Table Of ContentMinimax and Applications
Nonconvex Optimization and Its Applications
VolumeS
Managing Editors:
Panos Pardalos
University ofF lorida, U.S.A.
Reiner Horst
University ofT rier, Germany
Advisory Board:
Ding-ZhuDu
University ofM innesota, U.S.A.
C. A. Floudas
Princeton University, U.S.A.
G. Infanger
Stanford University, U.S.A.
J. Mockus
Lithuanian Academy ofS ciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, U.S.A.
The titles published in this series are listed at the end oft his volume.
Minimax
and Applications
Edited by
Ding-Zhu Du
University ofM innesota. U.SA..
and Institute ofA pplied Mathematics. Beijing. China
and
Panos M. Pardalos
Department ofI ndustrial and Systems Engineering.
University ofF lorida.
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.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
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In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved
e
1995 Kluwer Academic Publishers
Softcover reprint of the hardcover 1s t 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,
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"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
IIf
3. -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
