that (yj )j j1 ⊂ B1 for some j1 ∈ J. Taking B := B0 ∪ B1 , we obtain that y j ∈ (B + C) ∩ (B − C) for j  j1 . Since C is normal, the set (B + C) ∩ (B − C) is bounded, whence (y j )j j1 ⊂ Ω(U0• ) is bounded. By hypothesis, (y j )j j1 contains a subnet (y ψ(p) )p∈P converging to y ∈ Y . Since Γ is closed at x0 , it follows that y ∈ Γ (x0 ). Since kj = yj − y j ∈ C and C is closed, we obtain that y0 − y ∈ C, whence y0 = y (C being pointed and y0 ∈ Ω(x0 )). Therefore (y ψ(p) )p∈P → y. Using again Proposition 2.5.6 we
have that Ω is l.c. at x0 . Examples 3.5.8, 3.5.9, and 3.5.17 show that the conditions Γ is closed at x0 , Γ is l.c. at x0 , and (DP) holds for Γ around x0 are essential in the

3.5 Continuity properties

131

preceding theorem. Also, the hypothesis that Γ (U0• ) is bounded from below for some U ∈ VX (x0 ) is important, as the next example shows. Example 3.5.22. Let X = [0, ∞[, Y = R2 , C = R2+ , and let Γ be defined by  Γ (x) :=

{(0, 0)}∪ ] − ∞, 1] × {1} if x = 0, x 1 {(0, 0)} ∪ {(y1 , x+1 y1 + x+1 ) | y1 ∈ [− x1 , 1]} if x > 0.

Then Γ is closed and l.c. at x0 = 0, Γ (x) has the domination property for every x = 0, but Γ (U • ) is not bounded from below for every U ∈ VX (x0 ). Moreover,  {(0, 0)} if x = 0, Ω(x) = {(− x1 , 0)} if x > 0. Of course, Ω is not l.c. at x0 . Another version of the preceding theorem is given in the next result. Theorem 3.5.23. Suppose that Y is a normed vector space, C is closed and C ⊂ {y ∈ Y | y ≤ y, y ∗ } for some y ∗ ∈ Y ∗ , and Γ is l.c. and closed at x0 ∈ X. If there exists a neighborhood U0 of x0 such that Γ (x) is complete for x ∈ U0• , y ∗ (Γ (U0• )) is bounded from below, and the bounded subsets of Ω(U0• ) are relatively compact, then Ω is l.c. at x0 . Proof. Fix x ∈ U0• . Since Γ (x) is complete and y ∗ (Γ (x)) is bounded from below, using Theorem 3.2.36 of Attouch–Riahi (see also [10, Th. 2.5]; it is stated for Y a Banach space, but it is sufficient for the set to be complete), Γ (x) has the domination property. ' Using ' thesame notation as in the proof of Theorem 3.5.21, we have that 'yj − y j ' ≤ yj − y j , y ∗ ≤ yj , y ∗  − sup y ∗ (Γ (U0• )). It follows that (y j )j j1 is bounded. The conclusion follows as in the proof of Theorem 3.5.21.
The same examples, as for Theorem 3.5.21, show that all the conditions on Γ in the preceding theorem are essential. Corollary 3.5.24. Suppose that X is a topological vector space, C is closed and normal, Λ : U ⇒ X is H-u.c. and l.c. at u0 , and either f : X × U → Y is equicontinuous on Λ(u0 ) × {u0 } or f is continuous on Λ(u0 ) × {u0 } and  Λ : U ⇒ X × U, Λ(u) := Λ(u) × {u}, is u.c. at u0 . If (DP) holds for f Λ around u0 and either Λ(u0 ) is compact or Ω(u0 ) is compact, then Ω is l.c. at u0 . Proof. From Proposition 2.5.28 (iii) we have that f Λ is l.c. at u0 , while from (v) if f is equicontinuous on Λ(u0 ) × {u0 } or (i) if Λ is u.c. at u0 and f is continuous on Λ(u0 ) × {u0 } (of the same proposition), we have that f Λ is H-u.c., or u.c. at u0 , respectively. Moreover, f Λ(u0 ) is compact if Λ(u0 ) is so. The conclusion follows by applying the preceding theorem.


132

3 Optimization in partially ordered spaces

One can obtain the lower continuity of Ω at (x0 , y0 ) without using condition (CP) or compactness conditions if one knows that y0 is a strong proper efficient point of Γ (x0 ). One says that y is a strong proper efficient point of A ⊂ Y , the class of such points being denoted by SPEff(A; C), if y ∈ Eff(A; P ), where P is a proper convex cone such that the pair (C, P ) has the property ∀ W ∈ VY , ∃ V ∈ VY : (C \ W ) + V ⊂ P.

(3.36)

Proposition 3.5.25. Let P ⊂ Y be a convex cone such that (3.36) holds. Then (cl C, P ) satisfies (3.36), cl C \ {0} ⊂ int P , and ∀ W ∈ VY , ∃ V ∈ VY , ∀ y ∈ Y \ (W ∪ P ) : (y + V ) ∩ C = ∅.

(3.37)

If C has a bounded base B, then (C, P ) satisfies (3.36) if and only if there exists V ∈ VY such that B + V ⊂ P . So, if C has a bounded base and Y is a locally convex space, then there exists a proper convex cone P ⊂ Y such that (3.36) holds. The converse is true if Y is a normed vector space. Proof. Let (C, P ) satisfy (3.36) and take W ∈ VY . There exists W0 ∈ VY such that W0 + W0 ⊂ W . By (3.36) there exists V ∈ VY such that (C \ W0 ) + V ⊂ P . Consider V0 ∈ VY such that V0 + V0 ⊂ V . Then (cl C \ W ) + V0 ⊂ P . Indeed, let x = x + v with x ∈ cl C \ W and v ∈ V0 . Since x ∈ cl C ⊂ C + (V0 ∩ W0 ), x = k + w for some k ∈ C and w ∈ V0 ∩ W0 . Then k ∈ / W0 ; otherwise, x ∈ W0 + W0 ⊂ W . Moreover, w + v ∈ V0 + V0 ⊂ V , whence x ∈ (C \ W0 ) + V ⊂ P . Consider now y ∈ cl C \ {0}; there exists W ∈ VY such that y ∈ / W . Since (cl C, P ) satisfies (3.36), there exists V ∈ VY such that (cl C \ W ) + V ⊂ P , whence y + V ⊂ P . Therefore y ∈ int P . In order to prove (3.37), let W ∈ VY and take W1 ∈ VY with W1 +W1 ⊂ W . There exists V1 ∈ VY such that (C \ W1 ) + V1 ⊂ P . Consider V := V1 ∩ W1 . Suppose that for some y ∈ Y \ (W ∪ P ) there exists y
∈ (y + V ) ∩ C; then y
∈ / W1 (otherwise, y ∈ W1 −V ⊂ W ). Hence y ∈ y
+V ⊂ P , a contradiction. Suppose now that B is a (convex) base of C. Suppose first that (3.36) holds. Since 0 ∈ / cl B, there exists W ∈ VY such that B ∩ W = ∅. There exists V ∈ VY such that B + V ⊂ (C \ W ) + V ⊂ P . Suppose now that V0 ∈ VY and B+V0 ⊂ P . By contradiction, suppose that (3.36) does not hold. Then there exists W0 ∈ VY such that (C \ W0 ) + V ⊂ P for every V ∈ VY . Of course, we may suppose that B ∩ W0 = ∅. Since B is bounded, there exists λ > 0 such that B ⊂ λW0 . It follows that / W0 , tV bV + yV ∈ / P. ∀ V ∈ VY , ∃ tV > 0, ∃ bV ∈ B, ∃ yV ∈ V : tV bV ∈ Therefore bV +t−1 / P , whence t−1 / V0 for every V ∈ VY . Since tV bV ∈ / V yV ∈ V yV ∈ W0 and B ⊂ λW0 we get t−1 V < λ for every V ∈ VY . Since (yV )V ∈VY → 0, −1 / V0 for every it follows that (t−1 V yV ) → 0, contradicting the fact that tV yV ∈ V ∈ VY . Therefore (3.36) holds.

3.5 Continuity properties

133

Now let Y be a locally convex space and C have a bounded base B. Then there exists a convex V ∈ VY such that B ∩ V = ∅. Then 0 ∈ / B + V . Taking P = [0, ∞[·(B + V ), P is a proper convex cone and B + V ⊂ P . Suppose now that Y is a normed vector space and (3.36) holds for some proper convex cone P ⊂ Y . Let B0 = {y ∈ C | y = 1}. From (3.36) for W = B(0, 1) = {y ∈ Y | y < 1}, there exists r > 0 such that B0 + B(0, r) ⊂ P . Then B = conv B0 is a bounded base for C. Indeed, it is obvious that C = [0, ∞[·B. But, B + B(0, r) = conv (B0 + B(0, r)) ⊂ P, and so cl B + B(0, r/2) ⊂ P . If 0 ∈ cl B, then B(0, r/2) ⊂ P , whence P = Y , a contradiction. It follows that B is a (bounded) base for C.
In the definition of a strong proper efficient point we may suppose that P is pointed (otherwise, replace P by {0} ∪ int P ). Theorem 3.5.26. Consider y0 ∈ SPEff(Γ (x0 ); C) and suppose that Γ is C0 H-u.c. at x0 and l.c. at (x0 , y0 ), where C0 = C if C is normal and C0 = {0} otherwise. Then Ω is l.c. at (x0 , y0 ) if for some U0 ∈ VX (x0 ) one of the following conditions holds: (i) Γ (x) has the domination property for every x ∈ U0 ; (ii) Y is locally compact, C is closed, and Γ (u) is closed for every x ∈ U0 . Proof. Let P be the pointed convex cone with the property (3.37) for which y0 ∈ Eff(Γ (x0 ); P ). It follows that Γ (x0 ) ⊂ (y0 − P )c ∪ {y0 },

(3.38)

where Ac denotes Y \ A. Consider W ∈ VY such that (W + C0 ) ∩ (W − C0 ) = W , and W compact if Y is locally compact. Let W1 ∈ VY be such that W1 + W1 ⊂ W . From (3.37) we get V ∈ VY such that (y + V ) ∩ C = ∅ for every y ∈ Y \ (W ∪ P ). It follows that [((y0 − P )c \ (y0 + W1 )) + V ] ∩ (y0 − C) = ∅. Taking V1 ∈ VY such that V1 + V1 ⊂ V , we obtain that [((y0 − P )c \ (y0 + W1 )) + V1 ] ∩ [(y0 + V1 ) − C] = ∅.

(3.39)

From (3.38), taking into account that (y0 − P )c + C = (y0 − P )c , we obtain Γ (x0 ) + V1 ∩ W1 + C0 ⊂ [((y0 − P )c \ (y0 + W1 )) + V1 ∩ W1 ] ∪ (y0 + W + C0 ). Since Γ is C0 -H-u.c. at x0 , there exists U1 ∈ VX (x0 ) such that Γ (x) ⊂ Γ (x0 ) + V1 ∩ W1 + C0

∀ x ∈ U1 .

Therefore Γ (x) ⊂ [((y0 − P )c \ (y0 + W1 )) + V1 ∩ W1 ]∪(y0 +W +C0 )

∀ x ∈ U1 . (3.40)

134

3 Optimization in partially ordered spaces

Since Γ is l.c. at (x0 , y0 ), there exists U2 ∈ VX (x0 ) such that (y0 + V1 ∩ W1 ) ∩ Γ (x) = ∅ for every x ∈ U2 . For x ∈ U := U0 ∩ U1 ∩ U2 let yx ∈ (y0 + V1 ∩ W1 ) ∩ Γ (x). Then, by applying (3.39), we obtain that (yx − C) ∩ [((y0 − P )c \ (y0 + W1 )) + V1 ∩ W1 ] = ∅. Thus, from (3.40), we obtain that (yx − C) ∩ Γ (x) ⊂ (y0 + W + C0 ) ∩ (y0 + V1 ∩ W1 − C) ⊂ y0 + W

∀ x ∈ U.

For x ∈ U , if (i) holds, then there exists y x ∈ Ω(x) ∩ (yx − C), while if (ii) holds, (yx − C) ∩ Γ (x) is a compact set (as a closed subset of a compact set), and we obtain again y x ∈ Ω(x) ∩ (yx − C); in both cases we have that Ω(x) ∩ (y0 + W ) = ∅ for all x ∈ U . The proof is complete.
As a consequence of the preceding theorem we get the following result. Corollary 3.5.27. Let Y = Rm and let C be closed and pointed. Suppose that y0 is an element of SPEff(Γ (x0 ); C) and Γ is closed-valued, H-C-u.c. at x0 , and l.c. at (x0 , y0 ). Then Ω is l.c. at (x0 , y0 ). Proof. Note that C is normal in this case. Applying Theorem 3.5.26 (ii) we get the conclusion.
Examples 3.5.8, 3.5.9 (with y0 ∈ Ω(x0 )), and 3.5.17 (with y0 = (−2−1/2 , −2−1/2 )) show that the hypotheses Γ is H-C-u.c. at x0 , Γ is l.c. at (x0 , y0 ), and Γ is closed-valued, respectively, are essential for Theorem 3.5.26 and Corollary 3.5.27. Corollary 3.5.28. Suppose that Λ : U ⇒ X is l.c. and H-u.c. at u0 , f : X × U → Y is equicontinuous on Λ(u0 ) × {u0 }, and f Λ satisfies (DP) around u0 . Then Ω is l.c. at (u0 , y0 ) for all y0 ∈ SPEff (f Λ(u0 ); C). Proof. Applying again Proposition 2.5.28 (iii) and (vii), f Λ is l.c. and H-u.c.
at u0 . The conclusion follows by applying the preceding theorem. Note that Sterna-Karwat [333] established Proposition 3.5.1 for Γ with the domination property, C-mid-convex, bounded-valued, and H-C-l.c. at x0 ∈ dom Γ , and Proposition 3.5.2 for Γ with the domination property, C-midconvex, bounded-valued, and C-upper bounded, i.e., Im Γ ⊂ B − C for some bounded set B ⊂ Y . Corollary 3.5.4 was established by Ferro [114, Th. 4.1] under the following additional assumptions: U is a locally convex space ordered by closed convex cone D with nonempty interior, G is a function, Λ(u1 ) ⊂ Λ(u2 ) if u2 − u1 ∈ int D. Theorem 3.5.5, Propositions 3.5.10, 3.5.16, and Corollaries 3.5.11, 3.5.7, 3.5.18 were obtained by Penot and Sterna-Karwat in [291] and [292] even for more general relations then ≤C . Theorem 3.5.12 for P = {0} is stated by Bednarczuk in [17], while Theorem 3.5.12 for P = −C and Corollary 3.5.13 are stated in [19]. Theorem 3.5.19 with

3.5 Continuity properties

135

the stronger conditions that Γ is H-u.c. and l.c. at x0 , Theorem 3.5.20 under somewhat different conditions (for Y first countable, without asking that C be normal, but we doubt that this is true), Theorem 3.5.26 (except the case when C is normal) and its corollary are obtained in [19]. Also, Corollaries 3.5.15, 3.5.24, and 3.5.28 are from [19]. Theorem 3.5.23 was obtained by Attouch and Riahi [10] for X = N ∪ {∞}, x0 = ∞, and Y a Banach space. 3.5.2 Continuity Properties for the Optimal Multifunction in the Case of Moving Cones As in the preceding section, X is a topological space, Y is a topological vector space, and Γ : X ⇒ Y . In this section instead of a single convex cone C ⊂ Y we consider a multifunction C : X ⇒ Y whose values are pointed convex cones. We associate also the multifunction C c : X ⇒ Y defined by C c (x) := Y \ (C(x) \ {0}) = (Y \ C(x)) ∪ {0}. In this section Ω, Ωw : X ⇒ Y , Ω(x) := Eff(Γ (x);C(x)), and Ωw (x) := w Eff(Γ (x);C(x)). Theorem 3.5.29. Suppose that Γ (x0 ) ⊂ lim inf (Γ (x) + C(x)) and lim sup C c (x) ⊂ C c (x0 ). x→x0

(3.41)

x→x0

(i) If lim supx→x0 Γ (x) ⊂ Γ (x0 )+C(x0 ), then lim supx→x0 Ω(x) ⊂ Ω(x0 ). (ii) If Γ is closed at x0 and C(x0 ) \ {0} is open, then Ω is closed at x0 . (iii) If Γ is compact at x0 , then Ω is u.c. at x0 . Proof. (i) Let y ∈ lim supx→x0 Ω(x). By Proposition 2.5.3 (i), there exists a net ((xi , yi ))i∈I in gr Ω converging to (x0 , y) with xi = x0 for i ∈ I. Since gr Ω ⊂ gr Γ , from the hypothesis, we obtain that y ∈ Γ (x0 )+C(x0 ). Suppose that y ∈ / Ω(x0 ) = Eff (Γ (x0 ) + C(x0 ); C(x0 )); then there exists k ∈C(x0 ) \ {0} such that y
:= y − k ∈ Γ (x0 ). Since Γ (x0 ) ⊂ lim inf x→x0 (Γ (x) + C(x)), using again Proposition 2.5.3, there exist a subnet (xϕ(j) )j∈J of (xi ) and the nets (yj
), (kj ) with yj
∈ Γ (xϕ(j) ), kj ∈C(xϕ(j) ) for j ∈ J and (yj
+ kj ) → y
. Since (yϕ(j) − yj
− kj ) → y − y
= 0, we may suppose that yϕ(j) − yj
− kj = 0 for all j ∈ J. Taking into account that yϕ(j) ∈ Ω(xϕ(j) ), we have that yϕ(j) − yj
− kj ∈ C c (xϕ(j) ) for j ∈ J. Since lim supx→x0 C c (x) ⊂ C c (x0 ), we obtain that k ∈ C c (x0 ), a contradiction. Therefore y ∈ Ω(x0 ). (ii) If Γ is closed at x0 , then Γ (x0 ) is closed and lim supx→x0 Γ (x) ⊂ Γ (x0 ). From (i) we obtain that lim supx→x0 Ω(x) ⊂ Ω(x0 ). Since C(x0 ) \ {0} is open, Ω(x0 ) is closed. Using Proposition 2.5.10 we obtain that Ω is closed at x0 . (iii) Suppose that Ω is not u.c. at x0 . Then there exist an open set D0 ⊂ Y such that Ω(x0 ) ⊂ D0 and a net ((xi , yi ))i∈I ⊂ gr Ω (⊂ gr Γ ) with (xi ) → x0 and yi ∈ / D0 for every i ∈ I. Since Γ is compact at x0 , there exists a subnet (yϕ(j) )j∈J → y ∈ Γ (x0 ). Of course, y ∈ / D0 , and so y ∈ / Ω(x0 ). Hence there exists k ∈C(x0 ) \ {0} such that y
= y − k ∈ Γ (x0 ). Proceeding as in (i) we
obtain a contradiction. Therefore Ω is u.c. at x0 .

136

3 Optimization in partially ordered spaces

In the next example all the hypotheses of the preceding theorem are satisfied at x0 = −1, except Γ (x0 ) ⊂ lim inf x→x0 (Γ (x) + C(x)), and the conclusions of (i)–(iii) fail. Example 3.5.30. Let X = ] − ∞, 0[, Y = R2 , C(x) = {(0, 0)}∪ ]0, ∞[ × ]0, ∞[, and let Γ be defined by  {(y1 , y2 ) ∈ R2 | y2 ≥ xy1 , y1 ≤ 1, y2 ≤ 1} if x = −1, Γ (x) = {(y1 , y2 ) ∈ R2 | y2 ≥ −y1 − 1, y1 ≤ 1, y2 ≤ 1} if x = −1. Then

 Ω(x) =

if x = −1, {(y1 , y2 ) ∈ R2 | y2 = xy1 , x−1 ≤ y1 ≤ 1} {(y1 , y2 ) ∈ R2 | y2 = −y1 − 1, −2 ≤ y1 ≤ 1} if x = −1.

In the next example all the hypotheses of the preceding theorem are satisfied at x0 = 1, except lim supx→x0 C c (x) ⊂C c (x0 ), and the conclusions of (i)–(iii) fail. Example 3.5.31. ([356], modified) Let X = ]0, ∞[, Y = R2 , and Γ and C be defined by Γ (x) = {(y1 , y2 ) ∈ R2 | y12 + y22 ≤ 1},  {(0, 0)} ∪ {(y1 , y2 ) ∈ R2 | y2 > 0, y1 + y2 > 0} if x = 1, C(x) = {(0, 0)}∪ ]0, ∞[ × ]0, ∞[ if x = 1. Then



Ω(x) =

{(y1 , y2 ) ∈ R2 | y12 + y22 = 1, y1 ≤ 0, y1 − y2 ≥ 0} if x = 1, {(y1 , y2 ) ∈ R2 | y12 + y22 = 1, y1 ≤ 0, y2 ≤ 0} if x = 1.

In the next example condition (3.41) holds, but the hypotheses of (i) and (ii) fail at x0 = 0, as well as their conclusions. Example 3.5.32. ([356], modified) Let X = ]0, ∞[, Y = R2 , and let Γ and C be defined by Γ (x) = {(y1 , y2 ) ∈ R2 | y12 + y22 ≤ 1} \ {(−1, 0)}, C(x) = {(0, 0)} ∪ {(y1 , y2 ) ∈ R2 | y2 > 0, y1 + xy2 > 0}. Then

 Ω(x) =

{(y1 , y2 ) ∈ R2 | y12 + y22 = 1, y1 ≤ 0, y2 ≤ xy1 } if x = 0, {(y1 , y2 ) ∈ R2 | y12 + y22 = 1, y1 ≤ 0, y2 < 0} if x = 0.

Theorem 3.5.33. Consider (x0 , y0 ) ∈ gr Ω. Suppose that Γ + C is l.c. at (x0 , y0 ) and Γ (x) has (DP) w.r.t. C(x) for x ∈ U • , where U ∈ VX (x0 ). Then (i) y0 ∈ lim inf x→x0 (Ω(x) + C(x)).

3.5 Continuity properties

137

(ii) Moreover, assume that lim supx→x0 Γ (x) ⊂ Γ (x0 ) (or more generally, lim supx→x0 Ω(x) ⊂ Γ (x0 )+C(x0 )) and lim supx→x0 C(x) ⊂C(x0 ) and for all nets ((xi , yi ))i∈I ⊂ gr Γ , ((xi , zi ))i∈I ⊂ gr Ω with xi = x0 and zi ≤C(xi ) yi for i ∈ I, (zi ) contains a convergent subnet. Then Ω is l.c. at (x0 , y0 ). Proof. Let X \ {x0 } ⊃ (xi ) → x0 . We may suppose that (xi ) ⊂ U • . Since Γ +C is l.c. at (x0 , y0 ), there exist a subnet (xϕ(j) )j∈J and a net (yj +kj )j∈J → y0 with yj ∈ Γ (xϕ(j) ) and kj ∈ C(xϕ(j) ) for every j ∈ J. Since Γ (xϕ(j) ) has (DP) w.r.t. C(xϕ(j) ), there exist zj ∈ Ω(xϕ(j) ) and kj
∈C(xϕ(j) ) such that yj = zj + kj
. (i) It follows that yj + kj ∈ Ω(xϕ(j) )+C(xϕ(j) ) for every j ∈ J. Therefore y0 is in lim inf x→x0 (Ω(x) + C(x)). (ii) By hypothesis, there exists a subnet (zψ(l) )l∈L convergent to some z ∈
Y . It follows that z = z0 +k0 with z0 ∈ Γ (x0 ), k0 ∈ C(x0 ), and (kψ(l) +kψ(l) )→ y0 − z =: k ∈C(x0 ). Therefore y0 = z + k = z0 + k0 + k. Since y0 ∈ Ω(x0 ) and C(x0 ) is pointed, we obtain that z = z0 = y0 . Using Proposition 2.5.6, we have that Ω is l.c. at y0 .
Note that the last condition of the hypothesis of (ii) in the preceding theorem holds if Ω(U • ) is relatively compact for some U ∈ VX (x0 ). Taking C(x) = C and y0 ∈ Ω(x0 ) in Examples 3.5.9, 3.5.17, 3.5.8, and 3.5.22, we observe that the conditions of Theorem 3.5.33, Γ + C is l.c. at (x0 , y0 ), Γ (x) has (DP) w.r.t. C(x) for x ∈ U • , lim supx→x0 Γ (x) ⊂ Γ (x0 ), and the second hypothesis of (ii) are essential for obtaining (i) and the conclusion of (ii), respectively. In the following example all the hypotheses of Theorem 3.5.33 (ii) for x0 = 1 and y0 = (−1, 0) hold, except lim supx→x0 C(x) ⊂ C(x0 ); Ω is not l.c. at (x0 , y0 ). Example 3.5.34. Let X = ]0, ∞[, Y = R2 , and let the multifunctions Γ and C be defined by Γ (x) = {(y1 , y2 ) ∈ R2 | y12 + y22 ≤ 1},  {(y1 , y2 ) ∈ R2 | y2 ≥ 0, y1 + y2 ≥ 0} if x = 1, C(x) = R2+ if x = 1. Then Ω(x) =



{(y1 , y2 ) ∈ R2 | y12 + y22 = 1, y1 ≤ 0, y1 − y2 ≥ 0} if x = 1, {(y1 , y2 ) ∈ R2 | y12 + y22 = 1, y1 ≤ 0, y2 ≤ 0} if x = 1.

Before stating the next result we extend the notion of lower semicontinuous multifunction to the case of moving cones. So, if F : X × U ⇒ Y [G : U ⇒ Y ], we say that F [G] is C-lower semicontinuous (C-l.s.c. for short), where this time C : U ⇒ Y if the sets levF,C (y) := {(x, u) ∈ X ×U | F (x, u)∩(y − C(u)) = ∅} [levG,C (y) := {u ∈ U | G(u) ∩ (y − C(u)) = ∅}] are closed for every y ∈ Y . We have the following extension of Corollary 3.5.4.

138

3 Optimization in partially ordered spaces

Theorem 3.5.35. Let F : X × U ⇒ Y C-l.s.c. and Λ : U ⇒ X compact at every u ∈ U. If F (x, u)+C(u) is C(u)-complete for every u ∈ U and x ∈ Λ(u), then Γ (u) := F (Λ(u) × {u}) has the domination property w.r.t. C(u) for every u ∈ U, and the optimal multifunction Ω is C-l.s.c. Proof. Let us note first that Γ (u) has (DP) w.r.t. C(u) for every u ∈ U. The proof is the same as that of the corresponding part in the proof of Corollary 3.5.4; just replace G(x) by F (x, u) and C by C(u). Let y ∈ Y , and suppose that the net (ui ) ⊂ levΩ,C (y) converge to u ∈ U. Therefore, for every i ∈ I there exist xi ∈ Λ(ui ) and yi ∈ F (xi , ui ) such that yi ∈ y − C(ui ). It follows that ((xi , ui ))i∈I ⊂ levF,C (y). Since Λ is compact at u, there exists the subnet (xϕ(j) )j∈J of (xi )i∈I converging to x ∈ Λ(u). Since F is C-l.s.c. we get (x, u) ∈ levF,C (y), whence u ∈ levΓ,C (y). Since Γ (u) has (DP) w.r.t. C(u), we obtain that u ∈ levΓ,C (y). Therefore Ω is C-l.s.c.
Theorem 3.5.29 (i), (iii) was essentially obtained by Dolecki and Malivert [98], while (ii) was obtained by Penot and Sterna-Karwat [291]. From this theorem one obtains easily Theorem 2.1 of [245] (established for Y a normed vector space, X = N ∪ {∞}, x0 = ∞, and and C(x) \ {0} open) and Theorem 2.2 of [245] (established for Y a reflexive Banach space endowed with the weak topology and X = N ∪ {∞}, x0 = ∞). Theorem 3.5.33 (i) extends a result of Luc [244], while (ii) is obtained, essentially, in [291] for C a constant multifunction, in [98] for Ω compact at x0 , and in [245] for X = N ∪ {∞}, x0 = ∞. From this theorem one obtains easily Theorems 3.3 and 3.4 from [245]. Under the condition lim supx→x0 Ω(x) ⊂ Γ (x0 )+C(x0 ) with C(x0 ) = C a pointed closed convex cone, X = N ∪ {∞}, and x0 = ∞, Theorem 3.5.33 (ii) is obtained in [242, Th. 2.2]; note that [242, Th. 2.3] can be obtained from Theorem 3.5.33 (ii), too. The notion of lower semicontinuity w.r.t. moving cones is given by Ferro [115]. Theorem 3.5.35 was established by Ferro [115, Th. 4.1], too; in [115, Th. 4.2] a slightly different result is also stated. 3.5.3 Continuity Properties for the Solution Multifunction Consider U, X two topological spaces and Y a topological vector space. As in the preceding section, consider a multifunction C : U ⇒ Y whose values are pointed convex cones. With C we associate the multifunction C c : U ⇒ Y defined by C c (u) := (Y \ C(u))∪{0}. Take also Λ : U ⇒ X and f : X ×U → Y . For a fixed u0 ∈ U we consider the problem (P) C(u0 )-minimize f (x, u0 ) subject to x ∈ Λ(u0 ), and its perturbed problems (Pu ) C(u)-minimize f (x) subject to x ∈ Λ(u). Thus the initial problem is just (Pu0 ). We associate the optimal-value multifunction Ω : U ⇒ Y and the solution multifunction Σ : U ⇒ X defined, respectively, by

3.5 Continuity properties

Ω(u) := Eff (f (Λ(u) × {u}); C(u)) ,

139

Σ(u) := {x ∈ Λ(u) | f (x, u) ∈ Ω(u)}.

It is obvious that dom Σ = dom Ω. In the preceding section we obtained continuity properties of Ω in Corollaries 3.5.15, 3.5.24, and 3.5.28. In this section we are interested in continuity properties for the solution multifunction Σ. We begin with a closedness condition for Σw . Theorem 3.5.36. Suppose that f is continuous on Λ(u0 ) × {u0 }, Λ(u0 ) ⊂ lim inf Λ(u), and lim sup C c (u) ⊂ C c (u0 ). u→u0

(3.42)

u→u0

(i) If lim supu→u0 Λ(u) ⊂ Λ(u0 ), then lim supu→u0 Σ(u) ⊂ Σ(u0 ). (ii) If C(u0 ) \ {0} is open and Λ is closed at u0 , then Σ is closed at u0 . (iii) If C(u0 ) \ {0} is open and Λ is compact at u0 (Λ(u0 ) is compact and Λ is u.c. at u0 ), then Σ is compact at u0 (Σ is u.c. at u0 ). Proof. (i) Let x ∈ lim supu→u0 Σ(u). This means that there exists a net (ui , xi ) i∈I ⊂ gr Σ (⊂ gr Λ) converging to (u0 , x0 ), with ui = u0 for every i ∈ I. From the hypothesis we have that x0 ∈ Λ(u0 ). Suppose that x0 ∈ / Σ(u0 ); then there exists x ∈ Λ(u0 ) such that k := f (x0 , u0 ) − f (x, u0) ∈C(u0 ) \ {0}. Since x0 ∈ Λ(u0 ), from (3.42) there exist a subnet uϕ(j) j∈J of (ui ) and a net x
ϕ(j) j∈J converging to x. Since f is continu    ous on Λ(u0 ) × {u0 }, f x
ϕ(j) , uϕ(j) − f xϕ(j) , uϕ(j) → k = 0, so we may 
   suppose that f xϕ(j) , uϕ(j) − f xϕ(j) , uϕ(j) = 0 for every j ∈ J. Since     xϕ(j) ∈ Σ(uϕ(j) ), we have that f x
ϕ(j) , uϕ(j) − f xϕ(j) , uϕ(j) ∈C c (uϕ(j) ) for j ∈ J. From (3.42) we obtain that k ∈C c (u0 ), a contradiction. Therefore x0 ∈ Σ(u0 ). (ii) Using the fact that Λ(u0 ) is closed (Λ being closed at u0 ) and C(u0 )\{0} is open, one obtains easily that Σ(u0 ) is closed. The conclusion then follows from (i). (iii) If Λ is compact at u0 , the proof is similar to that of (ii). If Λ(u0 ) is compact and Λ is u.c. at u0 , then Λ is closed at u0 by Proposition 2.5.10 (iv). Noting that Σ(u) = Σ(u) ∩ Λ(u) for every u ∈ U, the upper continuity of Σ follows from (ii) and Proposition 2.5.11 (i).
Note that the preceding result can be stated for a multifunction F : X × U ⇒ Y instead of the function f ; the conclusions remain valid if we suppose that F is compact and l.c. at (x, u0 ) for every x ∈ Λ(u0 ). In the sequel in this section f (x, u) = g(x), where g : X → Y and C(u) = C for every u ∈ U. Theorem 3.5.37. Suppose that int C = ∅, g is continuous, Λ is u.c. at u0 , (P) is well-posed, and Ω is (−C)-H-u.c. at u0 . Then Σ is u.c. at u0 .

140

3 Optimization in partially ordered spaces

Proof. If Σ(u0 ) = ∅, then Ω(u0 ) = ∅. Since Ω is (−C)-H-u.c. at u0 , it follows that u0 ∈ int (U \ dom Ω) = int (U \ dom Σ), and so Σ is u.c. at u0 . Let Σ(u0 ) = ∅. Consider an open set D ⊂ Y such that Σ(u0 ) ⊂ D. Since (P) is well-posed, there exists W ∈ VY such that  {x ∈ Λ(u0 ) | g(x) ≤C η + ε} ⊂ D ∀ ε ∈ W ∩ C. η∈Ω(u0 )

Let L(A) :=( Ω(u0 ) + A − C for) A ⊂ Y . The above relation shows that Λ(u0 ) ⊂ D ∪ X \ g −1 (L(W ∩ C) . Fix ε ∈ W ∩ int C. There exists W1 ∈ VY with ε + W1 (⊂ C. It follows )that W1 ⊂ ε − C ⊂ (W ∩ C) − C, and so Λ(u0 ) ⊂ D ∪ X \ g −1 (L(W1 )) . Let W2 ∈ VY be such that W2 + W2 ⊂ W1 . Since cl L(W2 ) ⊂ W2 + L(W2 ) ⊂ L(W1 ), we have that ( ) Λ(u0 ) ⊂ D ∪ X \ g −1 (cl L(W2 )) =: D1 . Because g is continuous, g −1 (cl L(W2 )) is closed, and so D1 is open. Taking into account that Λ is u.c. at u0 , there exists U1 ∈ VU (u0 ) such that Λ(U1 ) ⊂ D1 . Since Ω is (−C)-H-u.c. at u0 , there exists U2 ∈ VU (u0 ) such that Ω(U2 ) ⊂ Ω(u0 ) + W2 − C = L(W2 ). Let u ∈ U1 ∩ U2 and x ∈ Σ(u) (⊂ Λ(u)). Then g(x) ∈ Ω(u) ⊂ L(W2 ), which means that x ∈ g −1 (L(W2 )) ⊂ g −1 (cl L(W2 )). It follows that x ∈ D. Therefore Σ(U1 ∩ U2 ) ⊂ D, which shows that Σ is u.c. at u0 .
Corollary 3.5.38. Suppose that int C =  ∅, g is continuous, Λ is u.c. at u0 , Ω(u0 ) is compact, (P) is η-well-posed, and Ω is (−C)-H-u.c. at u0 . Then Σ is u.c. at u0 . Proof. By Proposition 3.4.5 the problem (P) is well-posed. The conclusion follows from the preceding theorem.
The next result establishes sufficient conditions for Hausdorff upper continuity of Σ. Theorem 3.5.39. Suppose that X is a topological vector space, int C = ∅, g is uniformly continuous on X, Λ is H-u.c. at u0 , (P) is weakly well-posed, and Ω is (−C)-H-u.c. at u0 . Then Σ is H-u.c. at u0 . Proof. Let V ∈ VX be fixed and consider V1 ∈ VX such that V1 + V1 ⊂ V . Since (P) is weakly well-posed, there exists W ∈ VY such that  {x ∈ Λ(u0 ) | g(x) ≤C η + ε} ⊂ Σ(u0 ) + V1 ∀ ε ∈ W ∩ C; η∈Ω(u0 )

i.e., with the notation from the proof of the preceding theorem, Λ(u0 ) ∩ g −1 (L(W ∩ C)) ⊂ Σ(u0 ) + V1 .

3.6 Sensitivity of vector optimization problems

141

Taking ε, W1 , and W2 as in the proof of the preceding theorem, we obtain that ( ) Λ(u0 ) ⊂ (Σ(u0 ) + V1 ) ∪ X \ g −1 (L(W1 )) . (3.43) By the uniform continuity of g there exists V2 ∈ VX such that      g X \ g −1 (L(W1 )) + V2 ⊂ g X \ g −1 (L(W1 )) +W2 ⊂ (Y \ L(W1 ))+W2 . Since (Y \ L(W1 )) + W2 ⊂ Y \ L(W2 ), we obtain that   X \ g −1 (L(W1 )) + V2 ⊂ g −1 (Y \ L(W2 )) = X \ g −1 (L(W2 )) .

(3.44)

Since Λ is H-u.c. at u0 , there exists U1 ∈ VU (u0 ) such that Λ(U1 ) ⊂ Λ(u0 ) + V1 ∩ V2 . From (3.43) we have that ) (  Λ(u0 ) + V1 ∩ V2 ⊂ (Σ(u0 ) + V1 + V1 ∩ V2 ) ∪ X \ g −1 (L(W1 )) + V1 ∩ V2 . Taking into account (3.44) and that V1 + V1 ⊂ V , we get   Λ(u0 ) + V1 ∩ V2 ⊂ (Σ(u0 ) + V ) ∪ X \ g −1 (L(W2 )) . Therefore

  Λ(U1 ) ⊂ (Σ(u0 ) + V ) ∪ X \ g −1 (L(W2 )) .

(3.45)

Since Ω is (−C)-H-u.c. at u0 , there exists U2 ∈ VU (u0 ) such that Ω(U2 ) ⊂ Ω(u0 ) + W2 − C = L(W2 ). So, for u ∈ U1 ∩ U2 and x ∈ Σ(u) we have that g(x) ∈ L(W2 ). From (3.45) we obtain that x ∈ Σ(u0 ) + V . This shows that Σ(U1 ∩ U2 ) ⊂ Σ(u0 ) + V , and so Σ is u.c. at u0 .
Theorem 3.5.36 was stated, essentially, by Penot and Sterna-Karwat in [291] and Dolecki and Malivert [98]. Bednarczuk stated Theorems 3.5.37 and 3.5.39 and Corollary 3.5.38 in [18].

3.6 Sensitivity of Vector Optimization Problems Throughout this section X, Y are normed vector spaces and {0} = C ⊂ Y is a pointed closed convex cone; when we consider weak minimal points of a set we assume also that int C = ∅. Let D ⊂ Y ; recall that the efficient set of D w.r.t. C is Eff(D; C) = {y ∈ D | D ∩ (y − C) ⊂ y − C}, the set of Henig-proper efficient points of D w.r.t. C is  HEff(D; C) = {Eff(D; K) | Kconvex cone, C \ {0} ⊂ int K = Y }, the set of Benson-proper efficient points of D w.r.t. C is
 BEff(D; C) = y ∈ D | cone(D + C − y) ∩ (−C) = {0} ,

142

3 Optimization in partially ordered spaces

and the set of weakly efficient points of D w.r.t. C is the set w Eff(D; C) = Eff(D; C0 ), where C0 = {0} ∪ int C. It is obvious that SPEff(D; C) ⊂ HEff(D; C) ⊂ BEff(D; C) ⊂ Eff(D; C) ⊂ w Eff(D; C), the last inclusion if int C = ∅. Note that if C has a compact base, then SPEff(D; C) = HEff(D; C) = BEff(D; C). The first equality follows from Proposition 3.5.25. Let us prove the second one. Suppose that B is a compact base for C (i.e., B is compact, convex, 0 ∈ / B, and C = cone B), and y ∈ BEff(D; C). It follows that cone(D − y) ∩ (−B) = ∅, and so r := d(−B, cone(D − y)) > 0. Otherwise, there are (yn ) ⊂ cone(D − y) such that bn + yn → 0. There exists bnk → b ∈ B since B is compact, and so ynk → −b, which shows that cone(D − y) ∩ (−B) = ∅. Taking C := cone(B + 2r BY ), we have that C \ {0} ⊂ int C and (D − y) ∩ (−C) ⊂ cone(D − y) ∩ (−C) = {0}. Therefore y ∈ HEff(D; C). The equality of HEff(D; C) = BEff(D; C) in the case of finite-dimensional spaces is proven by Henig [161]. Consider the multivalued mapping Γ : X ⇒ Y and the following associated multivalued mappings: Ω, HΩ, Ωw : X ⇒ Y , given by Ω(x) := Eff(Γ (x); C),

HΩ(x) := HEff(Γ (x); C),

Ωw (x) := w Eff(F (u); C). In the sequel we shall also use the multivalued mapping ΓC defined by ΓC (u) := Γ (u) + C for u ∈ X. Of course, gr ΓC = gr Γ + {0} × C. Throughout this section we denote by DΓ (x, y)(u) the set DΓ (x, y)(u). Shi [319] introduced the following derivative of Γ at (x, y) ∈ gr Γ in the direction u: SΓ (x, y)(u) := {y ∈ Y | ∃ (tn ) ⊂ (0, ∞), ((xn , yn )) ⊂ gr Γ : xn → x, tn (xn − x, yn − y) → (u, y)}. It is obvious that for (x, y) ∈ gr Γ , gr DΓ (x, y) ⊂ gr SΓ (x, y) ⊂ cone (gr Γ − (x, y)) , with equality if gr Γ is a convex set, and SΓ (x, y)(u) = DΓ (x, y)(u)

∀ u ∈ X \ {0}.

In the sequel we shall use several times the following condition: SΓ (x, y)(0) ∩ (−C) = {0}.

(3.46)

Note that if (3.46) holds, then cone(Γ (x)− y) ∩ (−C) = {0}, while if C has a compact base, then y ∈ BEff (Γ (x); C), and therefore y ∈ HΩ(x). In the following theorem we give sufficient conditions for (3.46).

3.6 Sensitivity of vector optimization problems

143

Theorem 3.6.1. Let y ∈ Γ (x). Then each of the following conditions is sufficient for (3.46): (i) y ∈ BEff (Γ (x); C), Γ is upper Lipschitz at x; i.e., there exist L, ρ > 0 such that Γ (x) ⊂ Γ (x) + L x − x BY for all x ∈ B(x, ρ); (ii) y ∈ BEff (Γ (x); C), X, Y are finite-dimensional, x ∈ raint(dom F ), and Γ is C-convex; (iii) y ∈ HEff (Γ (x); C), X, Y are Banach spaces, x ∈ int(dom F ), and Γ is C-convex and C-closed (gr ΓC is closed). Proof. (i) Let y ∈ SΓ (x, y)(0) ∩ (−C). Then there exist (tn ) ⊂ (0, ∞), ((xn , yn )) ⊂ gr Γ such that xn → 0 and tn ((xn , yn ) − (x, y)) → (0, y). We may suppose that xn ∈ B(x, ρ) for every n ∈ N. It follows that for every n there exist y n ∈ Γ (x) and vn ∈ BY such that yn = y n + L xn − x · vn . Therefore tn (y n − y) − tn (yn − y) ≤ Ltn xn − x for every n ∈ N, whence tn (y n − y) → y. So y ∈ cone(Γ (x) − y) ∩ (−C) = {0}, a contradiction. (ii) Taking (x, y) = (0, 0) (replacing Γ by Γ − (x, y)) and replacing X by span(dom Γ ) if necessary, we may suppose that x ∈ int(dom Γ ). Since dim Y < ∞ and y ∈ BEff (Γ (x); C), we have that y ∈ HEff (Γ (x); C); therefore there exists a convex cone C such that C \ {0} ⊂ int C = Y and Γ (x) ∩ (−C) ⊂ C. Since gr ΓC = gr Γ + {0} × C + {0} × C, the set gr ΓC is convex. We have that (x, y) ∈ / raint gr ΓC ; in the contrary case for k ∈ C \ {0} there exists some λ > 0 such that (x, y) − λ(0, k) = (x, y − λk) ∈ gr Γ + {0} × C; i.e., −λk − c
∈ Γ (x) for some c
∈ C. This contradicts the hypothesis, since λk + c
∈ C \ {0} + C ⊂ int C. Using a separation theorem, there exists (u∗ , y ∗ ) ∈ X ∗ × Y ∗ \ {(0, 0)} such that x, u∗  + y + k, y ∗  ≤ x, u∗  + c + k, y ∗ 

∀ (x, y) ∈ gr Γ, c ∈ C.

(3.47)

If y ∗ = 0, then x, u∗  ≤ x, u∗  for every u ∈ dom Γ , and so u∗ = 0. Therefore y ∗ = 0. From (3.47) we obtain that c, y ∗  ≥ 0, whence c, y ∗  > 0 for every c ∈ int C ⊃ C \ {0}. Suppose that there exists y ∈ SΓ (x, y)(0) ∩ (−C) \ {0}. Then there are (tn ) ⊂ (0, ∞) and ((xn , yn )) ⊂ gr Γ such that xn → 0 and tn ((xn , yn ) − (x, y)) → (0, y). From (3.47) we obtain that tn (xn − x), u∗  + tn (yn − y), y ∗  ≥ 0 for every n, whence −y, y ∗  ≤ 0, a contradiction. (iii) Let C be a convex cone such that C \ {0} ⊂ int C = Y and y ∈ Eff(F (u); C). If we show that int gr ΓC = ∅, arguments like those in the proof of (ii) show that (3.46) holds. For this aim consider the relation R := {(y, x) | (x, y) ∈ gr ΓC }; R is closed, convex, and 0 ∈ int(Im R). Applying the Robinson–Ursescu theorem, we have that R(V ) is a neighborhood of x for every neighborhood V of y. Let c0 ∈ int C. There exists ρ > 0 such that B(c0 , ρ) ⊂ C. It follows that for some δ > 0 we have B(x, δ) ⊂ R (B(y, ρ/2)). Let us show that B(x, δ) × B(c0 + y, ρ/2) ⊂ gr ΓC . Indeed, let (u, y) belong to the first set. There exists y
∈ R−1 (u) such that y
− y ≤ ρ/2. It follows that y = y
+ (y − y
) and

c0 − (y − y
) ≤ (c0 + y − y) + y
− y ≤ ρ/2 + ρ/2 = ρ,

144

3 Optimization in partially ordered spaces

which shows that y − y
∈ B(c0 , ρ) ⊂ C. Therefore (x, y) ∈ R−1 + {0} × C = gr ΓC . The proof is complete.
Remark 3.6.2. Part (i) of the preceding theorem is proven in [319, Prop. 3.2]. The condition that Γ is upper Lipschitz at x in Theorem 3.6.1(i) is important, as the next example shows. Example 3.6.3. ([354]) Let X = Y = R, C = R+ , and let Γ : R ⇒ R be defined by  {0}√ if x ≤ 0, Γ (x) := {− x, 0} if x > 0. Then

 ΓC (x) =

[0,√ ∞[ if x ≤ 0, [− x, ∞[ if x > 0,

 Ω(x) =

{0}√ if x ≤ 0, {− x} if x > 0.

Take x = 0 and y = 0. Then  SΓ (x, y)(u) = DΓ (x, y)(u) =  DΓC (x, y)(u) =  Eff (DΓ (x, y)(u); C) =  Eff (DΓC (x, y)(u); C) =

C if u < 0, R if u ≥ 0,

{0} if u = 0, −C if u = 0,

⎧ ⎨ {0} if u < 0, DΩ(x, y)(u) = −C if u = 0, ⎩ ∅ if u > 0,

{0} if u = 0, ∅ if u = 0, {0} if u < 0, ∅ if u ≥ 0.

The condition y ∈ BEff (Γ (x); C) in the preceding theorem cannot be replaced by y ∈ Eff (Γ (x); C). In the next example Γ is upper Lipschitz at x ∈ int(dom Γ ), C-convex, and C-closed. Example 3.6.4. Let X = R, Y = R2 , C = R2+ , and Γ : X ⇒ Y be defined by Γ (x) := {(y1 , y2 ) | y2 ≥ y12 }. Take x = 0 and y = 0; then y ∈ Eff(Γ (x); C)\BEff(Γ (x); C). Moreover, SΓ (x, y)(u) = DΓ (x, y)(u) = R×R+ , and so SΓ (x, y)(u) ∩ (−C) = ] − ∞, 0] × {0} = {0}. If dim X = ∞ in Theorem 3.6.1(ii), the conclusion may be not true. Example 3.6.5. Consider X an infinite-dimensional normed vector space, Y := R, φ : X → R a noncontinuous linear functional, let Γ : X ⇒ Y be defined by Γ (x) := {φ(x)} and C := R+ . One obtains easily that SΓ (x, y)(u) = DΓ (x, y)(u) = R for all (x, y) ∈ gr Γ and u ∈ X. Theorem 3.6.6. Let (x, y) ∈ gr Γ . Then

3.6 Sensitivity of vector optimization problems

145

(i) for every u ∈ X we have DΓ (x, y)(u) + C ⊂ DΓC (x, y)(u). (ii) If the set {y ∈ C | y = 1} is compact, then for every u ∈ X, Eff (DΓC (x, y)(u); C) ⊂ Eff (DΓ (x, y)(u); C) ⊂ DΓ (x, y)(u).

(3.48)

Moreover, if Γ (x) is convex for x ∈ V , where V is some neighborhood of x, then Eff (DΓ (x, y)(u); C) = Eff (DΓC (x, y)(u); C)

∀ u ∈ X.

(3.49)

(iii) If the set {y ∈ C | y = 1} is compact and (3.46) holds, then DΓ (x, y)(u) + C = DΓC (x, y)(u)

∀ u ∈ X.

(3.50)

Proof. (i) Let y ∈ DΓ (x, y)(u) and k ∈ C. Then there exist (tn ) → 0+ and ((un , yn )) → (u, y) such that (x, y) + tn (un , yn ) ∈ gr Γ for every n ∈ N. It follows that (x, y) + tn (un , yn + k) ∈ gr ΓC and (un , yn + k) → (u, y + k). Therefore y + k ∈ DΓC (0, 0)(u). In the rest of the proof we suppose that Q := {y ∈ C | y = 1} is compact. Of course, C = cone Q. (ii) Let y ∈ Eff (DΓC (x, y)(u); C); there exists (tn ) → 0+ , (λn ) ⊂ [0, ∞), (qn ) ⊂ Q, ((un , yn )) ⊂ X × Y such that (un , yn + λn qn ) → (u, y) and (x, y) + tn (un , yn ) ∈ gr Γ for every n ∈ N. Since Q is compact, we may suppose that qn → q ∈ Q. Assume that there are a subsequence (λnk ) and γ > 0 such that λnk ≥ γ for every k ∈ N. Then (x, y)+tnk (unk , ynk +(λnk −γ)qnk ) ∈ gr ΓC and (unk , ynk +(λnk −γ)qnk ) → (u, y −λq). It follows that y −γq ∈ DΓC (x, y)(u)∩ (y − C) \ {y}, a contradiction. Therefore λn → 0, and so (un , yn ) → (u, y). It follows that y ∈ DΓ (x, y)(u). Using also (i), we have that for every u ∈ X, Eff (DΓC (x, y)(u); C) ⊂ DΓ (x, y)(u) ⊂ DΓC (x, y)(u). From the above inclusions we obtain immediately that (3.48) holds. Assume, furthermore, that Γ (x) is convex near x and let y be an element of Eff (DΓ (x, y)(u); C) (⊂ DΓ (x, y)(u) ⊂ DΓC (x, y)(u)). Suppose that y ∈ / Eff (DΓC (x, y)(u); C). Then there exists k ∈ C \ {0} such that y − k ∈ DΓC (x, y)(u). It follows that there exists ((tn , un , yn ))n∈N → (0+ , u, y − k) such that y + tn yn ∈ Γ (x + tn un ) + C for every n ∈ N; i.e., y + tn (yn − λn qn ) ∈ Γ (x+tn un ) for every n, with (qn ) ⊂ Q, (λn ) ⊂ [0, ∞[. There exists n0 ∈ N such that Γ (x+tn un ) is convex for every n ≥ n0 . Suppose that lim sup λn > 0. Then there exist (np )p∈N ⊂ N an increasing sequence and α > 0 such  λ np ≥ α  that for every p ∈ N. Since Q is compact, we may suppose that qnp → q ∈ Q. Since Γ (x + tnp unp ) is convex, we have that    α  α y + tnp y np − q np = y + tnp (ynp − λnp qnp ) λ np λ np   α (y + tnp ynp ) ∈ Γ (x + tnp unp ) + 1− λ np

146

3 Optimization in partially ordered spaces

  for every p. Since ynp − λαn qnp → y − k − αq, we obtain that y − (k + p αq) ∈ DΓ (x, y)(u), with k + αq ∈ C \ {0}, contradicting the choice of y. Hence (λn ) → 0, whence the contradiction y − k ∈ DΓ (x, y)(u). Therefore y ∈ Eff (DΓC (x, y)(u); C). It follows that (3.49) holds. (iii) The inclusion ⊃ being proven in (i), let us prove the converse one. Of course, we assume that (3.46) holds. Let y ∈ DΓC (x, y)(u); there exist (tn ) → 0+ , (λn ) ⊂ [0, ∞), (qn ) ⊂ Q, ((un , yn )) ⊂ X × Y such that (un , yn + λn qn ) → (u, y) and (x, y)+tn (un , yn ) ∈ gr Γ for every n ∈ N. We may suppose that qn → q ∈ Q. Taking also a subsequence if necessary, we may assume that λn → λ ∈ [0, ∞]. Suppose that λ = ∞. Since yn + λn qn → y, we obtain that λ−1 n yn → −q. Of course, x + tn un → x and   1 u n yn → (0, −q). ((x + tn un , y + tn yn ) − (x, y)) = , λ n tn λn λn It follows that −q ∈ SΓ (x, y)(0) ∩ (−C) \ {0}, a contradiction. Therefore λ < ∞, whence (un , yn ) → (u, y − λq). Hence y ∈ DΓ (x, y)(u) + C. The proof is complete.
Remark 3.6.7. (i) and the first part of (ii) of Theorem 3.6.6 are proved by Tanino [354, Prop. 2.1, Th. 2.1] (see also [212, Th. 3.4]), while part (iii) is proved by Shi [319, Prop. 3.1]. The next example shows that the hypothesis “{y ∈ C | y = 1} is compact” in the previous theorem is essential for having (3.48), (3.49), and (3.50). Example 3.6.8. Let X = R, Y = 2 , C = 2+ = {(xn )n∈N ∈ 2 | xn ≥ 0 ∀ n ∈ N}, and Γ : X ⇒ Y be defined by  {0} if x ∈ ] − ∞, 0] ∪ [1, ∞[, Γ (x) := 1 1 [−a − e , a] if x ∈ [ n+1 , n1 [, n ∈ N∗ , n n where a ∈ 2+ \ {0} and (en )n∈N is the canonical base of 2 . Then  C if x ∈ ] − ∞, 0] ∪ [1, ∞[, ΓC (x) : = 1 − n1 (a + en ) + C if x ∈ [ n+1 , n1 [, n ∈ N∗ ,  {0} if x ∈ ] − ∞, 0] ∪ [1, ∞[, Ω(x) : = 1 − n1 (a + en ) if x ∈ [ n+1 , n1 [, n ∈ N∗ . Take x = 0 and y = 0. Then



DΓ (x, y)(u) =  DΓC (x, y)(u) =  DΩ(x, y)(u) =

{0} if u ≤ 0, {ua} if u > 0, C if u ≤ 0, −ua + C if u > 0, {0} if u ≤ 0, ∅ if u > 0.

3.6 Sensitivity of vector optimization problems

147

So (3.48), (3.49), and (3.50) do not hold for u > 0. Let us prove the formulae for DΓ (x, y)(u) and DΓC (x, y)(u) for u ≥ 0, those for u < 0 being obvious. So,let u ≥ 0 and v ∈ DΓ (x, y)(u). Then there exist (tn ) ⊂ ]0, ∞[ and gr Γ ⊃ (xn , yn ) → (0, 0) such that (tn xn ) → u and (tn yn ) → v. If P := {n ∈ N | xn ≤ 0} is infinite, then yn = 0 for every n ∈ P , and so (u, v) = (0, 0). Suppose that P is finite (this is the case when u > 0). Since (xn ) → 0, there exists n0 ∈ N∗ such that xn ∈ ]0, 1[ for n ≥ n0 . For every n ≥ n0 there exists pn ∈ N∗ such that xn ∈ [ pn1+1 , p1n [; of course, (p  n ) → ∞. Then for  every n ≥ n0 there   exists λn ∈ [0, 1] such that yn = p1n (1 − 2λn )a − λn en . It follows that ptnn → u and ptnn ((1 − 2λn )a −  λn epn ) → v. If u = 0, it is obvious that v = 0. Suppose that u > 0. Taking eventually a subsequence, we may assume that (λn ) → λ ∈ [0, 1]. We obtain that (λn epn ) → (1 − 2λ)a − u−1 v. If λ = 0, we get the contradiction that (en ) contains a norm-convergent subsequence. Therefore (λn ) → 0, whence v = ua. It is easy to get that ua ∈ DΓ (x, y)(u) for u ≥ 0 (in fact, similar to the proof in the next paragraph). So the formula for DΓ (x, y)(u) holds. 1 Consider first u > 0 and v = −ua + k with k ∈ C. Taking xn = n+1 , yn = 1 1 −1 −1 k) = n (−a − en+ u k + n (−a + u  en ) (∈ ΓC (xn )),  and tn = (n + 1)u > 0 for n ∈ N∗ , we have that (xn , yn ) → (0, 0) and tn (xn , yn ) → (u, v); if u = 0 and v = k ∈ k just take xn = 0, yn = n1 k, and tn = n. Therefore v ∈ DΓC (x, y)(u). Let u ≥ 0 and exist (tn ) ⊂ ]0, ∞[ and  v ∈ DΓC (x, y)(u).  Then there  gr ΓC ⊃ (xn , yn ) → (0, 0) such that tn (xn , yn ) → (u, v). As in the first part we may suppose that xn ∈ ]0, 1[ for n ≥ n0 . So there exists N∗ ⊃ (pn ) → ∞ with xn ∈ [ pn1+1 , p1n [ for n ≥ n0 . Of course, yn = − p1n (a + epn ) + kn with     kn ∈ C. It follows that ptnn → u and − ptnn (a + epn ) + tn kn → v. Since 0 = w-lim en , it follows that v+ua = w-lim(tn kn ). Because C is weakly closed, v + ua ∈ C, whence v ∈ −ua + C. The formula for DΩ(x, y)(u) is obtained similarly. In Example 3.6.3, (3.46) is not satisfied and Γ (x) is not convex for x > 0; (3.50) and (3.49) do not hold for u > 0. Corollary 3.6.9. Assume that the conditions of Theorem 3.6.6 (iii) hold. Then the formulae (3.49) and (3.51) below hold: HEff (DΓ (x, y)(u); C) = HEff (DΓC (x, y)(u); C)

∀ u ∈ X.

(3.51)

Moreover, if int C = ∅ (so dim Y < ∞), then w Eff (DΓ (x, y)(u); C) ⊂ w Eff (DΓC (x, y)(u); C)   w Eff (DΓ (x, y)(u); C) = w Eff DΓC
(x, y)(u); C  is a closed convex cone such that C  ⊂ {0} ∪ int C. where C

∀ u ∈ X,

(3.52)

∀ u ∈ X,

(3.53)

148

3 Optimization in partially ordered spaces

Proof. Using the preceding theorem, equality (3.50) holds. The conclusion then follows by using Proposition 3.2.15 (iv) for the pair (C, K) (there) replaced by the pairs (C, C) and (C, K), respectively, where the last K is from the definition of Henson proper efficiency. Suppose now that int C = ∅. Then (3.54) follows immediately from (3.50).  be a closed convex cone with C  ⊂ {0} ∪ int C. It is obvious that Let C  C satisfies the conditions of Theorem 3.6.6 (iii), and so (3.50) holds with C  Applying again Proposition 3.2.15 (iv), now for (C, K) replaced replaced by C.  by (C, C0 ) (with C0 = {0} ∪ int C), we obtain (3.53), too.
Remark 3.6.10. When X, Y are finite-dimensional, formulae (3.49), (3.51), (3.54), and (3.53) are established in [220, Th. 2.1] under the conditions of Corollary 3.6.9, while formulae (3.51) and (3.53) are established in [221, Th. 2.1] for x ∈ int(dom Γ ) and Γ locally C-convex near x; i.e., Γ |V is C-convex for some neighborhood V of x [see also Theorem 3.6.1 (ii)]. In Example 3.6.3, (3.46) is not satisfied and (3.51), (3.52), (3.53) (for  = C) do not hold for u > 0. C Theorem 3.6.11. Let (x, y) ∈ gr Γ . Assume that {y ∈ C | y = 1} is compact and Γ is C-dominated by Ω near x, i.e., Γ (u) ⊂ Ω(u) + C for every u ∈ V for some neighborhood V of x. (i) If y ∈ Ω(x), then for every u ∈ X, Eff (DΓC (x, y)(u); C) ⊂ Eff (DΩ(x, y)(u); C) ⊂ DΩ(x, y)(u).

(3.54)

Moreover, if Γ (x) is convex for x ∈ V , with V a neighborhood of x, then Eff (DΓ (x, y)(u); C) ⊂ Eff (DΩ(x, y)(u); C)

∀ u ∈ X.

(3.55)

(ii) If (3.46) holds, then y ∈ Ω(x) and for every u ∈ X, Eff (DΓ (x, y)(u); C) = Eff (DΩ(x, y)(u); C) ⊂ DΩ(x, y)(u),

(3.56)

HEff (DΓ (x, y)(u); C) = HEff (DΩ(x, y)(u); C) ⊂ DΩ(x, y)(u). (3.57) Proof. By hypothesis we have that ΓC (u) = ΩC (u) for every u ∈ V , where V is a neighborhood of x. It follows that DΓC (x, y)(u) = DΩC (x, y)(u) for every u ∈ X. (i) Using Theorem 3.6.6 (ii), we obtain for every u ∈ X, Eff (DΓC (x, y)(u); C) = Eff (DΩC (x, y)(u); C) ⊂ Eff (DΩ(x, y)(u); C) ⊂ DΩ(x, y)(u). If Γ (x) is convex for x ∈ V , with V a neighborhood of x, by Theorem 3.6.6 (ii), (3.49) holds, whence the inclusion (3.55) follows immediately.

3.6 Sensitivity of vector optimization problems

149

(ii) Assume that (3.46) holds. We already remarked that y is an element of BEff(Γ (x); C), and so y ∈ Ω(x). Since (3.46) holds for Γ , it holds for Ω, too. It follows that equality (3.50) and the corresponding one for Ω hold. The conclusion is immediate from Proposition 3.2.15 (iv), taking K = C for (3.56) and (C, K) for (3.57), where K is the cone in the definition of Henig-proper efficient point.
In Example 3.6.8 the set {y ∈ C | y = 1} is not compact, but all the conditions on Γ in Theorem 3.6.11 hold, although the relations (3.54), (3.55), (3.56), and (3.57) do not hold for u > 0. In Example 3.6.3, (3.46) is not satisfied and Γ (x) is not convex for x > 0 (but Γ (x) is C-dominated for every x); (3.55), (3.56), and (3.57) do not hold for u > 0. In the next example Γ (x) is not C-dominated for x < 0, the other conditions of Theorem 3.6.11 being satisfied; (3.55), (3.56), and (3.57) do not hold for u < 0. Example 3.6.12. ([221]) Let X = R, Y = R2 , C = R2+ , and Γ : R ⇒ R2 be defined by  {(y1 , y2 ) | y1 ≥ 0, y2 ≥ y12 } if x ≥ 0, Γ (x) := {(y1 , y2 ) | y1 > 0, y2 ≥ y12 } if x < 0. 

Then Ω(x) :=

{(0, 0)} if x ≥ 0, ∅ if x < 0.

Take x = 0, y = (0, 0). Then DΓ (x, y)(u) = DΓC (x, y)(u) = C,

DΩ(x, y)(u) = Ω(u)

∀ u ∈ X.

Corollary 3.6.13. Suppose that y ∈ BEff (Γ (x); C) and Γ is C-dominated by Ω near x. Then (3.56) and (3.57) hold if one of the following conditions is satisfied: (i) Γ is upper Lipschitz at x and {y ∈ C | y = 1} is compact; (ii) X, Y are finite-dimensional, x ∈ int(dom Γ ), and Γ is C-convex near x. Proof. Applying Theorem 3.6.1, condition (3.46) holds if (i) or (ii) is satisfied. The conclusion follows by applying now part (ii) of the preceding theorem.
Theorem 3.6.14. Let (x, y) ∈ gr Γ . Assume that (3.46) holds, dim Y < ∞,  int C = ∅, and Γ is C-dominated by Ωw near x for some closed convex cone   C with C ⊂ {0} ∪ int C. Then for every u ∈ X, w Eff (DΓ (x, y)(u); C) = w Eff (DΩw (x, y)(u); C) ⊂ DΩw (x, y)(u).

(3.58)

Proof. Since (3.46) holds, we have that y ∈ Ω(x) ⊂ Ωw (x). Since the conditions of Corollary 3.6.9 hold, equality (3.53) holds, too. Since gr Ωw ⊂ gr Γ , condition (3.46) is satisfied with Γ replaced by Ωw , whence (3.53) holds

150

3 Optimization in partially ordered spaces

 for the same substitution. Because Γ is C-dominated by Ωw near x we have that ΓC
(u) = (Ωw )C
(u) for every u in a neighborhood of x, whence DΓC
(x, y) = D(Ωw )C
(x, y). The conclusion is obtained immediately by using  C0 ).
again Proposition 3.2.15 (iv) for the pair of cones (C, In Examples 3.6.3 and 3.6.12 relation (3.58) does not hold; in Example  3.6.3, (3.46) is not satisfied, while in Example 3.6.12, Γ (x) is not C-dominated   by Ωw (x) for x < 0 for some closed convex cone C with C ⊂ {0} ∪ int C. Corollary 3.6.15. Suppose that X, Y are finite-dimensional, x ∈ int(dom F ),  y ∈ BEff (Γ (x); C), and Γ is C-convex near x. If Γ is C-dominated by Ωw   near x for some closed convex cone C with C ⊂ {0} ∪ int C, then (3.58) holds. Proof. Applying Theorem 3.6.1, condition (3.46) holds. The conclusion follows from the preceding theorem.
Besides Examples 3.6.3 and 3.6.12 mentioned above, we consider the following example where Γ is not C-convex (near x); relation (3.58) does not hold for every u ∈ X. Example 3.6.16. ([320]) Let X = Y = R, C = R+ , and let Γ : R ⇒ R be defined by *  [−* |x| , ∞[ ∪{− |x|} if |x| ≤ 1, Γ (x) := [− |x|, ∞[ if |x| > 1. Then

* ΓC (x) = [− |x|, ∞[,

* Ω(x) = {− |x|}

∀ x ∈ X.

Take x = 0, y = 0. Then DΓ (x, y)(u) = [− |u| , ∞[, DΓC (x, y)(u) = X,  ] − ∞, 0] if u = 0, DΩ(x, y)(u) = ∅ if u = 0.

∀ u ∈ X,

Corollary 3.6.17. Suppose that Y is finite-dimensional, y ∈ BEff (Γ (x); C), Γ (u) is C-closed for all u in a neighborhood of x, and one of the following two conditions is satisfied: (i) (3.46) holds and Γ (u) is C-bounded (i.e., (Γ (u))∞ ∩ (−C) = {0}) for all u in a neighborhood of x; (ii) X is finite-dimensional, x ∈ int(dom Γ ), and Γ is C-convex and Cdominated by Ω near x. Then for every u ∈ X, Eff (DΓ (x, y)(u); C) = Eff (D(HΩ)(x, y)(u); C) ⊂ D(HΩ)(x, y)(u).

3.6 Sensitivity of vector optimization problems

151

Proof. We have that (ii) ⇒ (i). Indeed, condition (3.46) holds by Theorem 3.6.1. Since Γ is C-dominated by Ω near x ∈ int(dom F ), we have that Ω(u) = ∅ for all u in a neighborhood V of x; we may suppose that Γ (u) is also Cclosed and C-convex for u ∈ V . By a known result it follows that Γ (u) is C-bounded for u ∈ V . Suppose that (i) holds. Let V be a neighborhood of x such that Γ (u) is C-dominated by Ω(u) and C-bounded for every u ∈ V . By [161, Th. 5.1] we have that HΩ(u) ⊂ Ω(u) ⊂ cl (HΩ(u)) ∀ u ∈ V. It follows that gr HΩ|V ⊂ gr Ω|V ⊂ cl (gr HΩ|V ), and so DΩ(x, y) =
D(HΩ)(x, y). The conclusion follows by applying Theorem 3.6.11 (ii). In Example 3.6.16, SΓ (x, y)(0) = X. So (3.46) does not hold and Γ is not C-convex near x; the conclusion of the preceding corollary does not hold. Remark 3.6.18. Formula (3.54) of Theorem 3.6.11 is proved by Tanino [354, Th. 3.1] for Y finite-dimensional and by Klose [212, Th. 3.4] for arbitrary Y ; for X and Y finite-dimensional spaces, formula (3.55) is proved by Shi [320, Th. 4.1], Theorem 3.6.11 (ii) is proved by Shi [319, Th. 4.1], and Theorem 3.6.14 is proved by Kuk, Tanino, and Tanaka in [220, Th. 3.1] in the same conditions. Corollary 3.6.13 (i) was obtained by Tanino [354, Th. 3.2] for Y finite-dimensional, Shi [319, Cor. 4.1], and Klose [212, Th. 3.5], while part (ii) by Shi [320, Th. 4.2]. Corollary 3.6.15 was obtained by Shi in [320, Th. 5.2]. Formula (3.57) was obtained in [221, Th. 3.1] under the conditions of Corollary 3.6.13 (ii). Corollary 3.6.17 (i) and (ii) reinforce [220, Th. 3.2] and [221, Cor. 3.1] of Kuk, Tanino, and Tanaka. Theorem 3.6.19. Suppose that int C = ∅ and consider y ∈ Ωw (x). If one of the following conditions holds, (i) Γ is semidifferentiable at (x, y); (ii) Γ is C-convex and x ∈ int(dom Γ ); (iii) X, Y are finite-dimensional, Γ is C-convex and x ∈ raint(dom Γ ), then DΩw (x, y)(u) ⊂ w Eff (DΓ (x, y)(u); C)

∀ u ∈ X.

(3.59)

Suppose that either (a) (3.46) and (i) hold or (b) y ∈ BEff (Γ (x); C) and  (iii) hold. If {y ∈ C | y = 1} is compact and Γ is C-dominated by Ωw or Ω  near x, where C ⊂ {0} ∪ int C is a closed convex cone, then DΩw (x, y)(u) = w Eff (DΓ (x, y)(u); C)

∀u ∈ X

(3.60)

DΩ(x, y)(u) = w Eff (DΓ (x, y)(u); C)

∀ u ∈ X,

(3.61)

or respectively.

152

3 Optimization in partially ordered spaces

Proof. Assume that (i) holds and consider y ∈ DΩw (x, y)(u). If y does not belong to w Eff (DΓ (x, y)(u); C), there exists k ∈ int C such that y − k ∈ DΩw (x, y)(u). Therefore there are (tn ) → 0+ and ((un , yn )) → (u, y − k) such that (x, y) + tn (un , yn ) ∈ gr Ωw ⊂ gr Γ . Since Γ is semidifferentiable at (x, y) and (tn ) → 0+ , un → u, there exists (yn
) → y − k such that (x, y) + tn (un , yn
) ∈ gr Γ for every n ≥ n0 . Therefore y + tn yn
∈ Γ (x + tn un ) and y + tn yn ∈ Ωw (x + tn un ) ⊂ Γ (x + tn un ). Since   (y + tn yn
) − (y + tn yn ) /tn = yn
− yn → y − k − y = −k ∈ − int C, there exists some n1 ≥ n0 such that (y + tn yn
) − (y + tn yn ) ∈ − int C for every n ≥ n1 , contradicting the fact that y + tn yn ∈ Ωw (x + tn un ). Assume now that (ii) holds; since int C = ∅, (x, y, k) ∈ int(gr ΓC ) for k ∈ int C. By Theorem 2.7.6 (i) ΓC is semidifferentiable at (x, y). Taking y ∈ DΩw (x, y)(u), with the same argument as in the proof of (i), we obtain the same sequences, the sole difference being that y + tn yn
∈ Γ (x + tn un ) + C for n ≥ n0 . The same contradiction is obtained. If (iii) holds, by Theorem 2.7.6 (ii), ΓC is semidifferentiable at (x, y). The proof is the same as for (ii). (b) ⇒ (a). Indeed, if (iii) holds, using Theorem 3.6.1 (ii) we get that (3.46) is satisfied.  Assume that (a) is satisfied and Γ (u) is C-dominated by Ωw (u) for all  = Ωw (u) + C  u ∈ V , for some neighborhood V of x. It follows that Γ (u) + C  for u ∈ V . So, DΓC
(x, y) = D(Ωw )C
(x, y). Applying Theorem 3.6.6 for (Γ, C)  we obtain that DΓ (x, y)(u) + C  = DΩw (x, y)(u) + C  for every and (Ωw , C)  u ∈ X. Applying now Proposition 3.2.15 (iv) for the pair (C, C0 ) we get for every u ∈ X, w Eff (DΓ (x, y)(u); C) = w Eff (DΩw (x, y)(u); C) ⊂ DΩw (x, y)(u). From (3.59) and the above relation (3.60) follows.  If Γ is C-dominated by Ω near x, all we said above is true if we replace Ωw by Ω. Therefore (3.61) holds, too. Note that y ∈ Ω(x) because (3.46) holds.
In Example 3.6.3 Γ is neither semidifferentiable at x, nor C-convex; relation (3.59) does not hold for u = 0. In Example 3.6.12 (a) and (b) are satisfied  but Γ is not C-dominated by Ωw or Ω near x, for some closed convex cone  C ⊂ {0} ∪ int C; relations (3.60) and (3.61) do not hold for u < 0. In the next example neither (a) nor (b) is satisfied; relations (3.60) and (3.61) do not hold for u > 0. Example 3.6.20. ([354]) Let X = R, Y = R2 , C = R2+ , and let Γ : R ⇒ R2 be defined by Γ (x) := {(y1 , −y1 ) | y1 ≤ x} ∪ {(y1 , −1 − y1 ) | y1 > 0}.

3.6 Sensitivity of vector optimization problems

153

Then Ω(x) = Ωw (x) = {(y1 , −y1 ) | y1 ≤ min(0, x)} ∪ {(y1 , −1 − y1 ) | y1 > 0}. Let x = 0 and y = (0, 0). Then DΓ (x, y)(u) = Eff (DΓ (x, y)(u); C) = {(v1 , −v1 ) | v1 ≤ u}, DΩ(x, y)(u) = Eff (DΓ (x, y)(u); C) = {(v1 , −v1 ) | v1 ≤ min(0, u)}. Remark 3.6.21. For Ωw replaced by Ω and X, Y finite-dimensional, (3.59) was obtained by Shi [320, Th. 5.1] under condition (iii) and by Kuk, Tanino, and Tanaka [220, Th. 3.3] under (i). Kuk, Tanino, and Tanaka obtained (3.59) under condition (iii) and (3.60) under condition (b) in [221, Th. 3.2] and (3.61) under condition (b) in [221, Th. 3.3]. Consider now another normed vector space U, the multifunction Λ : U ⇒ X, the function f : X × U → Y , and the multifunction Γ : U ⇒ Y defined by Γ (u) := f (Λ(u) × {u}). Theorem 3.6.22. Suppose that the following conditions hold: (i) Λ is upper Lipschitz at u ∈ dom Λ and Λ(u) is compact for u ∈ U , with U a neighborhood of u; (ii) x ∈ Λ(u) and y = f (x, u) ∈ BEff(Γ (u); C); (iii) X is finite-dimensional, C has a compact base, f is Lipschitz on bounded sets and Fr´echet differentiable at (x, u); (iv) Λ(u) = {x} or the multifunction Λ : U × Y ⇒ X,

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

 y) = {x}. is upper Lipschitz at (u, y) and Λ(u, Then for every u ∈ X, Eff (∇x f (x, u) (DΛ(u, x)(u)) + ∇u f (x, u)(u); C) = Eff (DΩ(u, y)(u); C) . In order to prove this result we need the following lemma. Lemma 3.6.23. Let Λ, f , Γ be as above and u ∈ dom Λ. If f is Lipschitz on bounded sets, Λ(u) is bounded, and Λ is upper Lipschitz at u, then Γ is upper Lipschitz at u. Proof. Consider the box norm on U × X. Since Λ is upper Lipschitz at u, there exist L, ρ > 0 such that Λ(u) ⊂ Λ(u) + L u − u BX ∀ u ∈ B(u, ρ). (3.62) ! It follows that the set A := u∈B(u,ρ) (Λ(u) × {u}) is bounded. From hypothesis, there exists L
> 0 such that f is L
-Lipschitz on A. Let u ∈ B(u, ρ)

154

3 Optimization in partially ordered spaces

and y ∈ Γ (u). Hence there exists x ∈ Λ(u) such that y = f (x, u). From (3.62), there exists x
∈ Λ(u) such that x − x
≤ L u − u . Taking y
= f (x
, u) ∈ Γ (u), we obtain that

y − y
= f (x, u) − f (x
, u) ≤ L
(x − x
, u − u) ≤ L
max(L, 1) u − u . Taking L

> L
max(L, 1), we have that Γ (u) ⊂ Γ (u) + L

u − u BY for every u ∈ B(u, ρ). Therefore Γ is upper Lipschitz at u.
Of course, a sufficient condition for f to be Lipschitz on bounded sets is that f be of class C 1 and U and X have finite dimension. Proof of Theorem 3.6.22. From (i) and (iii), using Lemma 3.6.23, we obtain that Γ is upper Lipschitz at u. Since y ∈ BEff(Γ (u); C), from Theorem 3.6.1(i) we obtain that (3.46) holds for Γ at (u, y). Since Λ(u) is compact for u ∈ U and f is continuous, Γ (u) is compact for u ∈ U . Therefore Γ (u) ⊂ Ω(u) + C for u ∈ U . Now using Theorem 3.6.11(ii), we have that Eff (DΓ (u, y)(u); C) = Eff (DΩ(u, y)(u); C)

∀ u ∈ U.

But from the second part of Theorem 2.7.8, we have that DΓ (u, y)(u) = ∇x f (x, u) (DΛ(u, x)(u)) + ∇u f (x, u)(u)

∀ u ∈ U.

The conclusion follows.
Note that the preceding theorem (with a slightly weaker conclusion) is obtained by Tanino in [354, Th. 4.1] for U, X, Y finite-dimensional and f of class C 1 and by Klose [212, Th. 4.4] (observe that the compactness of Λ(u) for u ∈ U was used for having that Γ (u) ⊂ Ω(u) + C, while the Lipschitz condition of f on bounded sets for having that Γ is upper Lipschitz at u, conditions written directly in [212]). Variants of Lemma 3.6.23 are stated and proved in [354, Lemma 4.1] and [212, Lemmas 4.2, 4.3].

3.7 Duality It is an old idea to try to complement a given optimization problem (f (x) → min with minimal value I) by a dual problem (g(y) → max with supremal value S, S ≤ I); remember the dual variational principles of Dirichlet and Thompson (see Zeidler [391]) or, e.g., the paper of K.O. Friedrichs [121]. There are at least three reasons to look for useful dual problems: •

•

The dual problem has (under additional conditions) the same optimal value as the given “primal” optimization problem, but solving the dual problem could be done with other methods of analysis or numerical mathematics. An approximate solution of the given minimization problem gives an estimation of the minimal value I from above, whereas an approximate solution of the dual problem is an estimation of I from below, so that one gets intervals that contain I.

3.7 Duality

•

155

Recalling the Lagrange method, saddle points, equilibrium points of twoperson games, shadow prices in economics, perturbation methods or dual variational principles, it becomes clear that optimal dual variables often have a special meaning for the given problem.

Of course, the advantages just listed require a skillfully chosen dual program. Nevertheless, the mentioned points are motivation enough, to look for dual problems in multicriteria optimization too. There are many papers dedicated to that aim, as well as many survey papers (see Jahn [195], Luc [244]). So we shall concentrate firstly upon nonconvex multicriteria problems and derive theoretical duality assertions. These theoretical results will be used in order to derive duality assertions on the basis of the special structure of multicriteria approximation problems. There are different approaches to duality: Conjugation: Sch¨ onfeld [318], Breckner [43, 44], Zowe [394, 395], Nehse [268], Rosinger [311], Tanino and Sawaragi [355], Brumelle [50], Kawasaki [207, 208], Gerstewitz and G¨ opfert [128], Malivert [248], Sawaragi, Nakayama and Tanino [314], Luc [244], Z˘ alinescu [386]. Lagrangian: Corley [83, 84], Bitran [31], Nehse [268], Gerstewitz and Iwanow [129], G¨ opfert and Gerth [138], Jahn [190, 193, 195], Iwanow and Nehse [188], Nakayama [265, 266], Sawaragi, Nakayama and Tanino [314], Luc [244], Dolecki and Malivert [99]. Axiomatic Duality: Luc [244]. 3.7.1 Duality Without Scalarization Let (Y, C) be a (with the cone C) partially ordered linear topological space, P, D nonempty subsets of Y , and let us consider the following multicriteria problems using the solution concepts introduced in Remark 2.1.3 (P )

Eff Min (P, C),

(D)

Eff Max (D, C).

We speak of a pair of weakly dual problems, if P ∩ (D − (C \ {0})) = ∅.

(3.63)

Since C is pointed, this is equivalent to (P + (C \ {0})) ∩ (D − (C \ {0})) = ∅. (P) and (D) are called strongly dual, if (3.63) holds together with 0 ∈ cl(P − D) or equivalently (P + O) ∩ (D + O) = ∅ for all open neighborhoods O of zero in Y . So strong duality means that P and D touch each other. Otherwise, we speak of a pair of dual problems with a duality gap (in the aforementioned scalar case at the beginning of this chapter that would mean I > S).

156

3 Optimization in partially ordered spaces

Lemma 3.7.1. Assume (P), (D) to be weakly dual. If there are z 0 ∈ B, ζ 0 ∈ D such that z 0 = ζ 0 , then z 0 is minimal-efficient for (P), ζ 0 maximal-efficient for (D), and (P), (D) are strongly dual. Proof. z 0 not minimal-efficient means that there is z 1 ∈ P such that 1 0 z ∈ z − (C \ {0}) = ζ 0 − (C \ {0}), which contradicts (3.63). For ζ 0 to be maximal-efficient follows similarly. From z 0 = ζ 0 it follows that (P) and (D) are strongly dual.
As usual, if one deals with multicriteria optimization problems, the question is whether to apply scalarization methods. So at first we prove duality theorems completely without taking into account scalarization of the goal function. Instead of (P) we consider the following multicriteria problem with side conditions (P1 ) Eff Min (f [A], C) (3.64) with A := {x ∈ M, g(x) ∈ CV }, where X is a linear space, M a nonempty set in X, (V, CV ) a reflexively preordered linear topological space, (Y, C) as above, f : X → Y • , dom f = M , g : X → V • , and dom g = M . So a vector optimization problem is given as usual. Remark 3.7.2. To deal with duality in vector optimization, we suppose that for C in Y we choose a closed convex pointed cone B ⊂ Y such that B+(C\{0}) ⊂ B \ {0} and later, in Section 3.7.2, such that B + (C \ {0}) ⊂ int B,

(3.65)

and an element k 0 ∈ B with Rk 0 − B = Y . If we use in the definition of the scalarizing functional (2.23) the set B with condition (3.65), then the functional (2.23) is strictly C- monotone (see Theorem 2.3.1, (g)). Such a property is important for the proof of duality assertions via scalarization. We introduce the following generalized Lagrangian, having another set M 0∗ , L : M × M 0∗ −→ Y ∪ {+∞Y }, and assume L(x, y) ∈ f (x) − B for all x ∈ A and for all y ∈ M 0∗ . Now we are able to write down a problem (D1 ) that can be considered as a dual problem to (P1 ): (D1 ) Eff Max (f ∗ [M ∗ ], C),

(3.66)

3.7 Duality

with

157

M ∗ := {y ∈ M 0∗ : Eff Min (L[M, y], B) = ∅},

where is B as explained above, and f ∗ (y) ∈ Eff Min ({L(x, y) | x ∈ M }, B) = ∅ ⊂ Y . The problem (D1 ) is a so-called Lagrange dual problem to (P1 ), as is easy to see if we reduce to the finite-dimensional scalar case (Y, C) = (R1 , R1+ ), 1 x ∈ Rn , (V, CV ) = (Rm , Rm + ), B = R+ , and if we take y as linear mapping λ. Then instead of (P1 ) we get f (x) → min s.t. x ∈ A = {x ∈ M, g(x) ∈ Rm + }. T T So, with M ∗ = {λ ∈ −Rm + | ∃ x0 ∈ M : (f + λ g)(x) ≥ (f + λ g)(x0 ) ∀ x ∈ M ∗ T and f (λ) = min (f + λ g)(x) > −∞}, (D1 ) has the well-known maxmin x∈M

form

max f ∗ (λ) = max∗ min (f + λT g)(x).

λ∈M ∗

λ∈M x∈M

Lemma 3.7.3. Weak duality. (P1 ) and (D1 ) are weakly dual; i.e., f [A] ∩ f ∗ [M ∗ ] − (C \ {0}) = ∅.

(3.67)

Proof. Take y ∈ M ∗ ; therefore f ∗ (y) ∈ Eff Min ({L(x, y) | x ∈ M }, B). This means that {L(x, y) | x ∈ M } ∩ (f ∗ (y) − (B \ {0})) = ∅, and for A ⊂ M instead of M , {L(x, y) : x ∈ A)} ∩ f ∗ (y) − (B \ {0}) = ∅, and regarding a property of B, {L(x, y) | x ∈ A} ∩ f ∗ (y) − (B + (C \ {0})) = ∅ and

({L(x, y) | x ∈ A} + B) ∩ (f ∗ (y) − (C \ {0})) = ∅.

By definition of the Lagrangian L(x, y) we have for x ∈ A and y ∈ M ∗ , L(x, y) ∈ f (x) − B, and because y ∈ M ∗ is chosen arbitrarily, we can conclude that f [A]) ∩ f ∗ [M ∗ ] − (C \ {0})) = ∅, which yields (3.67).




A strong duality theorem holds as well. Therefore, we assume a condition (V1) (see Weidner [372]) that is related to the so-called domination property (cf. [244] and Section 3.4): (f + y ◦ g)[M ] ⊂ Eff Min ((f + y ◦ g)[M ]), B) + B for all y ∈ M ∗ . Since one can choose the wrapping cone B very close to C \ {0}, (V 1) means that (f + y ◦ g)[M ] cannot have improperly efficient elements with respect to C. This is reflected by the formulation of the next theorem, which takes into account only primarily efficient points with respect to a cone B.

158

3 Optimization in partially ordered spaces

Theorem 3.7.4. Strong duality theorem. For (P1 ) and (D1 ) as given above, let (V 1) be fulfilled and assume f (x) ∈ Eff Min (f [A], B) to be in Eff Min (f [A], C). Then f (x) ∈ Eff Max (f ∗ [M ∗ ], C). Proof. This time we take into account that the Lagrangian L : M × M ∗ −→ Y ∪ {+∞Y } for an element y ∈ M ∗ can be chosen as  f (x) : g(x) ∈ CV , L(x, y) = (+∞)Y : otherwise. Therefore y ∈ M ∗ and Eff Min ({L(x, y) : x ∈ A}, B) = Eff Min (f [A], B). The second setting yields Eff Min ({L(x, y) : x ∈ M \ A}, B) = ∅. Together these results give Eff Min ({L(x, y) | x ∈ M }, B) = Eff Min (f [A], B),

(3.68)

since Eff Min ({L(x, y) | x ∈ M }, B) = Eff Min (Eff Min ({L(x, y) | x ∈ A}, B) ∪ Eff Min ({L(x, y) | x ∈ M \ A}, B), B). As assumed, f (x) ∈ Eff Min (f [A], B), so (3.68) gives f (x) ∈ Eff Min ({L(x, y) | x ∈ M }, B); that is, f (x) ∈ f ∗ [M ∗ ]. For y¯ ∈ M ∗ we have ({L(x, y) | x ∈ M } + (B \ {0})) ∩ {f ∗ (y)} = ∅ in the proof of Lemma 3.7.3. For all cones B
⊂ Y with B + (B
\ {0}) ⊂ (B \ {0}) we get ({L(x, y) | x ∈ M } + B + (B
\ {0})) ∩ {f ∗ (y)} = ∅, and along the lines of the proof of Lemma 3.7.3 we have f [A] ∩ (f ∗ [M ∗ ] − (B
\ {0})) = ∅.

(3.69)

Because x ∈ A, we have that (f (x) + (B
\ {0})) ∩ f ∗ [M ∗ ] = ∅, and so f (x) ∈ Eff Max (f ∗ [M ∗ ], B
) for all B
⊂ Y as above. In particular, f (x) ∈ Eff Max (f ∗ [M ∗ ], C). The proof is complete.




Sometimes results like Theorem 3.7.4 are called a strong direct duality theorem, because a primal optimal solution is shown to be dually optimal. The converse direction is also interesting. This leads to converse duality theorems. To get such a theorem for our pair of optimization problems, we state a condition (V2): (V2) Every solution of (D1 ) is to be dominated by a properly efficient solution of (P1 ), which means that with a cone B as in (3.65), for all d˜ ∈ x) ∈ Eff Min (f [A], B) such that f (˜ x) ∈ d˜+ C. Eff Max (f ∗ [M ∗ ], C) there is an f (˜ Theorem 3.7.5. Converse duality theorem. Assume that (V1) and (V2) hold and d˜ ∈ Eff Max (f ∗ [M ∗ ], C). Then there are x ∈ A, and a cone B
⊂ Y with B + (B
\ {0}) ⊂ (B \ {0})such that d˜ = f (x) ∈ Eff Min (f [A], B
).

3.7 Duality

159

Proof. For d˜ as assumed, (V2) leads to an f (˜ x) ∈ Eff Min (f [A], B) with d˜ ∈ f (˜ x)−C. Then d˜ ∈ f [A]+C. Otherwise, d˜ ∈ f [A]+C, so that d˜ ∈ f (˜ x)−C can be satisfied only if d˜ ∈ f (˜ x) − (C \ {0}). (3.70) With f (˜ x) as above and (3.68) we obtain f (˜ x) ∈ f ∗ [M ∗ ]. Because of ∗ ∗ ˜ ˜ d ∈ Eff Max (f [M ], C) it follows that f (˜ x) ∈ d + (C \ {0}), which contradicts (3.70). Since d˜ ∈ f [A] + C, because of weak duality, d˜ ∈ f [A]. Therefore, there is x ∈ A such that d˜ = f (x). The last assertion follows from (3.69), which holds for (P1 ), (D1 ) for all B
. This means that f ∗ [M ∗ ] ∩ (f (x) + B
) = ∅, or equivalently, f (x) ∈ Eff Max (f ∗ [M ∗ ], B
), especially f (x) ∈ Eff Max (f ∗ [M ∗ ], C).
The duality in the last section is an abstract and not scalarized Lagrangian formalism for very general optimization problems. In the next section we will prove duality theorems with the help of scalarization. Here we again use the Lagrangian form of duality. For an equivalent method, using the Fenchel form of duality, see Wanka [366, 367]. 3.7.2 Duality by Scalarization To prove duality theorems we come back to the general case (P1 ) in the following form: (P2 ) Eff Min (f [A], C) withA = {x ∈ M | g(x) ∈ CV } = ∅, where f : M → Y, ∅ = M ⊂ X, g : M → V, CV a nonempty set in V and X, Y, V t.v.s., Y Hausdorff, Y partially ordered by C. Furthermore, we suppose that the assumptions mentioned in Remark 3.7.2 are satisfied. Then (compare Theorem 2.3.1), the set S of surjective continuous strictly monotone (with respect to C) functionals s : Y → R is nonempty. Again we introduce a generalized Lagrangian L : M × M ∗ −→ Y ∪ {+∞Y } and assume that for s ∈ S, sup s(L(x, y)) = s(f (x)) if g(x) ∈ CV .

y∈M ∗

Using the functionals s ∈ S we define a dual problem to (P2 ): (D2 ) Eff Max (AD , C),

(3.71)

where
 AD = h ∈ Y | ∃ s ∈ S, ∃ y ∈ M ∗ : s(h) = inf{s(L(x, y)) | x ∈ M } > −∞ . Lemma 3.7.6. Weak duality. The pair (P2 ), (D2 ) satisfies f [A] ∩ (AD − (C \ {0})) = ∅.

160

3 Optimization in partially ordered spaces

Proof. Otherwise, there would be x0 ∈ A, h0 ∈ AD such that f (x0 ) − h0 ∈ −C \ {0}. With s0 ∈ S corresponding to h0 we get s0 (f (x0 )) < s0 (h0 ) because s is strictly C-monotone. On the other hand, we consider with h0 a corresponding y0 ∈ M ∗ and derive regarding the definition of the dual feasible set AD , s0 (h0 ) = inf s0 (L(x, y0 )) ≤ s0 (L(x0 , y0 )) ≤ sup s0 (L(x0 , y)) = s0 (f (x0 )), x∈M

y∈M ∗




a contradiction.

For properly efficient elements z0 = f (x0 ), this means that there is an s0 ∈ S such that s0 (f (x0 )) = inf{s0 (f (x)) | x ∈ A} > −∞, we can prove a strong duality theorem: Theorem 3.7.7. Strong (direct) duality. If z0 = f (x0 ) with x0 ∈ A is properly efficient for (P2 ), then it is efficient for (D2 ) i.e., z0 ∈ Eff Max (D2 ). Proof. Let f (x0 ) be a properly efficient element of the primal problem (P2 ). With the corresponding element s0 ∈ S we can choose the Lagrangian L : M × M ∗ for an element y0 ∈ M ∗ such that  s0 (f (x)) : g(x) ∈ CV , s0 (L(x, y0 )) = +∞ : otherwise. Using this Lagrangian we get s0 (f (x0 )) = inf s0 (L(x, y0 )) = inf x∈A




inf s(L(x, y0 )) ,

x∈A

inf

x∈M \A



s(L(x, y0 ))

= inf s(L(x, y0 )). x∈M

This yields f (x0 ) ∈ AD . From the weak duality assertion in Lemma 3.7.6 we can conclude that f (x0 ) is an efficient element for (D2 ).
The point of the last proof is that there are functionals y0 and s0 such that a possibly nonlinear Lagrangian works. So an essential task is to find sufficiently practicable s0 and y0 for special optimization problems such as (P3 ) or (P4 ) below. Next we would like to complete the study of (P2 ) and (D2 ) with a converse duality theorem. For this we restrict Y to be an l.c.s. At first we show that efficient elements h ∈ AD can be characterized by scalarization. Lemma 3.7.8. h0 ∈ Eff Max (D2 ) iff sh (h) ≤ sh (h0 ) ∀ h ∈ AD where sh corresponds to h according to the definition of AD in (D2 ).

3.7 Duality

161

Proof. (a) If h0 ∈ Eff Max (D2 ) and h ∈ h0 +C \{0}), then h ∈ AD . Therefore s(h) > inf{s(f (x)) + y(g(x)) | x ∈ M } for all s ∈ S and y. Applying sh (belonging to h ∈ AD ) to h as above, we get sh (h0 ) = inf{sh (h) | h ∈ h0 + (C \ {0})} ≥ inf{sh (f (x)) + yh (g(x)) | x ∈ M } = sh (h), as claimed, where the inequality follows from the strict inequality above. (b) If h0 ∈ AD but h0 ∈ Eff Max (D2 ), then there is h ∈ AD such that h ∈ h0 + (C \ {0}). Therefore s(h) > s(h0 ) for all s ∈ S, contrary to our assumption.
Additionally, a strong converse duality statement holds. Theorem 3.7.9. Strong converse duality. Under the conditions given for the pair (P2 ), (D2 ) and additionally, if P := f [A] + C is closed and if for all s ∈ S such that inf{s(f (x)) | x ∈ A} > −∞ there is an x0 ∈ A with inf{s(f (x)) | x ∈ A} = s(f (x0 )), then a dually efficient element is properly primal efficient. Proof. Let h be an element of Eff Max (D2 ). If h were not in P , there would be an open convex neighborhood U of h with U ∩ P = ∅. Even (U − C) ∩ P = ∅; otherwise, there would exist u ∈ U, k ∈ C such that u − k ∈ P = f [A] + C, which means that u ∈ f [A] + C + k = P . Now we consider the set B := U − C. Obviously, B is open and convex and B −C = B, and so B −C ⊂ B − C ⊂ B. Theorems 2.3.1 and 2.3.6 deliver an s1 ∈ S such that s1 (p) ≥ 0 > s1 (u) ∀ p ∈ P ∀ u ∈ B. We have h ∈ B, since 0 ∈ C, so there is a number γ with s1 (h) < γ < 0 ≤ s1 (p) ∀ p ∈ P. From weak duality we get for s2 ∈ S, s2 (h) ≤ s2 (p) ∀ p ∈ P. Taking λ ∈ (0, 1) we consider sλ := λs2 + (1 − λ)s1 . Then sλ is again in S, and sλ (h) = s1 (h) + λ(s2 (h) − s1 (h)), ∀ x ∈ A : sλ (f (x)) = λ(s2 (f (x))) + (1 − λ)(s1 (f (x))), ≥ λs2 (f (x)) ≥ λs2 (h), since f (x) ∈ P . Now we choose λ sufficiently small; then not only s1 (h) < γ but sλ (h) < γ, and from γ < 0 it follows that γ < λs2 (h), so sλ (h) < γ < λs2 (h) ≤ sλ (f (x)) ∀ x ∈ A. This means that inf{sλ (f (x)) : x ∈ A} ≥ γ, and the assumptions give an x0 ∈ A such that sλ (h) < γ ≤ inf{sλ (f (x)) : x ∈ A} = sλ (f (x0 )). But strong duality now demands f (x0 ) ∈ AD , and Lemma 3.7.8 requires sλ (f (x0 )) ≤ sλ (h), a contradiction. So h ∈ P = f [A] + C. Weak duality gives h ∈ f [A].


162

3 Optimization in partially ordered spaces

3.7.3 Duality for Approximation Problems As an example for Section 3.7.2 and because of its practicability we now give duality theorems for a general approximation problem and a corresponding dual problem. To this end we consider (P1 ) or (P2 ) in the following form: (P3 ) Eff Min (f [A], C) withA := {x ∈ CX | B(x) − b ∈ CV }, where now X is a linear normed space, partially ordered by a closed convex cone CX , V is a reflexive Banach space, CV a cone in V that is convex and closed, B ∈ L(X, V ), Y = Rp , partially ordered by a closed convex cone K such that K + Rp+ ⊂ K and int K + = ∅, and f may have the form ⎛ ⎞ α1 A1 (x) − a1 ⎠ ... f (x) = C(x) + ⎝ (3.72) p p αp A (x) − a with C ∈ L(X, Rp ), αi ≥ 0 real (i = 1, . . . , p), ai ∈ U a given real normed space, Ai ∈ L(X, U ), i = 1, . . . , p, and A∗i denotes the adjoint operator to Ai . For brevity and clarity we sometimes leave out the parenthesis in connection with A∗i . On the one hand, (P3 ), although a special case of (P1 ) or (P2 ), itself contains many important special cases: (i) (P3 ) is a semi-infinite linear problem if αi = 0 ∀ i = 1, . . . , p. (ii) f can be interpreted as a Lipschitz perturbed linear problem. (iii) C = 0 gives a multicriteria location problem. (iv) Consider C(x) → min, x ∈ A0 = {x ∈ A = ∅ | Ai (x) = ai , i = 1, . . . , p} = ∅. Then (3.72) is a parameterized surrogate problem. (v) The general multicriteria approximation problem or location problem dealt with in Sections 4.1 and 4.3. On the other hand, (P3 ) can be generalized considerably if we assume ai to be variable in a nonempty set Wi ⊂ Ui (i = 1, . . . , p), Ui instead of a fixed space U , so that we get an optimization problem (P4 ): (P4 ) Eff Min (f [A], K) where A = {(x, a) | x ∈ CX , a = (a1 , . . . , ap ), ai ∈ Wi , i = 1, . . . , p, B(x)−b ∈ CV }, and

⎛

⎞ ... f (x, a) = C(x) + ⎝ αi Ai (x) − ai Ui ⎠ . ...

3.7 Duality

163

Using (3.71) we get for this multicriteria optimization problem a useful dual problem (D4 ), which satisfies together with (P4 ) the above defined conditions for weak and strong duality. Moreover, in Section 4.3 a practicable algorithmic procedure will be constructed and tested. In order to use the results of Section 3.7.2 we introduce a suitable Lagrangian Lλ0 to (P4 ) for a given λ0 ∈ int K + : ⎛⎛ ⎞ ⎛ ⎞⎞ ... ... Lλ0 (x, a, Y, u∗ ) = λ0 ⎝⎝αi Y i (ai − Ai (x))⎠ + ⎝C i (x)⎠⎠ + u∗ (b − B(x)), ... ... where (x, a) ∈ M = {(x, a) | x ∈ CX , ai ∈ Wi } and (Y, u∗ ) ∈ M 0∗ := {(Y, u∗ ) | Y = (Y 1 , . . . , Y p ), Y i ∈ L(Ui , R), u∗ ∈ L(V, R), u∗ ∈ CV+ , αi Y i ∗ ≤ αi , i = 1, . . . , p}. From this setting two results follow immediately: sup

(Y,u∗ )∈M 0∗

Lλ0 (x, a, Y, u∗ )  =

λ0i αi ai − Ai (x) + λ0 C(x) if B(x) − b ∈ CV , (3.73) +∞ otherwise, i

and inf Lλ0 (x, a, Y, u∗ )  0  0 + i i ∗ λi (−αi Ai∗ Y i + C i∗ ) − B ∗ (u∗ ) ∈ CX , i λi inf ai (αi Y (a )) + u (b) if = −∞ otherwise. (3.74)

(x,a)∈M

To obtain a dual problem to (P4 ) we recall (D2 ), (3.71), and define Mλ∗0 to be


+ Mλ∗0 := (Y, u∗ ) ∈ M 0∗ | . λ0i (−αi Ai∗ Y i + C i∗ ) − B ∗ (u∗ ) ∈ CX So we are able to give the following dual problem (D4 )
: Eff Max (AD , K) where
AD = d ∈ Y | ∃ λ ∈ int K + , ∃ (Y, u∗ ) ∈ Mλ∗ : λ(d) = It follows that λ(d) =

i

inf

(x,a)∈M

λi iinf (αi Y i (ai )) + u∗ (b). a ∈Wi

 Lλ (x, a, Y, u∗ ) .

164

3 Optimization in partially ordered spaces

In order to descalarize the last line we consider the set M ∗ and use Lemma 3.7.10 below, which is related to a result given by Jahn [195]: M ∗ = {(Y, Z) | Y = (Y 1 , . . . , Y p ), Y i ∈ L(Ui , R), i = 1, . . . , p, Z ∈ L(V, Rp ), ∃ λ∗ ∈ int K +

p

such that

+ λ∗i (−αi Ai∗ Y i + C i ) − (Z(B))∗ (λ∗ ) ∈ CX ,

i=1

Z ∗ (λ∗ ) ∈

CV+ , αi Y i ∗

≤ αi , i = 1, . . . , p}.

Moreover, consider the sets D1 and D2 : p | ∃ λ∗ ∈ int K + and ∃ (Y, u∗ ) ∈ Mλ∗∗ such that λ∗ (d) = • D 1p := ∗{d ∈ R i i λ (α Y (a )) + u∗ (b)}, i i=1 i ⎞ ⎛ ... + , • D2 := d ∈ Rp | ∃ (Y, Z) ∈ M ∗ such that d = ⎝ αi Y i (ai ) ⎠ + Z(b) , ...

where d in D2 satisfies a descalarized equation. Lemma 3.7.10. For the sets D1 , D2 as defined above, D2 ⊂ D1 . Moreover, if b = 0, even D2 = D1 . Proof. (a) Let d be in D2 . According to the definition of M ∗ there is λ∗ ∈ int K + , Y i ∈ L(Ui , R) (i = 1, . . . , p), Z ∈⎛L(V, Rp ), ⎞ Z ∗ (λ∗ ) ∈ CV+ , ... αi Y i ∗ ≤ αi (i = 1, . . . , p), such that t := d − ⎝ αi Y i (ai ) ⎠ = Z(b) and ... p + ∗ i∗ i i∗ ∗ ∗ λ (−α A Y + C ) − (Z(B)) (λ ) ∈ C . Then λ∗ (t) = u∗ (b) with i i=1 i  ∗ Xi i λi (αi Y (a )) + u∗ (b), which means u∗ = Z ∗ (λ∗ ) ∈ L(V, R), and so λ∗ (d) = that d ∈ D1 . (b) Let d be in D1 . Therefore there exist λ∗ ∈ int K + , Y i ∈ L(Ui , R) (i = 1, . . . , p), u∗ ∈ L(V, R), u∗ ∈ CV+ such that

+ (3.75) λ∗i (−αi Ai∗ Y i + C i∗ ) − B ∗ (u∗ ) ∈ CX and ∗

λ (d) =

p

λ∗i (αi Y i (ai )) + u∗ (b).

(3.76)

i=1

Since b = 0, (3.76) leads to the existence of Z ∈ L(V, Rp ) with ⎛ ⎞ ... d = ⎝ αi Y i (ai ) ⎠ + Z(b) ...

(3.77)

and Z ∗ (λ∗ ) = u∗ . Then from (3.75) and u∗ ∈ CV+ it is clear that d ∈ D2 . To understand where the existence of Z comes from, we consider once more

3.7 Duality

165

λ∗ (t) = u∗ (b), which follows from (3.76). It is well known that if b = 0 and λ∗ = 0, then one can define an operator T ∈ L(V, Rp ) such that T ∗ (λ∗ ) = u∗ ∗ (·) and t = T (b): If u∗ (b) = 0, take T (·) = uu∗ (b) t over V ; if u∗ (b) = 0, then ∗ ∗ consider λ = 0. For λ there exists t such that λ∗ (t0 ) = 1. Since b = 0, there is u ˜∗ ∈ L(V, R) that separates b and 0; i.e., u ˜∗ (b) = 1. Then take T (·) = u∗ (·)t0 + u ˜∗ (·)t over V.




If b = 0, we obtain from (D4 )
, regarding Lemma 3.7.10, the following dual problem (D4 ) to (P4 ): (D4 ) : where

Eff Max (f ∗ [M ∗ ], K),

⎛

⎞ ... ⎜ inf αi Y i (ai ) ⎟ ∗ ⎝ ai ∈Wi ⎠ + Z(b) =: f (Y, Z) ...

for (Y, Z) ∈ M ∗ .

For an example consider Section 4.3.2 or recall the dual problem for a common finite-dimensional linear multicriteria problem. Example 3.7.11. Consider instead of (P3 ) the following scalar optimization problem: p (P5 ) c(x) + i=1 αi Ai (x) − ai → min, s.t. x ∈ CX , B(x) − b ∈ CV , where X, V, CX , CV , Ai , ai (i = 1, . . . , p) as above, ai fixed, c ∈ L(X, R1 ), and αi > 0 ∀ i = 1, . . . , p. Since the sum in (P5 ) is really a norm, (P5 ) is an example of (P3 ). Then we get a dual problem (D5 ) as a special case of (D4 ): p i i (D5 ) i=1 αi y (a ) + z(b) → max 1 s.t. . . . , y p ) ∈ L(U, R)p , y i ∗ ≤ 1 ∀ i, z ∈ L(V, R), z ∈ CV+ , p y = (y , i∗ + i ∗ i=1 (−αi A (y ) + c) − B (z) ∈ CX . The problem (D5 ) follows immediately from (D4 ), since y ∗ ≤ 1 in (D4 ) means that maxi=1,...,p y i ∗ ≤ 1 and so y i ∗ ≤ 1 for all i, because the maximum norm is suitable as a dual norm to a sum of norms. Example 3.7.12. The ordinary scalar classic location problem is contained in (P5 ): X, α1 , . . . , αp , a1 , . . . , ap as above, a1 , . . . , ap fixed, p (P6 ) αi x − ai → minx∈X (Fermat’s problem), i=1 p p i i i ∗ i i (D6 ) i=1 αi y (a ) → max s.t. y ∈ X , y ∗ ≤ 1 ∀ i, 1 αi y = 0. Taking as A ⊂ X a closed linear subspace, we get (P7 ) x − a → min s.t. x ∈ A, (D7 ) y(a) → max s.t. y ∈ A⊥ , y ∗ ≤ 1, where A⊥ is the annihilator to A : A⊥ = {y ∈ X ∗ | y(x) = 0 ∀ x ∈ A}.

166

3 Optimization in partially ordered spaces

(P5 ) can be generalized in another way, too (see Wanka [366, 367]). Wanka considers the optimization problem (P8 ): p

αi x − ai βi → min s.t. x ∈ U ,

(3.78)

i=1

where βi ≥ 1 ∀ i = 1, . . . , p, αi > 0 ∀ i = 1, . . . , p, U a convex closed subset of a reflexive Banach space X, and a1 , . . . , ap given elements of X. Then he studies the dual problem (D8 ), which contains our problem (D6 ), using Fenchel–Rockafellar theory instead of Lagrange theory. He also considers the vectorial variant of (3.78) together with a linear mapping as in (P3 ). We come back to these general problems in Section 4.1.

(D8 ) :

p

αi (1 − βi ) y i

βi 1−βi

+

i=1 βi >1

p

αi βi y i (ai ) − sup

m

x∈U i=1

i=1

αi βi y i (x) → max (3.79)

s.t. y i ∈ X ∗ ∀ i and y i ∗ ≤ 1 for i with βi = 1.

Taking into account the fact that we constructed (D4 ) as in Section 3.7.2, weak, strong, and converse duality follow immediately: Theorem 3.7.13. Weak duality. f (A) ∩ f ∗ (M ∗ ) − (K \ {0}) = ∅.

(3.80)

Now we derive a strong duality assertion. We use the chosen Lagrangian and the saddle point theorem of convex programming. For simplicity we assume ai ∈ W i = {ai }; therefore, we write x ∈ A instead of (x, a) ∈ A. Theorem 3.7.14. Strong direct duality. Let U be a reflexive Banach space and b = 0. Then, for every properly efficient f (x0 ) of (P4 ) there exists an efficient element f ∗ (Y 0 , Z 0 ) of (D4 ) such that f (x0 ) = f ∗ (Y 0 , Z 0 )

(3.81)

if the following condition is satisfied: There is a feasible element x of (P4 ) with B(x) − b ∈ int CV , and for every λ∗ ∈ int K + there is a feasible pair + (Y , Z) of (D4 ) such that the CX -inclusion in (D4 ) is satisfied with respect to int K + . Proof. Since f (x0 ) is properly efficient, there is λ0 ∈ int K + such that λ0 (f (x0 )) = inf{λ0 (f (x)) : x ∈ A}. Then (3.73) gives λ0 (f (x0 )) = inf λ0 (f (x)) = inf x∈A

sup

x∈M (Y,u∗ )∈M 0∗

Lλ0 (x, Y, u∗ ).

3.7 Duality

167

We use the saddle point theorem of convex programming (see [391]) and have that there is (Y 0 , u∗0 ) contained in Mλ∗0 such that for x ∈ M and (Y, u∗ ) ∈ Mλ∗0 , Lλ0 (x, Y 0 , u∗0 ) ≥ Lλ0 (x0 , Y 0 , u∗0 ) ≥ Lλ0 (x0 , Y, u∗ ). (3.82)  λ∗i (αi Y i0 (ai )) + u∗0 (b), so we Then (3.74) gives inf x∈CX Lλ0 (x, Y 0 , u∗0 ) = get p

0 0 λ0i (αi Y i0 (ai )) + u∗0 (b) (3.83) λ (f (x )) = i=1

or using Lemma 3.7.10 its vectorial variant (3.81). Considering (D4 ), we see that the proof is complete.
Our next aim is a converse duality theorem. To attempt this we use the scalarization Lemma 3.7.8. In our convex case it reads as follows: Lemma 3.7.15. d0 ∈ Eff Max (D1 , K) iff λd (d0 ) ≥ λd (d) for all d ∈ D1 and λd ∈ int K + such that for every (Y, u∗ ) ∈ Mλ∗d ,

λd (d) = λd,i (αi Y i (ai )) + u∗ (b). With the help of the last lemma a converse duality theorem follows. We again suppose W i = {ai }, i = 1, . . . , p. Theorem 3.7.16. Converse duality. We assume b = 0, int K = ∅, ∃ x ∈ A with Bx−b ∈ int CV+ , for every λ∗ ∈ int K + with inf{λ∗ (f (x)) | x ∈ A} > −∞ there exists (Y , Z) ∈ M ∗ such that

i + λ∗i (−αi Ai∗ Y + C i∗ ) + (Zb)∗ (λ∗ ) ∈ int CX and additionally an xλ ∈ A with inf{λ∗ (f (x)) : x ∈ A} = λ∗ (f (xλ )). Then to every f ∗ (Y 0 , Z 0 ) that is optimal with respect of (D4 ) there is a properly efficient element f (x0 ) of (P4 ) such that f ∗ (Y 0 , Z 0 ) = f (x0 ).

(3.84)

We would like to emphasize that both strong duality Theorems 3.7.14 and 3.7.16 state duality in a nonscalarized form. Proof. Let f ∗ (Y 0 , Z 0 ) be maximal with respect to M ∗ . Then it is also maximal w.r.t. the set D2 , and since b = 0, also with respect to D1 ; hence Lemma 3.7.15 yields λd (f ∗ (Y 0 , Z 0 )) ≥ λd (d) (3.85) ∀ d ∈ D1 with λd ∈ int K + belonging to d. From (3.85) we get d0 := f ∗ (Y 0 , Z 0 ) ∈ P := f (A) + K.

(3.86)

Otherwise, d0 ∈ f (A) + K. Now, separating P and d0 by a continuous linear functional λ∗1 ∈ Rp \ {0}, we proceed as in the proof of Theorem 3.7.9 and get

168

3 Optimization in partially ordered spaces

d0 = f ∗ (Y 0 , Z 0 ) = f (x0 ) for a suitable x0 ∈ A. We observe that f (x0 ) is even properly minimal w.r.t. (P4 ) with a λ∗ ∈ int K + that belongs to (Y 0 , Z 0 ).
In Section 3.11 it is shown that the saddle point theory w.r.t. (P4 ) can be extended considerably: We use sublinear functionals instead of norms and approximate efficiency instead of efficiency.

3.8 Vector Equilibrium Problems and Related Topics It is well known that optimization and nonlinear analysis are two branches of modern mathematics much developed lately. The results obtained in these areas use certain kinds of differentiability (directional derivative, subgradient, generalized subgradients, etc.), certain generalizations of convexity, geometric methods (cone of feasible directions, normal and tangent cones, etc.), game theory, fixed point theory, topological degree, etc. The reader can find extensive expositions about these topics in a series of books of Zeidler [392] entitled Nonlinear Analysis and Its Applications, and those of Aubin and Ekeland [12], Aubin and Frankowska [13], and Rockafellar and Wets [309]. In the same way, equilibrium problems have come to play a central role in the development and unification of diverse areas of mathematics and physical sciences. Thus various problems of practical interest in optimization, variational inequalities, complementarity, economics, Nash equilibria, and engineering involve equilibrium in their description. The literature on equilibrium problems and treatment in these areas is vast; see [14], [34], [61], [384], [113], [203], [202], and so on. These sources provide extensive references and extended bibliographies for the reader wishing to explore the topic in depth. Our object in this section is to present a developed investigation in generalized vector equilibrium problems, which embody at least vector optimization problems and vector variational inequalities. The vector variational inequalities have been widely developed in recent years, and various solutions have been characterized and computed. These were first introduced by Giannessi [132] and further developed by many authors; see, for instance, [64], [67], [322], [2], [229], [321], [75], and precisely, [133] in different areas. Recent topics attracting considerable attention are equilibrium problems for vector-valued mappings, see [279], [280], [151]. Inspired by the scalar case, such problems have received different developments depending on the kind of order space where these have been considered. Some recent papers may be grouped in the following way: 1. Theory of vector optimization, vector equilibrium problems, and vector variational inequalities (Ansari [2, 3]; Ansari and Siddiqi [4]; Ansari, Siddiqi and Yao [5]; Bianchi, Hadjisvvas and Schaible [28]; Blum and Oettli [34, 33]; Chen and Craven [66, 67]; Fan [111]; Fu [122]; Kalmoun, Riahi

3.8 Vector equilibrium problems and related topics

2.

3.

4.

5.

169

and Tanaka [201, 202, 203, 204, 205]; Konnov and Yao [215]; Lee and Kum [231]; Lee, G.M., Kim and Lee, B.S. [228, 234, 235, 227]; Lee, G.M., Kim and Lee, B.S., Cho [229]; Lee and Kum [232]; Li, Yang and Chen [238]; Lin, Yang, Yao [239]; Nagurney, N. and Zhao, L., [264], Oettli [278]; Samuelson [313], Siddiqi, Ansari, Ahmad [321]; Yang [379]; Yang and Chen [380]; Yang and Goh [381], Yu and Yao [383]). Existence of solutions for generalized vector variational inequalities and complementarity problems (Chen [64]; Chen and Hou [70, 76]; Danilidis and Hadjisavvas [90]; Isac and Yuan [184]; Kazmi [209]; Lai and Yao [225]). Vector variational inequalities and vector equilibrium problems with multifunctions (Ding and Tarafdar [95]; Kalmoun and Riahi [201, 202, 203]; Siddiqi, Ansari, Khan [323]; Song [326]) Vector variational inequalities, vector optimization, and scalarization (Chen, G.-Y., Chen, G.M. [65]; Chen and Craven [67]; Giannessi, Mastroeni and Pellegrini [134]; Goh and Yang [135]; Lee, G.M., Kim, Lee, B.S., Yen [230]). Monotone vector variational inequalities (Chowdhury and Tan [78]; Hadjisavvas and Schaible [151], Ding and Tarafdar [97]).

We propose two approaches to establish the existence of solutions of equilibrium problems in the vector case. The first one directly uses a generalization of the well-known lemma of Knaster, Kuratowski, and Mazurkiewicz (KKMLemma) as proposed by Br´ezis, Nirenberg, and Stampacchia [47]. The second one, as proposed by Oettli [278], leads in a straightforward way to existence results for vector equilibrium problems from the results about scalar case. A key tool for the study of such problems is an appropriate gauge function. We will see in subsequent sections that an overwhelming majority of vector problems of potential interest in nonlinear analysis can be cast in the form of vector equilibrium problems. In this section our attention is focused on the existence of a generalized vector equilibrium. 3.8.1 Vector Equilibrium Problems Let us agree to define a standard scalar (= real) equilibrium problem as follows. Given a nonempty subset M of a real topological vector space X, and a real function f defined on M × M , (EP ) find at least one point x ∈ M satisfying f (x, y) ≤ 0 for all y ∈ M. Let us see whether this scalar equilibrium problem can be formulated for the case of vector-valued mappings. Given the following: • •

Y a real topological vector space; M a nonempty subset of X, called the set of feasible decisions;

170

• •

3 Optimization in partially ordered spaces

a multifunction P : M ⇒ Y that is used for an outcome y at a decision event x : for y1 , y2 ∈ Y , y1 dominates y2 at the decision event x ∈ M iff y1 − y2 ∈ P (x); a multifunction f : M × M ⇒ Y , called the criterion mapping.

We shall assume for the multifunctions P and f that dom P = M and dom f = M × M. Inspired by the scalar equilibrium problem, the generalized vector equilibrium problems (in short GVEP) we can consider can be generalized in several possible ways, for instance, to establish the existence of some feasible decision x ∈ M such that " 1. f (x, y) P (x) = ∅ for all y ∈ M ; 2. f (x, y) " ⊂ P (x) for all y ∈ M ; 3. f (x, y) − int P (x) = ∅ for all y ∈ M ; ⊂ − int P (x) for all y ∈ M ; 4. f (x, y) " 5. f (x, y) − (P (x) \ {0}) = ∅ for all y ∈ M ; 6. f (x, y) ⊂ − (P (x) \ {0}) for all y ∈ M . Proposition 3.8.1. If we denote the solution set of the problem (GVEPi ) by S(f,P)i , for i = 1, . . . , 6, respectively, we have (1) S(f, P )2 ⊂ S(f, P )1 , S(f, P )3 ⊂ S(f, P )4 and S(f, P )5 ⊂ S(f, P )6 ; (2) S(f, P )5 ⊂ S(f, P )3 and S(f, P )6 ⊂ S(f, P )4 if 0 ∈ / int P (x); (3) S(f, P )2 ⊂ S(f, P )3 and S(f, P )1 ⊂ S(f, P )4 , if ∀ x ∈ M , P (x) is wpointed, i.e., P (x) ∩ − int P (x) = ∅; (4) S(f, P )2 ⊂ S(f, P )5 and S(f, P )1 ⊂ S(f, P )6 , if ∀ x ∈ M , P (x) is pointed, i.e., P (x) ∩ −P (x) = {0}; (5) S(f, P )4 ⊂ S(f, P )1 and S(f, P )3 ⊂ S(f, P )2 if ∀ x ∈ M , the complement of P (x) is included in − int P (x). The proof of this proposition, which can be argued directly from the definition of the problems (GVEPi ), for i = 1, . . . , 6, can be omitted. Remark 3.8.2. (1) Combining (3) and (5) we can state, under the condition that the sets P (x) are connected, i.e., P (x) ∩ − int P (x) = ∅ and P (x) ∪ − int P (x) = Y , that S(f, P )2 = S(f, P )3 and S(f, P )1 = S(f, P )4 . (2) Taking M = Y = R, P (x) = [0, ∞[ and f (x, y) = {−1, 1} for all x, y ∈ Y , one can see that S(f, P )1 = S(f, P )4 = S(f, P )6 = Y and S(f, P )2 = S(f, P )3 = S(f, P )5 = ∅. This example contradicts the converse inclusions in (1). (3) If we suppose that P (x) \ {0} is open in Y , then the converse inclusions in (2) are valid; i.e., S(f, P )5 = S(f, P )3 and S(f, P )6 = S(f, P )4 . Remark 3.8.3. When f is a single-valued function, these problems (GVEPi ) can be reduced to

3.8 Vector equilibrium problems and related topics

(GVEP1 , GVEP2 ) f (x, y) ∈ P (x)

for all y ∈ M ;

(GVEP3 , GVEP4 ) f (x, y) ∈ − int P (x)

for all y ∈ M ;

171

(GVEP5 , GVEP6 ) f (x, y) ∈ − (P (x) \ {0}) for all y ∈ M. Remark 3.8.4. If f is as in the last remark, P =] − ∞, 0], and Y = R, all the problems (GVEPi ), for i = 1, . . . , 6, are reduced to the scalar equilibrium problem (EP). In order to emphasize the relationship to vector optimization we decide to consider here the fourth and sixth generalized vector equilibrium problems (WGVEP) find x ∈ M such that f (x, y) ⊂ − int P (x) ∀ y ∈ M and (SGVEP) find x ∈ M such that f (x, y) ⊂ − (P (x) \ {0}) ∀ y ∈ M . We mean by (WGVEP) (resp. (SGVEP)) weak (resp. strong) generalized vector equilibrium problem. 3.8.2 General Vector Monotonicity Recently, much attention has been given to the development of different monotonicity notions. These monotonicity notions permit one to lighten the topological requirements needed to establish a solution of vector equilibrium problems, as compared with the nonmonotone case. Let us recall some of them. Definition 3.8.5. Let X, Y be two vector spaces, M a nonempty subset of X. Let f and g be two multifunctions from M × M into Y . Let K and L be two multifunctions from M into Y . (a) The pair (f, g) is said to be (K, L)-monotone if for all x, y ∈ M, f (x, y) ⊂ K(y) =⇒ g(x, y) ⊂ L(y). (b) f is said to be K-monotone if for all x, y ∈ M , one has f (x, y)+f (y, x) ⊂ K(x) ∩ K(y). Remark 3.8.6. If g(x, y) = f (y, x) and K(y)+K(y) ⊂ K(y) for every x, y ∈ M , then f is K-monotone implies that the pair (f, g) is (−K, K)-monotone. The definition of (K, L)-monotonicity of a pair (f, g) unifies several notions of monotonicity in the literature. Let us cite some of them: Given a closed convex cone P with nonempty interior int P ; (1) if g(x, y) = f (y, x), then (int P, − int P )-monotonicity of the pair (f, g) is just the P -pseudomonotonicity of f , as introduced in [277], [47];

172

3 Optimization in partially ordered spaces

(2) in case g(x, y) = f (y, x) is a single-valued function, the above definition of (int P, −P )-monotonicity of the pair (f, g) reduces to that of P -quasimonotonicity of f , as introduced in [151], [28]. If we take in the last definition Y to be R and P = R− , we obtain the definitions, which have been introduced by Bianchi and Schaible in [29], of pseudomonotonicity, i.e., f (x, y) ≥ 0 implies f (y, x) ≤ 0 for each x, y ∈ K, and quasimonotonicity, i.e., f (x, y) > 0 implies f (y, x) ≤ 0 for each x, y ∈ K. 3.8.3 Existence of Vector Equilibria by Use of the Generalized KKM Lemma The classical Knaster–Kuratowski–Mazurkiewicz lemma (the KKM lemma) is of fundamental importance in nonlinear analysis, game theory, economics, optimization theory, and variational inequalities. Many generalizations of the KKM lemma, beginning with Fan’s result [111] (Fan–KKM lemma), have been given (see [64], [358], and the quoted bibliography). Most of the results obtained are based on the assumption that the considered multifunctions have closed values. If M is a nonempty subset and x ∈ M , we shall denote by F(M ) (respectively, F(M, x)) the family of all nonempty finite subsets of M (respectively, of M containing x). If X and Y are two subsets of a vector space and S a multifunction defined from X into Y , then S is said to be a KKM-mapping if for every A ∈ F(X), conv(A) ⊂ ∪x∈A S(x). Note that if S is a KKM-mapping, then x ∈ S(x) for all x ∈ X. We will next deal with the Fan–KKM lemma. Lemma 3.8.7. (Fan–KKM lemma, [111, Lemma 1]) Let X be a nonempty convex subset of a Hausdorff topological vector space E, and S a KKM-mapping defined from X into E. Suppose that (i) (compactness condition) ∃x0 ∈ X such that S(x0 ) is compact in Y ; (ii) (closedness condition) ∀x ∈ X, S(x) is closed in Y . " Then x∈X S(x) = ∅. Here, by relaxing the closedness condition, we begin by stating the following generalized Fan’s theorem, which will play a crucial role in proving the main existence result. Lemma 3.8.8. Let X be an arbitrary nonempty convex subset of a Hausdorff topological vector space E, and T a KKM-mapping defined from X into E. Suppose that for some x0 ∈ X and each A ∈ F(X, x0 ) one has: (i) T (x0 ) ∩ X is relatively compact in X;

3.8 Vector equilibrium problems and related topics

173

(ii) for every x ∈ conv(A), the intersection T (x) ∩ conv(A) is closed in conv(A); (iii) clX (X ∩ (∩x∈F T (x))) ∩ F = (∩x∈F T (x)) ∩ F , where F = conv(A). Then the intersection of all subsets T (x), for x ∈ X, is nonempty. Proof. For A ∈ F(X, x0 ) consider FA = conv(A) ⊂ X and SA a multifunction defined on FA by SA (x) := T (x) ∩ FA . Note that SA (x) is closed, and so SA is compact-valued. We claim that SA is a KKM-mapping. Indeed, for ∅ = B ⊂ FA , B finite, we have that conv(B) ⊂ (∪x∈B T (x)) ∩ FA = ∪x∈B (T (x) ∩ FA ) = ∪x∈B SA (x). Using the Fan–KKM lemma (Lemma 3.8.7), there exists xA ∈ ∩x∈FA SA (x) ⊂ (∩x∈FA T (x)) ∩ X ⊂ T (x0 ) ∩ X. Consider UA := clX {x B | B ∈ F(X, x0 ), "  A ⊂ B} ⊂ clX ( x∈FA T (x)) ∩ X ⊂ clX (T (x0 ) ∩ X). The family {UA | A ∈ F(X, x0 } has the finite intersection property. Since clX (T (x0 ) ∩ X) is compact, there exists some x ∈ ∩B∈F (X,x0 ) UB . Let x ∈ X and take A := {x, x, x0 } ∈ F(X, x0 ). Since x ∈ A ⊂ FA , then x ∈ UA ∩ FA ⊂ clX ((∩y∈FA T (y)) ∩ X) ∩ FA = (∩y∈FA T (y)) ∩ FA . Since x ∈ FA , we obtain x ∈ T (x). Therefore x ∈ ∩x∈X T (x).




Remark 3.8.9. (1) Observe that in the case where T is closed-valued, this result becomes the Fan–KKM lemma. (2) This result is in effect a generalization of the Br´ezis–Nirenberg–Stampacchia theorem; see [47, p. 295]. We are now in position to provide a first existence result. Theorem 3.8.10. Let X and Y be two real Hausdorff topological vector spaces. Let M be a nonempty convex subset of X, and consider two multifunctions K and L from M to Y . Suppose that f and g are two multifunctions from M × M to 2Y such that (H1) the !n pair (f, g) is (K, L)-monotone; (H2) i=1 g(xi , z) ⊂ L(z) ∀ (x1 , . . . ., xn ) ∈ M , ∀ z ∈ conv(x1 , . . . ., xn ); (H3) for every fixed x ∈ M and every A ∈ F(M ), the set {y ∈ conv(A) | f (x, y) ⊂ K(y)} is closed in conv(A); (H4) for every A ∈ F(M ), and every converging net (yα ) on M to y ∈ conv(A) f (z, yα ) ⊂ K(yα ) ∀ z ∈ conv(A) ⇒ f (z, y) ⊂ K(y) ∀ z ∈ conv(A); (3.87)

174

3 Optimization in partially ordered spaces

(H5) there is a nonempty subset B of X and x0 ∈ M ∩ B such that M ∩ B is relatively compact in M , and f (x0 , y) ⊂ K(y) for all y ∈ M \ B. Then there exists x ∈ M ∩ B such that f (y, x) ⊂ K(x) for all y ∈ M . Proof. For each y ∈ M , let T (y) := {x ∈ M | f (y, x) ⊂ K(x)} . From (H5), T (x0 ) ⊂ M ∩ B is relatively compact in M , and therefore (i) of Lemma 3.8.8 is satisfied. (H1) and (H2) lead clearly to the fact that T is a KKM-mapping, while property (ii) follows from (H3). It remains to check (iii) of Lemma 3.8.8. For this, let F be the convex hull of a finite subset A of M . We have to show that clM (∩z∈F T (z)) ∩ F = (∩z∈F T (z)) ∩ F. So let x ∈ clM (∩z∈F T (z)) ∩ F . Then there exists a net (xα ) on M that converges to x ∈ F such that f (z, xα ) ⊂ K(xα ) ∀ z ∈ F. From (H4) we get f (z, x) ⊂ K(x) ∀ z ∈ F, which implies that x ∈ (∩z∈F T (z)) ∩ F . Thus (iii) is satisfied. According to the above lemma we deduce that ∩y∈M T (y) = ∅, which is the conclusion of Theorem 3.8.10.
3.8.4 Existence by Scalarization of Vector Equilibrium Problems In this subsection, we derive the existence theorem of (GVEP) by way of solving an appropriate scalar one. Very recently, Oettli [278] proposed an approach to solving vector equilibrium problems by using results in the scalar case. A key tool for the study of such problems is an appropriate Minkowski’s gauge function. Scalar problems Before going into the study of vector equilibrium problems, let us consider the scalar case. Usually, we establish theorems in the theory of noncooperative games and in the theory of general equilibrium with the aid of the famous KKM lemma, which is equivalent to the Brouwer fixed point theorem. This method is more interesting; but since we have used this method in the first paragraph, we prefer for convenience to use partitions of unity and fixed point theorems to establish the existence of scalar equilibria. Theorem 3.8.11. Let X be a Hausdorff real topological vector space, M a nonempty closed convex subset. Consider two real functions ϕ and ψ defined on M × M such that:

3.8 Vector equilibrium problems and related topics

175

(1) φ(x, x) ≤ 0 for each x ∈ M ; (2) for each x, y ∈ M , if φ(x, y) ≤ 0, then ϕ(x, y) ≤ 0; (3) for each y ∈ M , {x ∈ M ∩ K | ϕ(x, y) ≤ 0} is closed for every compact subset K of M ; (4) for each y ∈ M , {x ∈ M | φ(x, y) > 0} is convex. (5) (Compactness assumption) there exists a compact convex subset B of M such that for all y ∈ M \ B, there exists x ∈ B such that ϕ(x, y) > 0. Then, there exists an equilibrium point x ∈ B, i.e. ϕ(y, x) ≤ 0 for each y ∈ M . Proof. We proceed in two steps. 1. Case of M = B; i.e., M is assumed compact. Suppose that a solution does not exist. Then the subset M can be covered by the family of open subsets O(y) := {x ∈ M | ϕ(y, x) > 0}, where y ∈ M . Since M is assumed to be compact, let {O(yi )} be a finite subcover of M (we set Oi = O(yi )). Let {pi | i = 1, . . . , n} be a continuous partition of unity subordinate to this finite covering. We introduce the function p : M −→ M defined by p(u) :=

n

pi (u)yi .

i=1

This mapping is continuous; we obtain, by using Tychonoff’s fixed point theorem, the existence of some u0 ∈ M such that p(u0 ) = u0 . Let I be the set of index i ∈ {1, . . . , n} for which pi (u0 ) > 0; then u0 ∈ Oi for all i ∈ I. From (2), we obtain φ(yi , u0 ) > 0 for all i ∈ I. Therefore, by using assumption (4), we obtain n  

φ(u0 , u0 ) = φ(p(u0 ), u0 ) = φ pi (u0 )yi , u0 > 0, i=1

a contradiction with condition (1). Hence ∃ x ∈ M such that ϕ(y, x) ≤ 0 for each y ∈ M . 2. General case. For every y ∈ M , consider A(y) := {x ∈ B | ϕ(y, x) ≤ 0}. Let {y1 , . . . , ym } be a fixed finite subset of M , and let M0 be the convex hull of B and {y1 , . . . , ym }. Then M0 is compact, since B is convex and compact and X is a Hausdorff topological vector space. By the first step, there exists some x0 ∈ M0 such that ϕ(y, x0 ) ≤ 0 for each y ∈ M0 . Using assumption (5), we have x0 ∈ B. Thus x0 ∈ ∩1≤i≤m A(yi ). Hence every finite subfamily of the sets A(y) has nonempty intersection. Observe that (3) justifies that A(y) is compact for each y ∈ M , since B is compact. Thus by virtue of the finite intersection property in a compact set, we deduce that there exists x ∈ B that belongs to all the sets A(y) for y ∈ M . This means that (EP ) has a solution in B.


176

3 Optimization in partially ordered spaces

Vector problems Now we prepare the treatment of the vector problem. First, let us recall the following propositions and notions on an ordered vector space. Let Y be a Hausdorff l.c.s., and let Y ∗ denote its topological dual space. Let P ,P = Y , be a closed convex proper cone with nonempty interior, and P + := {y ∗ ∈ Y ∗ | y, y ∗  ≥ 0 ∀ y ∈ P } its dual cone. Since we suppose Y a Hausdorff locally convex space, int P = ∅ and P = Y , the dual cone P + has a convex w∗ -compact base B ∗ ; i.e., 0 ∈ B ∗ and P + = ∪t≥0 tB ∗ . As has been mentioned in Lemma 2.2.17, we can choose, for instance, B ∗ = {y ∗ ∈ P + | b, y ∗  = 1} for some arbitrary b ∈ int P . We shall fix such a w∗ -compact base B ∗ in what follows. Let us consider the sup-support and inf-support functions α(y) := min{y, y ∗  | y ∗ ∈ B ∗ } and β(y) := max{y, y ∗  | y ∗ ∈ B ∗ }. Let us remark that β(y) = −α(−y), and that the infimum (resp. supremum) of linear and continuous functions α is superlinear and upper semicontinuous (resp. β is sublinear and lower semicontinuous) on the space Y . Note that in Y , see Section 2.2, the order relations between two elements x, y ∈ Y can be defined and characterized by x ≤P y ⇔ x ∈ y − P ⇔ x − y, y ∗  ≤ 0 ∀ y ∗ ∈ P + ⇔ x − y, y ∗  ≤ 0 ∀ y ∗ ∈ B ∗ ⇔ β(x − y) ≤ 0 ⇔ α(y − x) ≥ 0; and x

0. To produce the negation of each created relation for vector orders, it suffices to use negation of equivalent ones, i.e., x P y ⇔ x − y ∈ −P ⇔ β(x − y) > 0 ⇔ α(y − x) < 0 and x ≮P y ⇔ x − y ∈ − int P ⇔ β(x − y) ≥ 0 ⇔ α(y − x) ≤ 0. Now for a given vector-valued mapping f from M × M into Y , we define the real-valued functions F (x, y) := β(f (x, y)).

(3.88)

3.8 Vector equilibrium problems and related topics

177

Note that each inequality for F can be transformed into vector relations for f as follows: f (x, y) ≤P 0 ⇐⇒ F (x, y) ≤ 0 and f (x, y)

0 and f (x, y) ≮P 0 ⇐⇒ F (x, y) ≥ 0. A direct application of the above equivalences between vector orders and real inequalities to Theorem 3.8.11 gives the following theorem. Theorem 3.8.12. Under the same setting of Theorem 3.8.11, we suppose in addition that Y is a Hausdorff locally convex topological vector space and P a closed convex cone with int P = ∅ and P = Y . Consider f, g : M × M −→ Y satisfying (H1
) g(x, x) ∈ − int P for all x ∈ M ; (H2
) f (x, y) ∈ − int P implies g(x, y) ∈ − int P for every x, y ∈ M ; (H3
) for every y ∈ M and every compact subset K of X, the subset {x ∈ M ∩ K | f (x, y) ∈ − int P } is closed in M ; (H4
) for every y ∈ M , the subset {x ∈ M | g(x, y) ∈ − int P } is convex; (H5
) there exists a compact subset B of M such that for every x ∈ M \ B, there exists some y ∈ B for which f (y, x) ∈ − int P . Then there exists x ∈ B such that f (y, x) ∈ − int P . 3.8.5 Some Knowledge About the Assumptions The conditions of Theorems 3.8.10 and 3.8.22 are not particularly restrictive. We will now show, in the following lemmas, how to use these assumptions. Lemma 3.8.13. Suppose that (a) g(x, x) ⊂ L(x) for all x ∈ M , (b) for every fixed y ∈ M , the set {x ∈ M | g(x, y) ⊂ L(y)} is convex. Then the hypothesis (H2) is satisfied. Proof. Suppose by contradiction that for each i  = 1, . . . , n there exist n n xi ∈ M n and λi ∈ [0, 1] such that i=1 λi = 1 and g(xi , j=1 λj xj ) ⊂ L( j=1 λj xj ) n for every i = 1, . . . , n. Assumption (b) shows that if z = j=1 λj xj , then g(z, z) ⊂ L(z). This contradicts (a), and the proposition is proved.
Lemma 3.8.14. Condition (b) of Lemma 3.8.13, namely convexity of the subset {x ∈ M | g(x, y) ⊂ L(y)}, is satisfied if one of the following conditions is satisfied: (i) L(y) is convex and g(·, y) is concave; i.e., for all u, v ∈ M and t ∈ [0, 1], g(tu + (1 − t)v, y) ⊂ tg(u, y) + (1 − t)g(v, y).

178

3 Optimization in partially ordered spaces

(ii) L(y) is a convex open cone and g(·, y) is quasi L(y)-concave on M ; i.e., for all u, v ∈ M and t ∈ [0, 1], g(tu + (1 − t)v, y) ⊂ conv(g(u, y) ∪ g(v, y)) + L(y). Proof. Under assumption (i), the condition is trivially satisfied. For the second one, it suffices to use the inclusion L(y) + L(y) ⊂ L(y).
Remark 3.8.15. One can confirm that g satisfies condition (ii) of Lemma 3.8.14 if one supposes that g satisfies one of the following conditions: (a) g(·, y) is L(y)-concave: for all u, v, y ∈ M and t ∈ [0, 1], g(tu+(1−t)v, y) ⊂ tg(u, y) + (1 − t)g(v, y) + L(y); (b) for all u, v, y ∈ M and t ∈ [0, 1], either g(tu + (1 − t)v, y) ⊂ g(u, y) + L(y) or g(tu + (1 − t)v, y) ⊂ g(v, y) + L(y). The proof of this assertion is trivial. Lemma 3.8.16. Suppose that the set {y ∈ M | f (x, y) ⊂ K(y)} is closed for every x ∈ M ; then assumptions (H3) and (H4) are satisfied. The proof of this lemma is immediate. Lemma 3.8.17. If K(y) = K is open and independent of y ∈ M , then the assumption (H3) is satisfied if f (x, ·) is K-upper continuous on the convex hull of every finite subset of M . Proof. Let F be the convex hull of a finite subset of M , and set U (x) = f (x, ·)+1 (K) ∩ F := {y ∈ F | f (x, y) ⊂ K}. Let us show that U (x) is open in F . Fix y0 ∈ U (x). Then K is an open set in Y that contains f (x, y0 ). Since f (x, ·) is K-upper continuous on F , see Definition 2.33, it follows that for some U0 ∈ VF (y0 ), ∀y ∈ U0 f (x, y) ⊂ K + K ⊂ K. Thus U0 ⊂ U (x), and the proof is complete.
When we suppose f (x, ·) to be upper continuous on the convex hull of every finite subset of M , then since K is an open subset of Y , one can immediately justify that f (x, ·)+1 (K) ∩ F is open in F . Lemma 3.8.18. The assumption (H3) is satisfied when we suppose that for every fixed x ∈ M , f (x, ·), with compact values, is upper continuous on the convex hull of every finite subset of M , and the multifunction K has an open graph in M × Y . Proof. Let us fix arbitrarily x ∈ M and F the convex hull of a finite subset A of M . We claim that M (x) := {u ∈ F | f (x, u) ⊂ K(u)} is closed in F . Indeed, consider a net {ui | i ∈ I} in M (x) that converges to some u ∈ F ; then for every i ∈ I, there exists some zi ∈ f (x, ui ) such that zi ∈ / K(ui ). Suppose first that the net {zi } converges, and let z be the limit. Since K has an open graph in M × Y , we deduce that z ∈ / K(u). We claim that z ∈ f (x, u); otherwise, from the Hausdorff property of Y there exists V ∈

3.8 Vector equilibrium problems and related topics

179

NX (f (x, u)) such that z ∈ / V . The multifunction f (x, ·) is assumed to be upper continuous and ui → u. Then there exists i0 ∈ I such that ∀i ∈ I, i  i0 ⇒ f (x, ui ) ⊂ V , and zi ∈ / V , which is a contradiction. Thus u ∈ M (x). Suppose now that the net {zi } doesn’t converge. Then for every z ∈ f (x, u) one can find an open set V (z) and I(z) ⊂ I such that z ∈ V (z) and ∀i ∈ I(z), zi ∈ / V (z). Since f (x, u) is compact and f (x, u) ⊂ ∪z∈f (x,u) V (z), there exist z1 , . . . , zn ∈ f (x, u) such that f (x, u) ⊂ V (z1 ) ∪ · · · ∪ V (zn ). If we take V0 = ∪1≤k≤n V (zi ) and I0 = ∩1≤k≤n I(zi ), then one has ∀i ∈ I0 , zi ∈ / V0 , which gives f (x, ui ) ⊂ V0 . This contradicts our assumption on upper continuity of f (x, ·) on F , and completes the proof.
Lemma 3.8.19. The assumption (H4) of Theorem 3.8.10 is equivalent to the following: (H8) For every x, y ∈ M and (yα ) ⊂ M converging to y, then f (tx + (1 − t)y, yα ) ⊂ K(yα ) ∀ t ∈ [0, 1] implies f (x, y) ⊂ K(y). Proof. Let us first suppose (H4). Consider x, y ∈ M and (yα ) a net on M that converges to y. By taking D = conv{x, y} and using (H4), we have f (z, y) ⊂ K(y) for every z ∈ D; in particular, z = x does the job. For the converse, let A ∈ F(M ) and (yα ) a net on M converging to y ∈ conv(A). Suppose that f (z, yα ) ⊂ K(yα ) for all α and all z ∈ conv(A). Fix z ∈ conv(A). Then f (tz + (1 − t)y, yα ) ⊂ K(yα ) for all α and all t ∈ [0, 1], since tz + (1 − t)y ∈ conv(A). Using (H8), we deduce that f (z, y) ⊂ K(y). This is true for each z ∈ conv(A), and so (H4) is satisfied.
Consider the following problems: (I) find x ∈ M such that f (y, x) ⊂ K(x) for all y ∈ M ; and (II) find x ∈ M such that f (x, y) ⊂ L(x) for all y ∈ M . Lemma 3.8.20. (Generalized Minty’s Linearization) (i) If x solves Problem (II), then x is a solution of Problem (I) whenever the pair (f, g) is (K, L)-monotone with g(x, y) = f (y, x) for every x, y ∈ M . (ii) If x solves Problem (I), then x is a solution of Problem (II) whenever the following assumptions are satisfied: (1) f (x, x) ⊂ L(x) for all x ∈ M ; (2) if x, y ∈ M, x = y, then u ∈]x, y[ and f (u, y) ⊂ L(x) implies f (u, x) ⊂ K(x); (3) for all x, y ∈ M , the set {v ∈ [x, y] | f (v, y) ⊂ L(x)} is open in [x, y]. Proof. (i) The first assertion is derived from (K, L)-monotonicity of the pair (f, g); let us prove (ii). Assume, for contradiction, that f (x, z) ⊂ L(x) for some z ∈ M . By assumption (1), we have z = x. According to assumption (3), there exists some u ∈]x, z[ such that f (u, z) ⊂ L(x). This implies, see assumption (2), f (u, x) ⊂ K(x). Since f (u, x) ⊂ K(x), we obtain a contradiction.


180

3 Optimization in partially ordered spaces

Remark 3.8.21. This lemma remains true if we suppose instead of (2) the following: 1. if x = y ∈ M , then u ∈]x, y[, f (u, x) ⊂ K(x), and f (u, y) ⊂ L(x) imply that f (u, v) ⊂ K(x) for all v ∈]x, y]; 2. f (x, x) ⊂ K(y) for all x, y ∈ M . 3.8.6 Some Particular Cases Let us single out some particular cases of Theorem 3.8.10. First, Theorem 3.8.10 and Lemma 3.8.13 give the following result: Theorem 3.8.22. In addition to the assumptions (H1) and (H2) in Theorem 3.8.10, assume that M is closed and the following assumptions (H6) and (H7). Then the conclusion of Theorem 3.8.10 remains true. (H6) For every fixed x ∈ M and every convex compact subset K of X, the set {y ∈ K ∩ M | f (x, y) ⊂ K(y)} is closed in K ∩ M ; (H7) there is a convex compact subset B of X such that for all y ∈ M \ B, there exists x ∈ M ∩ B for which f (x, y) ⊂ K(y). Proof. Let us consider the multifunction S defined, for each y ∈ M , by S(y) := {x ∈ M ∩ B | f (y, x) ⊂ K(x)}. Let A ∈ F(M ) and consider M0 the convex hull of (B ∩ M ) ∪ A. Since B is compact and X is a Hausdorff topological vector space, M0 is compact. Note that the multifunction f from M0 × M0 into Y satisfies all assumptions of Theorem 3.8.10. Then there exists x1 ∈ M0 such that f (y, x1 ) ⊂ K(x1 ) for every y ∈ M0 . Using compactness assumption (H7), we have x1 ∈ B, and therefore x1 ∈ ∩y∈A S(y). We conclude that every finite-intersection subfamily of the set {S(y) | y ∈ M } has nonempty intersection. Observe that (H6) justifies that S(y) is compact for each y ∈ M . Thus by virtue of the finite-intersection property in a compact set, we deduce that there exists x ∈ B that belongs to the set S(y) for all y ∈ M . This means that x is a suitable solution in B.
Remark 3.8.23. It is clear that the condition (H7) of Theorem 3.8.22 is weaker than the corresponding condition (H5) of Theorem 3.8.10. Nevertheless, we restrict ourselves to the more stringent compactness condition in (H6). Theorem 3.8.22 and Lemma 3.8.13 give the following result: Theorem 3.8.24. Under the same setting of Theorem 3.8.22, we suppose that (H1), (H7) are satisfied, and (i) g(x, x) ⊂ L(x) for all x ∈ M ; (ii) for every fixed y ∈ M , the set {x ∈ M | g(x, y) ⊂ L(y)} is convex; (iii) for every fixed x ∈ M , the set {y ∈ M | f (x, y) ⊂ K(y)} is closed.

3.8 Vector equilibrium problems and related topics

181

Then we get the same conclusion of Theorem 3.8.10. To consider another particular case, we assume that K(x) = L(x) and f (x, y) = g(x, y) for all x, y ∈ M . Then, from Theorem 3.8.10 and Lemma 3.8.13 we obtain the following theorem: Theorem 3.8.25. Assume that (i) f (x, x) ⊂ K(x) for all x ∈ M ; (ii) for every fixed y ∈ M , the set {x ∈ M | f (x, y) ⊂ K(y)} is convex; (iii) for every fixed x ∈ M , the set {y ∈ M | f (x, y) ⊂ K(y)} is closed; (iv) there is a compact subset B of X and x0 ∈ M ∩ B such that f (x0 , y) ⊂ K(y) for all y ∈ M \ B. Then we get the same conclusion of Theorem 3.8.10. Remark 3.8.26. This theorem contains as a special case, when f is a singlevalued function, Lemma 2.1 from [228]. Remark 3.8.27. When K(x) = R+ for all x ∈ M and f (x, y) = h(x, y) − supx∈M h(x, x), we can state a version of the celebrated 1972 Fan’s minimax inequality; see [111, Theorem 1] and [11, Theorem 8.5]. Theorem 3.8.28. Under the same setting of Theorem 3.8.10. Assume that the multifunction f : M × M → Y satisfies (1) f (x, y) ⊂ − int P (x) implies f (y, x) ⊂ int P (x) for all x, y ∈ M ; (2) f (x, x) ⊂ − int P (x) for all x ∈ M ; (3) for every fixed x ∈ M , f (x, ·) satisfies condition (ii) of Lemma 3.8.14; (4) for every x ∈ M and A ∈ F(M ), the mapping f (x, ·) is upper continuous on conv(A); (5) for every x ∈ M and every A ∈ F(M ) whenever y ∈ conv(A) and (yα ) ⊂ M converges to y, then f (tx + (1 − t)y, yα ) ⊂ int P (yα ) ∀ α, ∀ t ∈ [0, 1] implies f (x, y) ⊂ int P (y) ; (6) there is a compact set B ⊂ X and x0 ∈ M ∩ B such that f (x0 , y) ⊂ int P (y) ∀y ∈ M \ B; (7) if x = y ∈ M and u ∈]x, y[, then f (u, x) ⊂ int P (x) or f (u, y) ⊂ − int P (x); (8) for all x, y ∈ M , the set {v ∈ [x, y] | f (v, y) ⊂ − int P (x)} is open in [x, y]. Then there exists x ∈ M such that f (x, y) ⊂ − int P (x) for all y ∈ M . Proof. All conditions of Theorem 3.8.10 are satisfied by assumptions (1)– (6) if we set K(x) = −L(x) = int P (x) and g(x, y) = f (y, x). Indeed, (H1) and (H5) follow from (1) and (6); combining Lemmas 3.8.13, 3.8.14 with assumptions (2)–(3) we get (H2); using Lemma 3.8.17 and assumption (3) we obtain (H3); and (H4) follows from Lemma 3.8.19 and assumption (5). Therefore there exists x ∈ M such that f (y, x) ⊂ K(x) ∀ y ∈ M . Then we

182

3 Optimization in partially ordered spaces

apply Lemma 3.8.20 to say that assumptions (1), (7), (8) lead to f (x, y) ⊂ L(x) ∀ y ∈ M .
A generalized Minty’s linearization lemma for single-valued functions can be formulated as follows. Lemma 3.8.29. Suppose that the single-valued function f is P -pseudomonotone; i.e., for all x, y ∈ M , f (x, y) ∈ − int P implies f (y, x) ∈ int P ; then f (x, y) ∈ − int P ∀ y ∈ M =⇒ f (y, x) ∈ int P ∀ y ∈ M. The converse in true whenever for every x ∈ M , f (x, ·) satisfies condition (ii) of Lemma 3.8.14, f (·, x) is P -lower continuous on every line segment in M , and f (x, x) ∈ P . Proof. As mentioned previously, the proof of the first statement is straightforward. Let us prove the second one. Fix y ∈ M and set yt := ty + (1 − t)x, which is in M for every t ∈ [0, 1]. Thus f (yt , x) ∈ / int P, ∀ t ∈ [0, 1].

(3.89)

Taking into account condition (ii) of Lemma 3.8.14 and Remark 3.8.15, we get for some s ∈ [0, 1[, −f (yt , yt ) + (sf (yt , x) + (1 − s)f (yt , y)) ∈ P.

(3.90)

Combining relations (3.89), (3.90), and f (yt , yt ) ∈ P , and using Y \int P −P ⊂ Y \ int P , we have (1 − s)f (yt , y) ∈ / − int P ⇔ f (yt , y) ∈ / − int P.

(3.91)

We deduce that f (x, y) ∈ / − int P ; otherwise, since f (·, y) is P -lower continuous on [x, y[ and V = −f (x, y) − int P ∈ VY , one can find t near zero such that f (yt , y) ∈ f (x, y) + V − P = f (x, y) − (f (x, y) + int P ) − P ⊂ − int P, which contradicts (3.91); and the result follows.




Just after this lemma we shall give a theorem of existence of vector equilibria for vector single-valued functions. Let us set g(x, y) = f (y, x), K(x) = int P and L(x) = − int P for all x, y ∈ M , where P is a closed convex and w-pointed cone. Then by using Theorem 3.8.24, Remark 3.8.15 and Lemmas 3.8.13, 3.8.14, 3.8.17, and 3.8.29, we can state the following theorem: Theorem 3.8.30. If f : M × M → Y satisfies (1) f is P -pseudomonotone; i.e., f (x, y) ∈ − int P implies f (y, x) ∈ int P ; (2) f (x, x) = 0 for all x ∈ M ;

3.8 Vector equilibrium problems and related topics

183

(3) for every x ∈ M , f (x, ·) is P -upper continuous and quasi P -concave on every convex compact subset of M ; (4) for every y ∈ M , f (·, y) is P -lower continuous on every line segment in M; (5) there exists a convex compact subset B of X such that for all y ∈ M \ B, there is x ∈ M ∩ B for which f (x, y) ∈ int P . Then there exists x ∈ M such that f (x, y) ∈ − int P for all y ∈ M . 3.8.7 Mixed Vector Equilibrium Problems In this section, we establish results for vector equilibrium problems where the monotone criterion mappings are supposed to be perturbed by nonmonotone mappings. Definition 3.8.31. Let X, Y be two Hausdorff real topological vector spaces, f a mapping from M × M to Y , P a convex closed cone of Y , and P + its dual cone. The mapping f is said to be P -pseudomonotone in the topological sense whenever (xi ) is a net on M converging to x ∈ M such that ∀V ∈ VY , ∃αx satisfying ∀i  αx , (f (xi , x) − V ) ⊂ f (x, x) − int P , then ∀y ∈ M and ∀W ∈ VY , ∃αxy such that ∀i  αxy , (f (xi , y) − W ) ∩ (f (x, y) − P ) = ∅. In the case where Y = R and P = R+ , the definition of P -pseudomonotonicity in the topological sense coincides with the classical Br´ezis– Browder pseudomonotonicity: M

xi → x and lim inf i f (xi , x) ≤ f (x, x) ⇒ lim supi f (xi , y) ≥ f (x, y) ∀y ∈ M . Observe that if f (·, y) is upper P -semicontinuous for all y ∈ M , then f is P -pseudomonotone in the topological sense. We give now the following result concerning a vector equilibrium problem with a nonmonotone perturbation of a P -monotone mapping. Theorem 3.8.32. Let X, Y be two real Hausdorff topological vector spaces, M a nonempty convex subset of X, and let P be a closed convex w-pointed cone in Y with a nonempty interior. Consider two mappings f and g from M × M to Y such that (1) g is −P -monotone; (2) for every x ∈ M , g(x, x) ∈ P and f (x, x) = 0; (3) for every x ∈ M , f (x, ·) and g(x, ·) are P -convex; (4) for every x ∈ M , g(x, ·) is P -lower continuous; (5) for every y ∈ M , f (·, y) is P -upper continuous on the convex hull of every finite subset of M ; (6) f is P -pseudomonotone in the topological sense; (7) for every y ∈ M , g(·, y) is P -upper continuous on each line segment in M ; (8) there is a compact subset B of X and x0 ∈ M ∩ B such that ∀y ∈ M \ B, g(x0 , y) − f (y, x0 ) ∈ int P .

184

3 Optimization in partially ordered spaces

Then there exists x ∈ M such that f (x, y) + g(x, y) ∈ − int P ∀y ∈ M , which is equivalent to f (x, y) − g(y, x) ∈ − int P ∀y ∈ M . Proof. Set ϕ(x, y) = g(x, y) − f (y, x), φ(x, y) = f (y, x) + g(y, x) and K(x) = int P and L(x) = − int P for all x ∈ M . We begin by proving that all assumptions of Theorem 3.8.10 are satisfied. (H1): Let x, y ∈ M with g(x, y) − f (y, x) ∈ int P ; then f (y, x) + g(y, x) = f (y, x) − g(x, y) + g(x, y) + g(y, x) ∈ g(x, y) + g(y, x) − int P. From assumption (1), we conclude that f (y, x) + g(y, x) ∈ −P − int P ⊂ − int P ; this is our claim. (H2): Let x1 , x2 , y ∈ M . Then by P -convexity of f (y, ·) and g(y, ·), for every t ∈]0, 1[ one has g(y, tx1 + (1 − t)x2 ) + f (y, tx1 + (1 − t)x2 ) ∈ t [g(y, x1 ) + f (y, x1 )] + (1 − t) [g(y, x2 ) + f (y, x2 )] − P. Since − int P is convex and −P − int P ⊂ − int P , we conclude that {x ∈ M : φ(x, y) ∈ − int P } is convex for all y ∈ M . (H3): Since g(x, ·) and −f (·, x) are P -lower continuous on the convex hull of every finite subset of M for all fixed x ∈ M , then so is −f (·, x) + g(x, ·). Proposition 2.5.23(ii) means that (H3) is satisfied. (H4): Let x, y ∈ M and {yi } ⊂ M , which converges to y. Suppose that for every i, g(tx + (1 − t)y, yi ) − f (yi , tx + (1 − t)y) ∈ / int P ∀t ∈ [0, 1].

(3.92)

Suppose, contrary to our claim, that g(x, y) − f (y, x) ∈ int P. Hence int P is a neighborhood of g(x, y)−f (y, x) in Y , and then we can find two neighborhoods V1 , V2 ∈ VY (0) such that (g(x, y) + V1 ) − (f (y, x) + V2 ) ⊂ int P . Since g(x, ·) is P -lower continuous, there exists an index αxy such that g(x, yi ) ∈ g(x, y) + V1 + P, ∀i  αxy .

(3.93)

For t = 0 in (3.92), one has g(y, yi ) − f (yi , y) ∈ / int P.

(3.94)

Let V ∈ VY . Since g(y, ·) is P -lower continuous, there exists αy such that g(y, yi ) ∈ g(y, y) + V + P for any i  αy . Combining this with assumption (2), the relation (3.94) yields (f (yi , y) − V ) ⊂ − int P ∀i  αy .
By topological P -pseudomonotonicity of f , for V2 there exists αxy such that

3.8 Vector equilibrium problems and related topics
f (yi , x) ∈ f (y, x) + V2 − P ∀i  αxy .

185

(3.95)


) and combining relations (3.93) and (3.95), we Setting α0 = max(αxy , αy , αxy deduce that ∀i  α0 ,

g(x, yi ) − f (yi , x) ∈ (g(x, y) + V1 ) − (f (y, x) + V2 ) + P ⊂ int P + P ⊂ int P, which contradicts (3.92) for t = 1. Thus (H4) is satisfied. All assumptions (H1)–(H5) of Theorem 3.8.10 are satisfied, so we deduce that there exists x ∈ M such that / int P for all y ∈ M. g(y, x) − f (x, y) ∈

(3.96)

Let y ∈ M be a fixed point and set yt = ty + (1 − t)x for t ∈ (0, 1). Since g(yt , ·) is P -convex and g(yt , yt ) ∈ P , then tg(yt , y) + (1 − t)g(yt , x) ∈ P.

(3.97)

/ int P . Combining From relation (3.96), one has (1−t)g(yt , x)−(1−t)f (x, yt ) ∈ with (3.97), it follows that tg(yt , y) + (1 − t)f (x, yt ) ∈ / − int P . Using the convexity of f (x, ·), we can write tg(yt , y) + t(1 − t)f (x, y) + (1 − t)2 f (x, x) ∈ / / − int P because f (x, x) = 0. The − int P , or else g(yt , y) + (1 − t)f (x, y) ∈ P -upper continuity of g(·, y) on line segments allows us to write / − int P. g(x, y) + f (x, y) ∈ The converse follows immediately from −P -monotonicity of g, and the proof is complete.
Remark 3.8.33. The P -monotonicity of g can be replaced by the following (f, P )-pseudomonotonicity: For all x, y ∈ M g(x, y) + f (x, y) ∈ int P ⇒ g(y, x) − f (x, y) ∈ int P. In that case, Theorem 3.8.32 appears as a generalization of Theorem 5.1 in [28]. Remark 3.8.34. The conclusion of Theorem 3.8.32 is valid (see Theorem 3.8.22) when instead of assumptions (5), (6), and (8) we only suppose that (8
) there is B ⊂ X convex and compact for which ∀y ∈ M \ B, ∃x ∈ M ∩ B such that g(x0 , y) − f (y, x0 ) ∈ int P ; in return, we must suppose that (5
) for every y ∈ M , f (·, y) is P -lower continuous on every convex compact subset of M . In the next section we present some typical situations in which the vector equilibrium structure occurs quite naturally.

186

3 Optimization in partially ordered spaces

3.9 Applications to Vector Variational Inequalities As a first example of particular case of the (GVE) problem, let us consider the following generalized vector variational inequality: / − int P (x). (GVVI) find x ∈ K s.t. ∀ y ∈ K, ∃ t ∈ T (x) : t(η(x, y)) ∈ Here T is a multifunction from X into the space L(X, Y ) of all linear continuous mappings from X into Y , η : K × K → K is a mapping, and t(x) denotes the evaluation of the linear mapping t at x. Thus (GVVI) is a particular case of the (WGVEP) problem if we take f (x, y) = T (x)(η(x, y)). This problem is an extension of the single-valued (GVVI) problem introduced by Siddiqi, Ansari, and Khaliq [322]. Note that for η(x, y) = y − g(x), (GVVI) is equivalent to the vector variational problem introduced and studied by Konnov and Yao [215]: / − int P (x) ∀ y ∈ K. (VVI) find x ∈ K such that T (x)(y − g(x)) ∈ When P (x) = P for all x ∈ K, the (GVVI) problem becomes the vector variational inequality problem introduced by Chen and Yang [75]. If Y = R, X = Rn , L(X, Y ) = X ∗ , P (x) = R+ ∀ x ∈ K, and g is the identity mapping, the problem (GVVI) becomes the usual scalar variational inequality considered and studied by Hartman and Stampacchia [159]. The vector variational inequalities (VVI) were introduced by Giannessi [132] in a finite-dimensional Euclidean space in 1980. Chen and several authors (see [65], [75], [66], [67], [383], [229], [64]) have intensively studied the vector variational inequalities in abstract infinite-dimensional spaces. Recently, the equivalence between a (VVI) and vector optimization problems and the equivalence between (VVI) and vector complementarity problems have been studied; see e.g., [75]. 3.9.1 Vector Variational-Like Inequalities We assume: • X and Y are two Hausdorff topological vector spaces; • L(X, Y ) is the set of all continuous linear operators from X into Y ; • M ⊆ X is a nonempty convex set; • {P (x) | x ∈ M } is a family of closed convex cones in Y with int P (x) = ∅ for all x ∈ M ; • η : M × M → M is continuous and affine with η(x, x) = 0, for every x ∈ M. For Π ⊆ L(X, Y ) we write Π(x) := {π(x) | π ∈ Π}, for all x ∈ M . We have to assume that L(X, Y ) is topologized in such a way that the bilinear form (π, x) → π(x) is continuous from L(X, Y ) × M into Y ; see the next lemma for some examples.

3.9 Applications to vector variational inequalities

187

Lemma 3.9.1. Let X, Y be two Banach spaces, M a closed convex subset of X, and let {(αi , xi )} be a net in L(X, Y ) × M , and {(α, x)} in L(X, Y ) × M . w

(i) If αi − α → 0 in L(X, Y ) and xi  x in X, then {αi (xi )} converges weakly to α(x) in Y . (ii) If (αi − α)(u), y ∗  → 0 for every u ∈ X, y ∗ ∈ Y ∗ and xi − x → 0 in X, then {αi (xi )} converges weakly to α(x) in Y . (iii) If αi − α → 0 in L(X, Y ) and xi − x → 0 in X, then αi (xi )} − α(x) → 0 in Y . Proof. (i) Fix y ∗ ∈ Y ∗ . We have |(αi − α)(xi ), y ∗ | ≤ y ∗ · αi − α · xi . w

Our assumption xi  x in X just says that x∗ (xi ) → x∗ (x) for all x∗ ∈ X ∗ . Then, according to [100, Theorem II.3.20], {xi } is bounded in X, and thus lim sup|αi (xi ) − α(x), y ∗ | i

≤ lim sup |αi (xi ) − α(xi ), y ∗ | + lim sup |α(xi ) − α(x), y ∗ | i ∗

i

≤ y · sup xi · lim sup αi − α i

i

+ lim sup |xi − x, α∗ y ∗ | (α∗ is the adjoint operator of α) i

≤ 0. This shows that {αi (xi )} converges weakly to α(x) in Y . (ii) Using [100, Corollary II.3.21] and supi αi (u), y ∗  < +∞ for every u ∈ X, y ∗ ∈ Y ∗ , we have, that {αi } is bounded in L(X, Y ), and lim sup|αi (xi ) − α(x), y ∗ | i

≤ lim sup |αi (xi ) − αi (x), y ∗ | + lim sup |(αi − α)(x), y ∗ | i ∗

i

≤ y · sup αi · lim sup xi − x + lim sup |(αi − α)(x), y ∗ | i

i

i

≤ 0. (iii) is obvious.




Definition 3.9.2. Let T be a multifunction from M to L(X, Y ). (1) T is said to be (P, η)-monotone if T y(η(x, y))−T x(η(x, y)) ⊂ P (x)∩P (y). (2) T : M → L(X, Y ) is said to be (P, η)-pseudomonotone if T x(η(x, y)) ⊂ − int P (y) implies T y(η(x, y)) ⊂ − int P (y). (3) T is said to be V-hemicontinuous if for any x, y ∈ M T (tx+(1−t)y) → T y as t → 0+ (i.e., ∀zt ∈ T (tx+(1−t)y) ∃z ∈ T y such that ∀a ∈ M, zt (a) → z(a) as t → 0+ ).

188

3 Optimization in partially ordered spaces

Then, according to Theorem 3.8.32, we can formulate the following existence result concerning a vector variational-like inequality. Theorem 3.9.3. Let T be a multifunction from M to L(X, Y ). If one has (i) T is compact-valued, (P, η)-pseudomonotone and V-hemicontinuous; (ii) for some convex compact subset B of X, we have for every y ∈ M \ B there exists x ∈ M ∩ B such that T x0 (η(x0 , y)) ⊂ − int P (y). Then there exists x ∈ B satisfying

T x(η(y, x)) ⊂ − int P (x) ∀y ∈ M .

Proof. Set f (x, y) = −T x(η(x, y)), g(x, y) = −T y(η(x, y)), and K(x) = int P (x) for x, y ∈ M . Then assumptions (H1), (i), (ii), and (H7) of Theorem 3.8.24 are easily satisfied. Let us show that (H6) of Theorem 3.8.24 is also satisfied. For this, let x ∈ M be fixed and let (yi ) be a net on M converging to y ∈ M such that T x(η(x, yi )) ⊂ − int P (yi ). Therefore there exists zi ∈ T x satisfying zi (η(x, yi )) ∈ / − int P (yi ). Since T x is compact, then, passing to a subnet if necessary, we may assume that zi converges to z ∈ T x. By the continuity of the mapping η and the bilinear form on L(X, Y ) × M , we get zi (η(x, yi )) → z(η(x, y)). Hence z(η(x, y)) ∈ / − int P (y) or else T x(η(x, y)) ⊂ − int P (y). Then all assumptions of Theorem 3.8.24 are satisfied; thus, there exists x ∈ M such that T y(η(y, x)) ⊂ − int P (x) for all y ∈ M. Let y ∈ M be fixed and set yt = ty +(1−t)x. Then T yt (η(yt , x)) ⊂ − int P (x). Hence T yt (η(y, x)) ⊂ − int P (x). By the V-hemicontinuity of T and the closed
ness of Y \ (− int P (x)), it follows that T x(η(y, x)) ⊂ − int P (x). 3.9.2 Perturbed Vector Variational Inequalities Now, if we take P (x) = P for all x ∈ M , then we obtain from Theorem 3.8.32 the following proposition: Proposition 3.9.4. Let ϕ : M → Y be a P -lower continuous and P -convex function, and let S, T : M −→ L(X, Y ) satisfy the following assumptions: (i) S is continuous on M ; (ii) T is P -monotone and continuous on each line segment on M ; (iii) there is a compact subset B of X and x0 ∈ M ∩ B such that (Sy + T x0 )(y − x0 ) + ϕ(y) − ϕ(x0 ) ∈ int P

for all y ∈ M \ B.

Then there exists x ∈ M such that (S + T )x(y − x) + ϕ(y) − ϕ(x) ∈ / − int P for all y ∈ M.

3.9 Applications to vector variational inequalities

189

Proof. It suffices to set f (x, y) = Sx(y − x) and g(x, y) = T x(y − x) + ϕ(y) − ϕ(x) and to check that all assumptions of Theorem 3.8.32 are satisfied. We observe that assumptions (1)–(3) and (8) arise automatically from P -convexity of ϕ, P -monotonicity of T , and condition (iii). Using P -lower continuity of ϕ, (i)–(ii), and the condition on the bilinear form on L(X, Y ) × M , one deduces (4)–(7).
3.9.3 Hemivariational Inequality Systems In 1981, by using the generalized Clarke’s gradient, Panagiotopoulos introduced the notion of nonconvex super-potential. In the absence of convexity, a new type of inequality was obtained, the so-called hemivariational inequalities. These look like a variational formulation of mechanic problems. Let us take a brief look at this concept of nonsmooth analysis. Let X be a Banach space, ϕ : X → R, and x ∈ X. We say that ϕ is locally Lipschitz at x if there is some ε > 0 and k > 0 such that

u − x < ε and v − x < ε =⇒ |ϕ(u) − ϕ(v)| ≤ k u − v . The Clarke’s generalized directional derivative of ϕ at x in the direction h ∈ X is defined by ϕ(y + th) − ϕ(y) ϕo (x; h) := lim sup , y→x t t0

and the associated subdifferential is defined at x by ∂ϕ(x) := {x∗ ∈ X ∗ | ϕo (x; h) ≥ h, x∗  ∀h ∈ X}. Proposition 3.9.5. [81, Prop. 1.1 and Prop. 1.5] Let ϕ : X → R be locally Lipschitz of rank k at x. Then (i) the function h −→ ϕo (x; h) is finite, positively homogeneous, subadditive on X, and for every h ∈ X, |ϕo (x; h)| ≤ k h ; (ii) the function (y, h) −→ ϕo (y; h) is upper semicontinuous at (x, h) for every h ∈ X; (iii) ∂ϕ(x) is nonempty, convex, weak∗ -compact, and x∗ ≤ k for every x∗ ∈ ∂ϕ(x); (iv) for every h ∈ X we have ϕo (x; h) = max{h, x∗  | x∗ ∈ ∂ϕ(x)}. Definition 3.9.6. We say that a multifunction R from X to X ∗ is (Browder– Hess) BH-pseudomonotone, (quasi-pseudomonotone), or (of class (S)+ ) if (xn ) converges weakly to x and lim inf n→∞ supx∗n ∈Rxn x−xn , x∗n  ≥ 0 imply lim supn→∞ supx∗n ∈Rxn y − xn , x∗n  ≤ supx∗ ∈Rx y − x, x∗  ∀y ∈ X, (limn→∞ supx∗n ∈Rxn x − xn , x∗n  = 0), or (xn → x and (Rxn ) is bounded for n sufficiently large).

190

3 Optimization in partially ordered spaces

Proposition 3.9.7. Consider two multifunctions R1 and R2 from X to X ∗ . Then R = R1 + R2 is BH-pseudomonotone if either R1 or R2 is BH-pseudomonotone or if one of the pair (R1 , R2 ) is the Clarke’s subdifferential ∂ϕ of a locally Lipschitz function ϕ that is assumed quasi-pseudomonotone and the other one is of class (S)+ . Proof. Let us suppose that xn  x and lim inf n→∞ supx∗n ∈Rxn x−xn , x∗n  ≥ 0. Then   sup x − xn , u∗n  +

lim inf n→∞

u∗ n ∈R1 xn

sup x − xn , vn∗ , 

∗ ∈R x vn 2 n

≥ 0.

(3.98)

We claim that lim inf

sup x − xn , u∗n  ≥ 0 and lim inf

n→∞ u∗ ∈R1 xn n

sup x − xn , x∗n  ≥ 0. (3.99)

n→∞ v ∗ ∈R2 xn n

To the contrary, suppose, for instance, that the first is false. Then for a subsequence r := limp→∞ supu∗n ∈R1 xnp x − xnp , u∗np  < 0. Using (3.98), it follows p

lim inf

sup

p→∞ v ∗ ∈R2 xn np p

x − xnp , vn∗ p  ≥ −r > 0.

(3.100)

• If R2 is BH-pseudomonotone, then limp→∞ supvn∗ ∈R2 xnp x−xnp , vn∗ p  = p 0, a contradiction. Thus (3.99) is satisfied. We conclude from BH-pseudomonotonicity of R1 and R2 that ∀y ∈ X, lim sup sup y − xn , x∗n  n→∞ x∗ n ∈Rxn

≤ lim sup

sup y − xn , u∗n  + lim sup

n→∞ u∗ n ∈R1 xn

sup y − xn , vn∗ 

∗ ∈R x n→∞ vn 2 n

≤ sup y − x, u∗  + sup y − x, v ∗  u∗ ∈R1 x

v ∗ ∈R2 x

∗

= sup y − x, x . x∗ ∈Rx

This gives BH-pseudomonotonicity of R. • If R2 is of class (S)+ , then (3.100) gives xnp → x and (R2 xnp ) is bounded. Thus limp→∞ supvn∗ ∈R2 xnp x − xnp , vn∗ p  = 0, which contradicts (3.100) and p establishes (3.99). Using that R2 is of class (S)+ , then xn → x and (R2 xn ) is bounded. We conclude that ∀y ∈ X, lim supn→∞ supvn∗ ∈R2 xn y−xn , vn∗  ≤ sup y−x, v ∗ . v ∗ ∈R2 x

Using Proposition 3.9.5 (ii) (iv), we deduce

3.9 Applications to vector variational inequalities

191

lim sup sup y − xn , x∗n  n→∞ x∗ n ∈Rxn

≤ lim sup ϕ0 (xn , y − xn ) + lim sup

sup y − xn , vn∗ 

∗ ∈R x n→∞ vn 2 n

n→∞

≤ ϕ0 (x, y − x) + sup y − x, v ∗  v ∗ ∈R2 x ∗

= sup y − x, u  + sup y − x, v ∗  u∗ ∈R1 x

v ∗ ∈R2 x

∗

= sup y − x, x . x∗ ∈Rx

• The same proof remains valid when we suppose that R2 = ∂ϕ is quasi
pseudomonotone and R1 is of class (S)+ . As the maximal monotone operators for the variational inequalities, the generalized pseudomonotone operators are a mathematically important tool for the formulation of existence results concerning the hemivariational mappings. In this subsection, we are interested in some hemivariational inequality systems. Assume that we are given X a real reflexive Banach space, M a nonempty convex subset of X. Let Sk , Tk : M → X ∗ , for k = 1, . . . , m, and let J : X → Rm be locally Lipschitz near M . Let us introduce the following mappings: S,T : M −→ X ∗ × X ∗ × · · · × -m m X ∗ defined by Sx := k=1 Sk x and T x := k=1 Tk x, and set J o (x; h) := o (J1o (x; h), . . . , Jm (x; h)). We consider the following hemivariational inequality system (HVIS), which consists in finding x ∈ M such that ∀ y ∈ M , ∃ky ∈ {1, . . . , m} with y − x, Sky x + y − x, Tky x + Jkoy (x; y − x) ≥ 0.

(3.101)

By making use of Theorem 3.8.32 and Remark 3.8.34 we are able to give conditions that guarantee the existence of a solution to the inequality (HVIS). Theorem 3.9.8. Let us impose the following conditions: (i) for every k = 1, . . . , m, Sk is BH-pseudomonotone and locally bounded; (ii) for every k = 1, . . . , m, either ∂Jk is BH-pseudomonotone, or ∂Jk is quasipseudomonotone and S is of class (S)+ ; (iii) for every k = 1, . . . , m, Tk is monotone and continuous from each line segment of M to the weak topology on X ∗ ; (iv) there is a weak-compact subset B ⊂ X such that for all y ∈ M \ B there exists x ∈ M ∩ B such that ∀ k ∈ {1, . . . , m} Jko (y, x − y) < y − x, Sk y + T x. Then the problem (HVIS) has at least one solution. Proof. Let us first remark that setting, for each k ∈ {1, . . . , m} and x, y ∈ M , fk (x, y) := y − x, Sk x + max y − x, ξ and gk (x, y) := y − x, Tk x, ξ∈∂Jk (x)

192

3 Optimization in partially ordered spaces

the system (3.101) is equivalent to f (x, y) + g(x, y) ∈ / − int Rm + ∀ y ∈ M. Let Y = Rm and P = Rm + . We shall verify the conditions of Theorem 3.8.32 when X and X ∗ are respectively endowed with the weak and weak∗ topologies: (1), (2), (3), (4), and (8) are obviously satisfied; (6) holds by using Propositions 3.9.7 and 3.9.5 (iv), since J and S satisfy conditions (i) and (ii). (5): Fix y ∈ M and F a convex hull of a finite subset of M . We have to show that for every k = 1, . . . , m, f (·, y) is Rm + -lower continuous on F , which is equivalent to fk (·, y) is R+ -upper continuous on F ; i.e., fk (·, y) is R+ -upper semicontinuous1 . Let {xn } be a sequence in F weak-converging to x0 ∈ F . Since F is a closed subset of a finite-dimensional space, we assert that {xn } converges in norm to x0 . By the upper semicontinuity of Jko on M × X, see Proposition 3.9.5 (ii), and the local boundedness of Sk we obtain that f (·, y) is R+ -upper semicontinuous at x0 . (7): holds by using a similar argument to that used in proving (5).
Remark 3.9.9. The hemivariational inequalities studied in Chapter 4 of [267] correspond to the case where k = 1 and for all x ∈ M, T x = l ∈ X ∗ . Remark 3.9.10. The study of variational–hemivariational inequalities is a particular case of the study of the inequalities system (3.101). It corresponds to the case in which k = 1 and y − x, T x = ϕ(y) − ϕ(x) − y − x, l for all x, y ∈ M, with l ∈ X ∗ and ϕ is a real lower semicontinuous convex function on M . Therefore, a solution of this inequality exists without recourse to a condition of quasi- or strong quasi-boundedness on ∂ϕ, as was made in [267]. 3.9.4 Vector Complementarity Problems To introduce a vector complementarity problem, let X, Y be two topological vector spaces, M ⊂ X, and let T be a function from X into L(X, Y ). Suppose Y is ordered by the family of cones {int P (x) : x ∈ M } ∪ {0}. We define the following weak and strong vector complementarity problems: (WVCP) find x ∈ M with T x(x) ∈ / int P (x), T x(y) ∈ / − int P (x) ∀y ∈ M. (SVCP) find x ∈ M with T x(x) ∈ / int P (x), T x(y) ∈ P (x) ∀y ∈ M . If Y = R, P (x) = R+ for each x ∈ M , the vector complementarity problems (WVCP) and (SVCP) coincide with the scalar complementarity problem: (CP) find x ∈ M such that x, T x = 0 and T x ∈ M + , where M + := {x∗ ∈ X ∗ | x, x∗  ≥ 0 ∀x ∈ M }. 1

Let us recall that vector R+ -upper (lower) semicontinuity of a real function coincides with the usual scalar upper (lower) semicontinuity.

3.9 Applications to vector variational inequalities

193

If we restrict ourselves to X = Rn and M = Rn+ , then the space L(X, Y ) becomes equal to Rn , and (CP) becomes, find x ≥ 0 such that x, T x = 0 and T x ≥ 0. In the following, conditions under which solutions of vector complementarity problems exist are presented. To derive these conditions, Lemma 3.9.11 will be used. Lemma 3.9.11. Let us consider the following vector variational inequality (VVI): find x ∈ M such that T x(y − x) ∈ / − int P (x) ∀ y ∈ M.

(3.102)

(i) Suppose that the cone P (x) is w-pointed (i.e., int P (x) ∩ −P (x) = ∅) and T is a single-valued function; then (SVCP) =⇒ (VVI).2 (ii) Suppose that M is a convex cone; then (VVI) =⇒ (WVCP). (iii) Suppose that int P (x) ∪ −P (x) = X; then (WVCP) =⇒ (SVCP). Proof. (i): Since P (x) is w-pointed, (SVCP) =⇒ (VVI) follows from the im/ int P (x) and T x(y) ∈ P (x) ⇒ T x(y) − T x(x) ∈ / − int P (x). plication T x(x) ∈ (ii): Let us take y = 0 in (3.102); we have T x(x) ∈ / int P (x). Together, fix z ∈ M and take y = z + x (∈ M , since M is a convex cone) in (3.102); then T x(z) = T x(y − x) ∈ / − int P (x). (iii): It is immediate from condition int P (x) ∪ −P (x) = X.
Theorem 3.9.12. Suppose that (i) for every x, y ∈ M, T x(y − x) ∈ / − int P (x) implies T y(y − x) ∈ / − int P (x); (ii) the graph of int P is open on M × Y ; (iii) T is continuous from each line segment of M to a topology on L(X, Y ) for which the bilinear form (π, x) → π(x) is continuous from L(X, Y )×M into Y ; (iv) there is a convex weakly compact subset B of X such that for every y ∈ M \ B there exists x ∈ M ∩ B such that T x(y − x) ∈ int P (x). Then the problem (WVCP) has at least one solution. Proof. The assumptions (1)–(5), (7), and (8) of Theorem 3.8.28 hold when applied to f (x, y) = T x(y − x) for x, y ∈ M . To verify (6), fix x, y ∈ M such that x = y, and suppose to the contrary that there exists some u ∈]x, y[ such that T u(x − u) ∈ / int P (x) and T u(y − u) ∈ − int P (x). According to Y \ int P (x) − int P (x) ⊂ Y \ P (x), we deduce that for every t ∈]0, 1[, T u(tx + (1 − t)y − u) = tT u(x − u) + (1 − t)T u(y − u) ∈ / P (x), which is impossible for t such that u = tx + (1 − t)y. Therefore, (VVI) has a solution; and invoking Lemma 3.9.11, we finish the proof of theorem.
2

This means: x is a solution of (SVCP) =⇒ x is a solution of (VVI).

194

3 Optimization in partially ordered spaces

3.9.5 Application to Vector Optimization Problems Consider φ : M −→ Y and the following vector optimization problems (WVOP) find x ∈ M such that φ (y) ∈ / φ (x) − int P ∀ y ∈ M; and (VOP) find x ∈ M such that φ (y) ∈ / φ (x) − (P \ {0}) ∀ y ∈ M. General vector mappings We first obtain a direct existence result for the weak minimum of a vector optimization problem by setting f (x, y) = φ(x) − φ(y). It is trivial to check that x ∈ M is a solution of (GVEP) (and (WGVEP)) iff x is a solution of (VOP) (and (WVOP)). Theorem 3.9.13. Suppose that (i) φ is quasi P -convex and P -lower semicontinuous; (ii) there exists B ⊆ M convex and compact such that ∀y ∈ M \ B, ∃x ∈ B such that φ(y) ∈ φ(x) + int P . Then (WVOP) admits at least one solution x ∈ B. Proof. We apply Theorem 3.8.25 with f (x, y) = φ(x) − φ(y) and K(x) = − int P .
Remark 3.9.14. In the particular case where Y = Rm and P = Rm − we have under assumptions (i) and (ii) of Theorem 3.9.13 the existence of a weak Pareto optimum, i.e., ϕ (M ) ∩ (ϕ (x) + int P ∪ {0}) = {ϕ (x)}. Remark 3.9.15. Theorem 3.9.13 is a generalization, in the case of single-valued functions, of a result [244, Corollary 5-6 p. 59] on existence of solutions of vector optimization problems. Smooth vector mappings In this paragraph, by considering smooth vector mappings, we prove the existence of weak minimum for vector optimization problems by means of a vector variational-like inequality and preinvex mappings. For our analysis, the following concepts are necessary. Definition 3.9.16. We say that a Fr´echet differentiable function3 φ : M → Y is P -invex with respect to η : M × M → X if φ(x) − φ(y) − ∇φ(y)(η(x, y)) ∈ P ∀x, y ∈ M . Here ∇φ denotes the Fr´echet derivative of φ. 3

φ must be defined in a neighborhood of M .

3.9 Applications to vector variational inequalities

195

Definition 3.9.17. We say that φ : M → Y is P -preinvex with respect to η : M × M → X if for all x, y ∈ M, t ∈ [0, 1], y + tη(x, y) ∈ M and t(φ(x) − φ(y)) + φ(y) − φ(y + tη(x, y)) ∈ P. Remark 3.9.18. If we suppose that φ is Fr´echet differentiable, then P -invexity of φ is satisfied whenever φ is P -preinvex with respect to the same η. Indeed, suppose that t(φ(x) − φ(y)) + φ(y) − φ(y + tη(x, y)) ∈ P . Dividing by t and letting t goes to zero, we obtain φ(x) − φ(y) − ∇φ(y)(η(x, y)) ∈ P ; that is, P -invexity of φ.
Remark 3.9.19. We remark that if we take R and R+ respectively in place of Y and P , we obtain the definitions of scalar invex and preinvex functions. Note that invex functions were first introduced by Hanson [157]. To characterize the invexity, Craven and Glover showed that the class of invex functions is equivalent to the class of functions whose starting points are global minima. Now, we prove the following existence theorem of weak vector minimum points. Theorem 3.9.20. Let M be a nonempty closed, convex subset of a real topological vector space. Let φ : M → Y be a Fr´echet differentiable and P -preinvex mapping such that its derivative ∇φ is continuous from each line segment of M to L(X, Y ), i.e., ∀x, y ∈ M, ∀z ∈ X ∇φ(x + t(y − x))(z) → ∇φ(x)(z) in Y if t → 0+ . Let η : M × M → X be such that η(x, x) = 0 for every x ∈ K. Suppose that, for every x, y ∈ M , (1) ∇φ(x)(η(y, x)) ∈ / − int P =⇒ ∇φ(y)(η(y, x)) ∈ / − int P ; (2) η is linear in the first argument and continuous in the second one; (3) there exists a compact subset B of M , and x0 ∈ B such that ∇φ(x0 )(η(y, x0 )) ∈ − int P for all y ∈ M \ B. Then the vector optimization problem (WVOP) has a global minimum x ∈ B. Proof. By Theorem 3.8.24, with K(x) = L(x) = − int P, f (x, y) = ∇φ(x)(η(x, y)), and g(x, y) = ∇φ(y)(η(x, y)), there exists x ∈ B such that ∇φ(y)(η(y, x)) ∈ / − int P ∀ y ∈ M. For y ∈ M fixed, set yt = ty +(1−t)x, where t ∈ [0, 1]; then ∇φ(yt )(η(yt , x)) ∈ / − int P ∀ t ∈ [0, 1]. Using (2) and η(x, x) = 0, we obtain ∇φ(yt )(η(y, x)) = 1/t∇φ(yt )(η(yt , x)) ∈ / − int P. Since ∇φ is continuous from each line segment of M to L(X, Y ), it follows that the vector variational-like inequality

196

3 Optimization in partially ordered spaces

∇φ(x)(η(y, x)) ∈ / − int P for all y ∈ M

(3.103)

has a solution. Because of the P -preinvexity, which implies P -invexity, of φ we deduce φ(y) − φ(x) − ∇φ(x)(η(y, x)) ∈ P.

(3.104)

Combining (3.103) and (3.104) we obtain for every y ∈ M , φ(y) − φ(x) = (φ(y) − φ(x) − ∇φ(x)(η(y, x))) + ∇φ(x)(η(y, x)) ∈ / − int P, which completes the proof.




3.9.6 Minimax Theorem for Vector-Valued Mappings Let us first introduce the notions of cone saddle point of vector-valued mappings. Assume that we are given X = X1 × X2 a product of two real topological vector spaces, and U , V two nonempty subsets of X1 and X2 , respectively. Let F : M = U × V −→ Y be a vector-valued mapping. Definition 3.9.21. (1) A point (u0 , v0 ) ∈ M is said to be a weak P -saddle point of F with respect to M = U × V , a WVSP for short, if  F (u0 , V ) ∩ (F (u0 , v0 ) + int P ∪ {0}) = {F (u0 , v0 )} F (U, v0 ) ∩ (F (u0 , v0 ) − int P ∪ {0}) = {F (u0 , v0 )}. (2) A point (u0 , v0 ) ∈ M is said to be a P -saddle point of F with respect to M = U × V , a VSP for short, if  F (u0 , V ) ∩ (F (u0 , v0 ) + P ) = {F (u0 , v0 )} F (U, v0 ) ∩ (F (u0 , v0 ) − P ) = {F (u0 , v0 )}. It should be remarked that a VSP is also a WVSP. In the case where int P ∪ {0} = P , the two concepts are coincident. Lemma 3.9.22. Assume that we are given F a vector-valued mapping from M = U × V into Y , P a nonempty cone with int P = ∅, and f : M × M → Y defined, for each (u1 , v1 ), (u2 , v2 ) ∈ M , by f ((u1 , v1 ), (u2 , v2 )) := F (u1 , v2 ) − F (u2 , v1 ). If (u0 , v0 ) satisfies f ((u, v), (u0 , v0 )) ∈ / − int P (resp. −P \ {0}) ∀(u, v) ∈ M , then (u0 , v0 ) is a WVSP (resp. VSP) of F . Proof. First, we remark that  / int P ∀ v ∈ V, F (u0 , v) − F (u0 , v0 ) ∈ (u0 , v0 ) is a WVSP ⇔ F (u, v0 ) − F (u0 , v0 ) ∈ / − int P ∀ u ∈ U,

3.10 Minimal point theorems in product spaces

and

 (u0 , v0 ) is a VSP ⇔

197

/ P \ {0} ∀ v ∈ V, F (u0 , v) − F (u0 , v0 ) ∈ F (u, v0 ) − F (u0 , v0 ) ∈ / −P \ {0} ∀ u ∈ U.

The end of the proof is immediate from these equivalences and the definition of f .
This lemma, combined with Theorem 3.8.25, Lemma 3.8.17, and Remark 3.8.15, leads to the following vector minimax theorem. Theorem 3.9.23. Suppose that U and V are compact and convex subsets of two Hausdorff topological vector spaces, P a closed convex cone with nonempty interior in a Hausdorff locally convex topological vector space. Let F : U ×V → Y satisfy the following assumptions: (1) for every u ∈ U , F (u, ·) is P -concave and P -upper semicontinuous; (2) for every u ∈ V , F (·, v) is P -convex and P -lower semicontinuous. Then F admits a weak vector saddle point in U × V . For further references see [112], [352], and [274].

3.10 Minimal-Point Theorems in Product Spaces and Corresponding Variational Principles The importance of the Ekeland variational principle in nonlinear analysis is well known. Below we recall a versatile variant. Proposition 3.10.1. Ekeland’s variational principle. Let (X, d) be a complete metric space and f : X → R• := R ∪ {∞} a proper, lower semicontinuous function bounded below. Consider ε > 0 and x0 ∈ X such that f (x0 ) ≤ inf f + ε. Then for every λ > 0 there exists x ∈ dom f such that f (x) + λ−1 εd(x, x0 ) ≤ f (x0 ), and

f (x) < f (x) + λ−1 εd(x, x)

d(x, x0 ) ≤ λ, ∀ x ∈ X \ {x}.

(3.105) (3.106)

This means that for λ, ε > 0 and x0 an ε-approximate solution of the minimization problem f (x) → min s.t. x ∈ X, (3.107) there exists a new point x that is not worse than x0 and belongs to a λ-neighborhood of x0 , and especially, x satisfies the variational inequality (3.106). Relation (3.106) says, in fact, that x minimizes globally f + λ−1 εd(x, ·), which is nothing else than a Lipschitz perturbation of f (for

198

3 Optimization in partially ordered spaces

√ “smooth” principles, see [41]). Note that λ = ε gives a useful compromise in Proposition 3.10.1. For applications see Section 4.1 and 4.5 and, e.g., [116, 117, 310, 339, 343]. There are several statements that are equivalent to Ekeland’s variational principle (EVP); see, e.g., [87, 126, 290, 297, 296, 48, 73, 154, 140, 142, 141, 144, 143, 71, 72, 124]. We mention explicitly a result of Attouch and Riahi [10], who showed that (in Banach spaces) EVP is equivalent to the existence of minimal points with respect to cones satisfying some additional conditions. Proposition 3.10.2. Phelps minimal-point theorem. Let X be a Banach space, C ⊂ X a closed convex cone such that C ⊂ Kx∗ := {x ∈ X | x ≤ x, x∗ } for some x∗ ∈ X ∗ , and A ⊂ X a nonempty closed set such that x∗ (A) is bounded from below. Then for every x0 ∈ A there exists x ∈ A ∩ (x0 − C) such that x is a minimal element of A with respect to the partial order induced by C. Obviously, Kx∗ is a pointed closed convex cone, and so C is pointed, too. Moreover, Kx∗ is well-based, a bounded base being B := {x ∈ Kx∗ | x, x∗  = 1}, and so C is well-based, too. Sometimes the cone Kx∗ is called a Phelps cone. Since in this book we are mainly interested in multicriteria problems, we look for multicriteria (or vector) variants of EVP and minimal-point theorems. Loridan [240], then Khanh [210], Nemeth [269], and Tammer [337], were the first to prove vectorial EVP. We illustrate this kind of result with the following one, which is very close to a result stated by Tammer [337] (see also Corollary 3.10.19). Proposition 3.10.3. Vector EVP. Let M be a nonempty closed subset of the Banach space (X, · ), Y a t.v.s., C ⊂ Y a proper closed convex cone, k 0 ∈ int C, and f : M → Y . Assume that {x ∈ M | f (x) ≤C rk 0 } is closed for every r ∈ R and for some ε > 0 and x0 ∈ dom f , we have that  f (M ) ∩ f (x0 ) − εk 0 − (C \ {0}) = ∅. Then for every λ > 0 there exists x ∈ dom f such that   f (M ) ∩ f (x) − λ−1 εk 0 − int C = ∅, (3.108)

x − x0 ≤ λ, and

f (x) + λ−1 εk 0 x − x ≤C f (x) ⇒ x = x.

(3.109)

In particular, x ∈ Eff(fλ−1 εk0 ,x , C), where fk,x := f + · −x · k. Note that for Y = R the conditions of the preceding proposition are very close to those of the Ekeland variational principle (Proposition 3.10.1). Before starting with more general minimal-point theorems let us have a look at (3.106). If we write it in the form f (x) − f (x) + ελ−1 d(x, x) ≤ 0 ⇒ x = x,

3.10 Minimal point theorems in product spaces

199

then (x, f (x)) is a minimal point of epi f with respect to the binary relation defined by (x1 , t1 )  (x2 , t2 ) ⇔ t1 − t2 ≥ ελ−1 d(x1 , x2 ). (3.110) If X is a normed space, this binary relation is determined by the cone C := {(x, t) ∈ X × R | t ≥ ελ−1 x }. The binary relations defined as in (3.110) will play a decisive role in this section. It is worth mentioning that a weaker result than a full (= authentic) minimal-point theorem gives an EVP, as shown in Section 3.10.1 below. 3.10.1 Not Authentic Minimal-Point Theorems Throughout the section (X, d) is a complete metric space, Y is a separated locally convex space, Y ∗ is its topological dual, C ⊂ Y is a convex cone; as usual, C + = {y ∗ ∈ Y ∗ | y ∗ (y) ≥ 0 ∀ y ∈ C} is the dual cone of C and C # = {y ∗ ∈ Y ∗ | y ∗ (y) > 0 ∀ y ∈ C \ {0}}. Consider also k 0 ∈ C \ (−C). We notice that there exists z ∗ ∈ C + such that z ∗ (k 0 ) = 1. Indeed, using a separation theorem for k 0 and −C we get z1∗ ∈ C + and a real α > 0 such that z1∗ (k 0 ) > α > z1∗ (y) for all y ∈ C, then take z ∗ := α−1 z1∗ . The cone C determines a preorder on Y denoted, as usual, by ≤C ; so, for y1 , y2 ∈ Y , y1 ≤C y2 iff y2 − y1 ∈ C. It is known (see Section 2.1) that ≤C is reflexive and transitive; ≤C is antisymmetric iff C is pointed, i.e., C ∩ −C = {0}. Using the element k 0 we introduce a preorder on X × Y , denoted by k0 , in the following manner: (x1 , y1 ) k0 (x2 , y2 ) ⇐⇒ y1 + k 0 d(x1 , x2 ) ≤C y2 .

(3.111)

Note that k0 is reflexive and transitive; if (x1 , y1 ) k0 (x2 , y2 ) and (x2 , y2 ) k0 (x1 , y1 ), then x1 = x2 . If C is pointed then k0 is antisymmetric, too. Note that if (X, · ) is a normed vector space (n.v.s.), then k0 is determined by the convex cone Ck0 := {(x, y) ∈ X × Y | x · k 0 ≤C y}; Ck0 is pointed if C is. Consider a nonempty set A ⊂ X × Y . For (x, y) ∈ A we denote by A(x, y) the lower section of A with respect to k0 : A(x, y) = {(x
, y
) ∈ A | (x
, y
) k0 (x, y)}. In the sequel we shall use the following condition on A: (H1) for every k0 -decreasing sequence ((xn , yn )) ⊂ A with xn → x ∈ X there exists y ∈ Y such that (x, y) ∈ A and (x, y) k0 (xn , yn ) for every n ∈ N. A related condition is

  (H2) for every sequence (xn , yn ) ⊂ A with xn → x ∈ X and (yn ) ≤C decreasing there exists y ∈ Y such that (x, y) ∈ A and y ≤C yn for every n ∈ N.

200

3 Optimization in partially ordered spaces

These two conditions are motivated by conditions used by Isac [179] and Nemeth [269], respectively, as we shall see below. Note that if A satisfies (H2) and C has closed lower sections with respect to R+ k 0 , i.e., C ∩ (y − R+ k 0 ) is closed for every y ∈ C, then (H1) is also satisfied.   Indeed, let (xn , yn ) ⊂ A be a k0 -decreasing sequence with xn → x. It is obvious that (yn ) is ≤C -decreasing. By (H2), there exists y ∈ Y such that (x, y) ∈ A and y ≤C yn for every n ∈ N. It follows that y + k 0 d(xn+p , xn ) ≤C yn+p + k 0 d(xn+p , xn ) ≤C yn

∀ n, p ∈ N.

Fixing n and letting p → ∞, by the closedness of C in the direction k 0 , one obtains that y +k 0 d(x, xn ) ≤C yn , i.e., (x, y) k0 (xn , yn ) for every n ∈ N. Note also that (H2) holds if A is closed and every ≤C -decreasing sequence in C is convergent (i.e., C is a sequential Daniell cone). This is the case (even for nets) if Y is a Banach space and C has a closed (convex) and bounded base (see [38, Prop. 3.6] and Section 2.2 for bases of cones). We establish now our first minimal-point theorem. In the sequel PX and PY will denote the projections of X × Y onto X and Y , respectively; so PX (x, y) = x for every (x, y) ∈ X × Y . Theorem 3.10.4. Let A ⊂ X × Y satisfy (H1) and suppose that there exists y˜ ∈ Y such that PY (A) ⊂ y˜ + C. Then for every (x0 , y0 ) ∈ A there exists (¯ x, y¯) ∈ A such that (¯ x, y¯) k0 (x0 , y0 ) and if (x
, y
) ∈ A is such that


(x , y ) k0 (¯ x, y¯) then x = x ¯. Proof. First of all, note that for (x, y) ∈ A the set PX (A(x, y)) is bounded. Indeed, as noticed at the beginning of this section, there exists z ∗ ∈ C + with z ∗ (k 0 ) = 1. Let x
∈ PX (A(x, y)); there exists y
∈ Y such that (x
, y
) ∈ A y ), and (x
, y
) k0 (x, y). Then d(x, x
) ≤ z ∗ (y) − z ∗ (y
) ≤ z ∗ (y) − z ∗ (˜ which shows that PX (A(x, y)) is bounded. Let us construct a sequence ((xn , yn ))n∈N ⊂ A in the following way: Having (xn , yn ) ∈ A, where n ∈ N, by the a