Minimax Theory and Applications

Nonconvex Optimization and Its Applications Volume 26

Managing Editors:

Panos Pardalos University ofFlorida, U.S.A.

Reiner Horst University of Trier, Germany

Advisory Board:

Ding-ZhuDu University ofMinnesota, U.S.A.

C.A. Floudas Princeton University, U.S.A.

G. Infanger Stanford University, U.S.A.

J.Mockus Lithuanian Academy ofSciences, Lithuania

P.D. Panagiotopoulos Aristotle University, Greece

H.D. Sherali Virginia Polytechnic Institute and State University, U.S.A.

The titles published in this series are listed at the end of this volume.

Minimax Theory and Applications Edited by

Biagio Ricceri Department ofMathematics, University of Catania Catania, Italy

and

Stephen Simons Department ofMathematics, University of California at Santa Barbara, Santa Barbara, California, U.S.A.

• Springer-Science+Business Media, B.V

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5030-4

ISBN 978-94-015-9113-3 (eBook)

DOI 10.1007/978-94-015-9113-3

Printed on acid-free paper

All Rights Reserved © 1998Springer Science +Busin ess Media Dordrecht

Originally published by Kluwer Academic Publishers in 1998.

Softcover reprintof thehardcover 1st edition1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Contents Preface

Nonlinear Two Functions Minimax Theorems Cao-Zong Cheng and Bor-Luh Lin

xi

1

1. Introduction 2. Nonlinear Minimax Theorems 3. Two Functions Minimax Theorems of Type A/Type B 4. Two Functions Minimax Theorems of Mixed Type References

1 5 10 18 19

Weakly Upward-Downward Minimax Theorem Cao-Zong Cheng, Bor-Luh Lin and Feng-Shuo Yu

21

References

28

A Two-Function Minimax Theorem Antonia Chinni

29

1. Introduction 2. The Main Result 3. Remarks and Examples Related to Theorem 2.2 References

29 30 31 33

Generalized Fixed-Points and Systems of Minimax Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paul Deguire

35

1. Introduction 2. Applications References

35 36 40

VI

CONTENTS

A Minimax Inequality for Marginally Semicontinuous Functions Gabriele H. Greco and Maria Pia Mosch en References On Variational Minimax Problems under Relaxed Coercivity Assumptions Joachim Gwin ner 1. Introduction 2. Some Preliminary Remarks 3. A Unilateral Boundary Value P roblem and its Variational Mimimax Formulation ............................... 4. The Semicoercive Case 5. Lagrangian Minimax Problems References

41 50 53 53 55 57 60 64 69

A Topological Investigation of the Finite Intersection Property Charles D. Horva th

71

1. Introduction 2. The Finite Intersection Property 3. Topological Spaces with a Convexity Structure 4. Conclusion References

71 74 81 88 89

Minimax Results and Randomization for Certain Stochastic Games.. . ......... .. ..... . . ...... . . . . .. .. . . . . . . . ... .... ... Albrecht Irle

91

1. Introduction 2. Randomization of Stopping Times 3. Compact Embedding and Equivalence of Randomization 4. Minimax Results in Discrete Time 5. A Minimax Result in Continuous Time References

91 93 95 98 99 103

Intersection Theorems, Minimax Theorems and Abstract Connectedness Jiirqen. Kindler

105

1. Introduction 2. Abstract Continuity

105 107

CONTENTS

vii

3. Abstract Connectedness 4. Intersection Theorems 5. Minimax Theorems References

108 110 113 120

K-K-M-S Type Theorems in Infinite Dimensional Spaces Hidetoshi Komiya

121

1. Introduction 2. Selection of Base Spaces and Preliminaries 3. Balanced Families 4. K-K-M -S Type Theorems in Infinite Dimensional Spaces 5. Application to Game Theory 6. Extensions of K-K-M-S Theorem References

121 122 123 127 130 132 134

Hahn-Banach Theorems for Convex Functions Marc Lassonde

135

1. Separation of Convex Functions 2. Continuity of Convex Functions References

137 140 144

Two Functions Generalization of Horvath's Minimax Theorem.. ...... ... . ............ ....... ... ... . ... . .. .. . .. . Bor-Luh Lin and Feng-Shuo Yu

147

References

156

Some Remarks on a Minimax Formulation of a Variational Inequality 157 Giandomenico Mastroeni 1. Saddle Point Conditions and Variational Inequalities 2. Applications to the Classical Variational Inequality 3. Connections with Complementarity Problems 4. Vector Variational Inequalities 5. Further Developments References

157 159 161 162· 164 166

Network Analysis Michael M. Neumann and Maria Victoria Velasco

167

1. Introduction 2. From Finite to Infinite Networks

167 167

viii

CONTENTS

3. Tools from Functional Analysis 4. Existence of Flows 5. Existence of Potentials 6. Symmetric, Antisymmetric and Net Flows 7. Marginal Problems 8. Concluding Remarks References

170 173 178 180 185 186 188

On a Topological Minimax Theorem and its Applications Biagio Ricceri

191

1. 2. 3. 4.

191 193 196

Introduction Preliminaries Proof of Theorem 1.1 An Application of Theorem 1.1 to the Problem inf x f = infax f 5. A Variational Property of Integral Functionals References

198 203 216

Three Lectures on Minimax and Monotonicity Stephen Simons

217

O. Introduction 1. Multifunctions and Monotonicity 2. A Convexification of E x E* and the Three Affine Maps 3. Monotone Subsets and their "Pictures" . . . . . . . . . . . . . . . . . . . .. 4. For Reflexive Spaces Only 5. The Convex Function Determined by a Multifunction 6. Surrounding Sets and the Dom-Dom Lemma 7. The "Dom-Dom Constraint Qualification" 8. A "Sum Theorem" for Reflexive Spaces References

217 219 221 222 224 227 228 234 236 239

Fan's Existence Theorem for Inequalities Concerning Convex Functions and its Applications 241 Wataru Takahashi 1. Introduction 2. Generalization of Fan's System Theorem 3. Basic Results in Functional Analysis 4. Applications References

241 242 248 252 259

CONTENTS

ix

An Algorithim for the Multi-Access Channel Problem Peng-Jung Wan, Ding-Zu Du and Panos M. Pardalos

261

1. Introduction 2. The Algorithm 3. Analysis 4. Conclusion References

262 262 265 269 269

Author Index

271

Preface The present volume contains the proceedings of the workshop on "Minimax Theory and Applications" that was held during the week 30 September - 6 October 1996 at the "G. Stampacchia" International School of Mathematics of the "E. Majorana" Centre for Scientific Culture in Erice (Italy) . The main theme of t he workshop was minimax theory in its most classical meaning. That is to say, given a real-valued function f on a product space X x Y , one tries to find conditions that ensure the validity of the equality sup inf f(x , y) = inf sup f( x, y). yEY x E X

xEX yEY

This is not an appropriate place to enter into the technical details of the proofs of minimax theorems, or into the history of the contributions to t he solution of t his basic problem in the last 7 decades. But we do want to stress its intrinsic interest and point out that, in spite of its extremely simple formulation , it conceals a great wealth of ideas. This is clearly shown by the large variety of methods and tools that have been used to study it. The applications of minimax theory are also extremely interesting. In fact , the need for the ability to "switch quantifiers" arises in a seemingly boundless range of different sit uations. So, the good quality of a minimax theorem can also be judged by its applicability. We hop e that this volume will offer a ra th er complete account of the st at e of the art of the subject . We first thank the "E . Majorana" Centre for the perfect organization of the workshop. We also thank the participants to the workshop and the contributors to this volume. Thanks are also due to Kluwer Academic Publishers for having offered us the opportunity to publish this volume. Finally, we wish to express our deepe st gratitude to our colleague Professor Michele Frasca of the University of Catania who, with great patience and expertise, has taken upon himself the task of doing the technical work needed to prepare the volume. Biagio Ricceri , University of Catania Stephen Simons, University of California at Santa Barbara