Minimax and Applications

Nonconvex Optimization and Its Applications

VolumeS

Managing Editors:

Panos Pardalos University of Florida, U.S.A.

Reiner Horst University of Trier, Germany

Advisory Board:

Ding-ZhuDu University of Minnesota, U.S.A.

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

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

J. Mockus Lithuanian Academy of Sciences, Lithuania

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 and Applications

Edited by

Ding-Zhu Du University of Minnesota. U.SA.. and Institute of Applied Mathematics. Beijing. China

and

Panos M. Pardalos Department of Industrial and Systems Engineering. University of Florida. Gainesville. Florida. U.S.A.

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

Library of Congress Cataloging-in-Publication Data

MInImax and applIcatIons I edIted by Dlng-Zhu Du and Panos M. Pardalos.

p. cm. -- (Nonconvax optImIzatIon and Its applIcatIons v. 4)

Includes bIblIographIcal references and Index.

1. MaxIma and mlnll •. 2. Mathelatlcal optlllzatlon. I. Du, Dlngzhu. II. Pardalos, P. M. (Panos M.), 1954- . III. SerIes. QA306.M54 1995 515·.64--dc20 95-30189

ISBN-13: 978-1-4613-3559-7 e-ISBN-13: 978-1-4613-3557-3 DOl: 10.1007/978-1-4613-3557-3

Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands.

Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.

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All Rights Reserved e 1995 Kluwer Academic Publishers Softcover reprint of the hardcover 1 st edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any fonn 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.

"You cannot step twice into the same river, for other waters are continually flowing on"

"Time is a child moving counters in a game; the royal power is a child's"

- Heraclitus

Contents

Preface ..................................................................... xiii

Minimax Theorems and Their Proofs ...................................... 1 Stephen Simons

1. Introduction ................................................................. 1 2. The First Minimax Theorem ................................................ 2 3. Infinite Dimensional Bilinear Results ........................................ 2 4. Minimax Theorems When X and Y Are More General Convex Sets .......... 2 5. Minimax Theorems for Separately Semicontinuous Functions ................ .4 6. Topological Minimax Theorems .............................................. 5 7. Quantitative Minimax Theorems ............................................ 8 8. Mixed Minimax Theorems .................................................. 12 9. Unifying Metaminimax Theorems .......................................... 13 10. Connections with Weak Compactness ...................................... 15 11. Minimax Inequalities for Two or More Functions .......................... 17 12. Coincidence Theorems .................................................... 19 References .................................................................... 19

A Survey on Minimax Trees and Associated Algorithms ............... 25 Claude G. Diderich and Marc Gengler

1. Introduction ............................................................... 25 2. Minimax Trees and the Theory Behind Them ............................... 26 3. Sequential Minimax Game Tree Algorithms ................................. 31 4. Parallel Minimax Tree Algorithms .......................................... 42 5. Open Problems and Conclusion ............................................. 51 References .................................................................... 52

An Iterative Method for the Minimax Problem ......................... 55 Liqun Qi and Wenyu Sun

1. Introduction ............................................................... 55 2. An ELQP Problem as a Subproblem ........................................ 56 3. Local and Superlinear Convergence ......................................... 60 References ..................................................................... 66

viii CONTENTS

A Dual and Interior Point Approach to Solve Convex Min-Max Problems ..................................... 69

Jos F. Sturm and Shuzhong Zhang

1. Introduction ............................................................... 69 2. The Scaling Supergradient Method ......................................... 70 3. Convergence Analysis ...................................................... 73 4. Concluding Remarks ....................................................... 77 References .................................................................... 77

Determining the Performance Ratio of Algorithm MULTIFIT for Scheduling .............................................................. 79

Feng Cao

1. Introduction ............................................................... 79 2. !l. > 56 ..................................................................... 81 3. ~56~!l.~1~6 .............................................................. 83 4. 26 ~ !l. < T6 .............................................................. 84 5. !l. < 2.56 ................................................................... 87 References .................................................................... 96

A Study of On-Line Scheduling Two-Stage Shops ....................... 97 Eo Chen and Gerhard J. Woeginger

1. Introduction ............................................................... 97 2. Definitions and Preliminaries ............................................... 98 3. A Lower Bound for 0211Cm~ ............................................... 99 4. An Algorithm for 0211Cmax ............................................... 100 5. A Best Algorithm for 021pmtniCmax •••••••••••••••••••.••••••••••••••••••• 103 6. On Flow and Job Shops ................................................... 105 7. Discussions ............................................................... 106 References ....................................................... ' ............ 106

Maxmin Formulation of the Apportionments of Seats to a Parliament .................................................... 109

Thorltell Helgason, Kurt Jiirnsten, and Athanasios Migdalas

1. Introduction .............................................................. 109 2. Concepts and models ...................................................... 110 3. Illustrative examples ...................................................... 115 4. Discussion ................................................................ 117 References ................................................................... 118

On Shortest It-Edge Connected Steiner Networks with Rectilinear Distance ............................................... '.119

D. Frank Hsu, Xiao-Dong Hu, and Yoji Kajitani

1. Introduction .............................................................. 119 2. Technical Preliminaries .................................................... 120 3. Main Results ................... ' ........................................... 122 References ................................................................... 127

CONTENTS ix

Mutually Repellant Sampling ............................................ 129 Shang-Hua Teng

1. Introduction .............................................................. 129 2. Mutually Repellant Sampling .............................................. 131 3. Max-Min Distance Sampling .............................................. 132 4. Max-Min-Selection Distance Sampling ..................................... 134 5. Max-Average Distance Sampling .......................................... 136 6. Lower Bounds ............................................................. 137 7. Applications and Open Questions ......................................... 139 References ................................................................... 140

Geometry and Local Optimality Conditions for Bilevel Programs with Quadratic Strictly Convex Lower Levels .......................... 141

Luis N. Vicente and Paul H. Calamai

1. Introduction .............................................................. 141 2. Problem Statement and (jeometry ......................................... 142 3. Computing the Convex Cones ............................................. 146 4. Number of Convex Cones ................................................. 147 5. Stationary Points and Local Minima ....................................... 148 6. Conclusions and Future Work ............................................. 150 References ................................................................... 150

The Spherical One-Center Problem ...................................... 153 Guoliang Xue and Shangzhi Sun

1. Introduction .............................................................. 153 2. Main Result .............................................................. 154 3. Conclusions ............................................................... 156 References ................................................................... 156

On Min-max Optimization of a Collection of Classical Discrete Optimization Problems .................................................. 157

Gang Yu and Panagiotis K ouvelis

1. Introduction .............................................................. 157 2. The Min-max Spanning Tree Problem ..................................... 159 3. The Min-max Resource Allocation Problem ................................ 162 4. The Min-max Production Control Problem ................................ 167 5. Summary and Extensions ................................................. 169 References ................................................................... 170

Heilbronn Problem for Six Points in a Planar Convex Body .......... 173 Andreas W.M. Dress, Lu Yang, and Zhenbing Zeng

1. Introduction .............................................................. 173 2. Prerequisites .............................................................. 175 3. Proof of the Main Theorem ............................................... 179 References ................................................................... 188

x CONTENTS

Heilbronn Problem for Seven Points in a Planar Convex Body ....... 191 Lu Yang and Zhenbing Zeng

1. Introduction .............................................................. 191 2. Propositions and Proofs for Easier Cases .................................. 192 3. Configurations with Stability .............................................. 195 4. Computing the Smallest Triangle .......................................... 203 5. Open Problems ........................................................... 217 References ................................................................... 218

On the Complexity of Min-Max Optimization Problems and Their Approximation ..................................................... 219

Ker-I Ko and Chih-Long Lin

1. Introduction .............................................................. 219 2. Definition ................................................................. 221 3. IIf -Completeness Results ................................................. 223 4. Approximation Problems and Their Hardness .............................. 231 5. Nonapproximability Results ............................................... 233 6. Conclusion and Open Questions ........................................... 238 References ................................................................... 239

A Competitive Algorithm for the Counterfeit Coin Problem ......... 241 Xiao-Dong Hu and Frank K. Hwang

1. Introduction .............................................................. 241 2. Some Lower Bounds of M(n : d) ........................................... 242 3. A Competitive Algorithm ................................................. 244 4. Analysis of Competitiveness ............................................... 247 5. Conclusion ................................................................ 249 References ................................................................... 250

A Minimax af3 Relaxation for Global Optimization .................... 251 Jun Gu

1. Introduction .............................................................. 251 2. Problem Model ........................................................... 252 3. Relaxation Approach ...................................................... 253 4. A General af3 Relaxation Algorithm ....................................... 254 5. A Minimax af3 Relaxation Algorithm for COP ............................. 258 6. Experimental Results ..................................................... 263 References ................................................................... 268

Minimax Problems in Combinatorial Optimization .................... 269 Feng Cao, Ding-Zhu Du, Biao Gao, P.M. Pardalos, and Peng-Jun Wan

1. Introduction .............................................................. 269 2. Algorithmic Problems ..................................................... 270 3. Geometric Problems ...................................................... 274 4. Graph Problems .......................................................... 281 5. Management Problems .................................................... 287

CONTENTS xi

6. Miscellaneous ............................................................. 291

Author Index .............................................................. 293

Preface

Techniques and principles of minimax theory play a key role in many areas of research, including game theory, optimization, and computational complexity. In general, a minimax problem can be formulated as

min max f(x, y) ",EX !lEY

(1)

where f(x, y) is a function defined on the product of X and Y spaces. There are two basic issues regarding minimax problems:

The first issue concerns the establishment of sufficient and necessary conditions for equality

minmaxf(x,y) = maxminf(x,y). "'EX !lEY !lEY "'EX

(2)

The classical minimax theorem of von Neumann is a result of this type. Duality theory in linear and convex quadratic programming interprets minimax theory in a different way.

The second issue concerns the establishment of sufficient and necessary conditions for values of the variables x and y that achieve the global minimax function value

f(x*, y*) = minmaxf(x, y). "'EX !lEY

(3)

There are two developments in minimax theory that we would like to mention. First, it has been shown that some minimax problems (which are NP-hard in gen­eral) are polynomially solvable when condition (2) holds [see e.g., Andras Recski, Minimax Results and Polynomial Algorithms in VLSI Routing, in Proceedings of the Fourth Czechoslovakian Symposium on Combinatorics, Graphs, and Complex­ity (Prachatice, Czechoslovakia, June 1990) North-Holland, 1992 (Editors M. Fiedler and J. Nesetril), pp. 261-273]. Secondly, a long-standing open problem on minimum networks, the Gilbert-Pollak conjecture on the Steiner ratio, was solved by using a minimax approach [see e.g., D.-Z. Du and F.K. Hwang, An approach for proving lower bounds: solution of Gilbert-Pollak's conjecture on the Steiner ratio, In Pro­ceedings of 31th FOCS Conference (1990), pp. 76-85]. Central to this approach is a result regarding the characterization of global optima of (3). These developments indicate that minimax theory will continue to be an important tool for solving dif­ficult and interesting problems. In addition, minimax methods provide a paradigm for investigating analogous problems. An exciting future with new unified theories may be expected.

A remark on terminology may be necessary. Many researchers reserve the term "minimax" for results of von Neumann's type. However, for study of the second issue, the term "min-max" or "minmax" is used. Since the two types of problems discussed above are of the same nature, we suggest that the term "minimax" is used.

The collection of papers in this book covers a diverse number of topics and provides a good picture of recent research in minimax theory. Most of the papers in the first group present results on classical minimax theorems and optimization problems using duality theory. The papers in the second group are mainly concerned with optimality and approximate algorithms of minimax problems. Instead of a

xiii

xiv CONTENTS

postface, the final paper provides a brief survey and open questions on minimax problems in combinatorial optimization.

The book will be a valuable source of information to faculty, students and re­searchers in optimization, computer sciences, and related areas. We would like to take the opportunity to thank the authors of the papers, the referees, and the pub­lisher for helping us to produce this book.

Ding-Zhu Du and Panos M. Pardalos

University of Minnesota and University of Florida March 1995