Semi-Infinite Programming

417 Pages·1998·12.81 MB·English

by  Miguel A. Goberna, Marco A. López (auth.), Rembert Reemtsen, Jan-J. Rückmann (eds.)

Semi-Infinite Programming
Nonconvex Optimization and Its Applications
Volume 25
Managing Editors:
Panos Pardalos
University ofF lorida, U.S.A.
Reiner Horst
University of Trier, Germany
Advisory Board:
Ding-ZhuDu
University ofM innesota, U.S.A.
C.A. Floudas
Princeton University, U.S.A.
G.lnfanger
Stanford University, U.S.A.
J. Mockus
Lithuanian Academy of Sciences, 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.
Semi-Infinite
Programming
Edited by
Rembert Reemtsen
Institute ofM athematics,
Brandenburg Technical University ofCottbus
and
Jan-J. Riickmann
Institute for Applied Mathematics,
University ofE rlangen-Nuremberg
Springer-Science+Business Media, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4795-6 ISBN 978-1-4757-2868-2 (eBook)
DOI 10.1007/978-1-4757-2868-2
Printed on acid-free paper
All Rights Reserved
© 1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998.
Softcover reprint of the hardcover 1s t edition 1998
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 xi
CONTRIBUTERS xv
Part I THEORY 1
1 A COMPREHENSIVE SURVEY OF LINEAR
SEMI-INFINITE OPTIMIZATION THEORY
Miguel A. Goberna and Marco A. Lopez 3
1 Introduction 3
2 Existence theorems for the LSIS 5
3 Geometry of the feasible set 6
4 Optimality 10
5 Duality theorems and discretization 12
6 Stability of the LSIS 14
7 Stability and well-posedness of the LSIP problem 19
8 Optimal solution unicity 23
REFERENCES 25
2 ON STABILITY AND DEFORMATION IN
SEMI-INFINITE OPTIMIZATION
Hubertus Th. Jongen and Jan-J. Riickmann 29
1 Introduction 29
2 Structure of the feasible set 32
3 Stability of the feasible set 40
4 Stability of stationary points 44
5 Global stability 53
v
VI SEMI-INFINITE PROGRAMMING
6 Global deformations 57
REFERENCES 63
3 REGULARITY AND STABILITY
IN NONLINEAR SEMI-INFINITE
OPTIMIZATION
Diethard Klatte and Rene Henrion 69
1 Introduction 69
2 Upper semicontinuity of stationary points 73
3 Metric regularity of the feasible set mapping 83
4 Stability of local minimizers 95
5 Concluding remarks 98
REFERENCES 99
4 FIRST AND SECOND ORDER OPTIMALITY
CONDITIONS AND PERTURBATION
ANALYSIS OF SEMI-INFINITE
PROGRAMMING PROBLEMS
Alexander Shapiro 103
1 Introduction 103
2 Duality and first order optimality conditions 106
3 Second order optimality conditions 115
4 Directional differentiability of the optimal value function 122
5 Stability and sensitivity of optimal solutions 127
REFERENCES 130
Part II NUMERICAL METHODS 135
5 EXACT PENALTY FUNCTION METHODS
FOR NONLINEAR SEMI-INFINITE
PROGRAMMING
Ian D. Coope and Christopher J. Price 137
1 Introduction 137
2 Exact penalty functions for semi-infinite programming 143
3 Trust region versus line search algorithms 145
4 The multi-local optimization subproblem 148
Contents Vll
5 Final comments 154
REFERENCES 155
6 FEASIBLE SEQUENTIAL QUADRATIC
PROGRAMMING FOR FINELY
DISCRETIZED PROBLEMS FROM SIP
Craig T. Lawrence and Andre L. Tits 159
1 Introduction 159
2 Algorithm 163
3 Convergence analysis 167
4 Extension to constrained minimax 177
5 Implementation and numerical results 180
6 Conclusions 186
REFERENCES 186
APPENDIX A Proofs 189
7 NUMERICAL METHODS FOR SEMI-
INFINITE PROGRAMMING: A SURVEY
Rembert Reemtsen and Stephan Gomer 195
1 Introduction 195
2 Fundamentals 196
3 Linear problems 219
4 Convex problems 234
5 Nonlinear problems 243
REFERENCES 262
8 CONNECTIONS BETWEEN SEMI-INFINITE
AND SEMIDEFINITE PROGRAMMING
Lieven Vandenberghe and Stephen Boyd 277
1 Introduction 277
2 Duality 280
3 Ellipsoidal approximation 281
4 Experiment design 285
5 Problems involving power moments 289
6 Positive-real lemma 291
7 Conclusion 292
VIll SEMI-INFINITE PROGRAMMING
REFERENCES 292
Part III APPLICATIONS 295
9 RELIABILITY TESTING AND SEMI-
INFINITE LINEAR PROGRAMMING
1. Kuban Altmel and Siileyman Ozekici 297
1 Introduction 297
2 Testing systems with independent component failures 301
3 Solution procedure 306
4 Testing systems with dependent component failures 311
5 A series system working in a random environment 318
6 Conclusions 320
REFERENCES 321
10 SEMI-INFINITE PROGRAMMING IN
ORTHOGONAL WAVELET FILTER DESIGN
Ken O. Kortanek and Pierre Moulin 323
1 Quadrature mirror filters: a functional analysis view 324
2 Design implications from the property of perfect reconstruc-
tion 332
3 The perfect reconstruction semi-infinite optimization prob-
lem 339
4 Characterization of optimal filters through SIP duality 342
5 On some SIP algorithms for quadrature mirror filter design 346
6 Numerical results 351
7 Regularity constraints 353
8 Conclusions 354
REFERENCES 355
11 THE DESIGN OF NONRECURSIVE
DIGITAL FILTERS VIA CONVEX
OPTIMIZATION
Alexander W. Potchinkov 361
1 Introduction 361
2 Characteristics of FIR filters 364
Contents ix
3 Application fields 368
4 Approximation problems 371
5 The optimization problem 374
6 Numerical examples 378
7 Conclusion 385
REFERENCES 386
12 SEMI-INFINITE PROGRAMMING IN
CONTROL
Ekkehard W. Sachs 389
1 Optimal control problems 390
2 Sterilization of food 395
3 Flutter control 401
REFERENCES 411
PREFACE
Semi-infinite programming (briefly: SIP) is an exciting part of mathematical
programming. SIP problems include finitely many variables and, in contrast
to finite optimization problems, infinitely many inequality constraints. Prob-
lems of this type naturally arise in approximation theory, optimal control, and
at numerous engineering applications where the model contains at least one
inequality constraint for each value of a parameter and the parameter, repre-
senting time, space, frequency etc., varies in a given domain. The treatment of
such problems requires particular theoretical and numerical techniques.
The theory in SIP as well as the number of numerical SIP methods and appli-
cations have expanded very fast during the last years. Therefore, the main goal
of this monograph is to provide a collection of tutorial and survey type articles
which represent a substantial part of the contemporary body of knowledge in
SIP. We are glad that leading researchers have contributed to this volume and
that their articles are covering a wide range of important topics in this subject.
It is our hope that both experienced students and scientists will be well advised
to consult this volume.
We got the idea for this volume when we were organizing the semi-infinite pro-
gramming workshop which was held in Cottbus, Germany, in September 1996.
About forty scientists from fourteen countries participated in this workshop
and presented surveys or new results concerning the field. At the same time,
an up-to-date monograph on SIP was much missing so that we invited several
of the participants to contribute to such volume. The result is the present
collection of articles.
The volume is divided into the three parts Theory, Numerical Methods, and
Applications, each of them consisting of four articles. Part I: Theory starts
with a review by Goberna and Lopez on fundamentals and properties of linear
SIP, including optimality conditions, duality theory, well-posedness, and geo-
metrical properties of the feasible and the optimal set. Subsequently, Jongen
and Ruckmann survey the structure and stability properties of SIP problems,
where, in particular, the topological structure of the feasible set, the strong sta-
xi