Semi-Infinite Programming

Nonconvex Optimization and Its Applications

Volume 25

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.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 of Mathematics, Brandenburg Technical University ofCottbus

and

Jan-J. Riickmann

Institute for Applied Mathematics, University of Erlangen-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 1 st 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

CONTRIBUTERS

Part I THEORY

1 A COMPREHENSIVE SURVEY OF LINEAR SEMI-INFINITE OPTIMIZATION THEORY Miguel A. Goberna and Marco A. Lopez 1 Introduction 2 Existence theorems for the LSIS 3 Geometry of the feasible set 4 Optimality

5 Duality theorems and discretization 6 Stability of the LSIS 7 Stability and well-posedness of the LSIP problem 8 Optimal solution unicity REFERENCES

2 ON STABILITY AND DEFORMATION IN SEMI-INFINITE OPTIMIZATION Hubertus Th. Jongen and Jan-J. Riickmann 1 Introduction

2 Structure of the feasible set

3 Stability of the feasible set 4 Stability of stationary points

5 Global stability

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3 3 5 6

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29 29 32 40

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6 Global deformations REFERENCES

3 REGULARITY AND STABILITY IN NONLINEAR SEMI-INFINITE OPTIMIZATION Diethard Klatte and Rene Henrion 1 Introduction 2 Upper semicontinuity of stationary points 3 Metric regularity of the feasible set mapping 4 Stability of local minimizers 5 Concluding remarks REFERENCES

4 FIRST AND SECOND ORDER OPTIMALITY CONDITIONS AND PERTURBATION ANALYSIS OF SEMI-INFINITE PROGRAMMING PROBLEMS Alexander Shapiro 1 Introduction

57 63

69 69 73 83 95 98 99

103 103

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 REFERENCES

Part II NUMERICAL METHODS

5 EXACT PENALTY FUNCTION METHODS FOR NONLINEAR SEMI-INFINITE PROGRAMMING Ian D. Coope and Christopher J. Price 1 Introduction

127 130

135

137 137

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. Prob­lems 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, repre­senting 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 appli­cations 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 pro­gramming 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 geo­metrical 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-

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bility of stationary points, and one-parametric deformations are investigated. Eventually, in the contributions by Klatte and Henrion and by Shapiro, SIP problems are considered which depend on additional parameters, where Klatte and Henrion provide an overview of interrelations between metric regularity, constraint qualifications, local boundedness of multipliers, and upper semicon­tinuity of the stationary solution mapping, while Shapiro discusses properties as duality, optimality conditions, stability, and sensitivity of problems which are described by cone constraints.

In the first chapter of Part II: Numerical Methods, Coope and Price trace and study the application of exact penalty function methods to the solution of SIP problems. Next, in an article by Lawrence and Tits, the convergence and the effectiveness of a new sequential quadratic programming algorithm for the solution of finely discretized nonlinear SIP problems or other problems with many inequality constraints are verified. Afterwards, Reemtsen and Gorner describe the fundamental ideas for the numerical solution of SIP problems and provide a comprehensive survey of existing methods. Connections between SIP and semi-definite programming are finally explored by Vandenberghe and Boyd, who especially study a number of applications, including such from signal processing, computational geometry, and statistics.

The last part, Part III: Applications, begins with an article by Altinel and Ozekici, who investigate an approach in reliability testing of a complex system, which leads to the solution of a parameterized linear SIP problem. It follows an article by Kortanek and Moulin who focus on certain wavelet design problems which lead to linear SIP models and can be solved in this way. Afterwards, Potchinkov gives an introduction to FIR filter design and shows that the main design problems also in this field can be best modelled and solved as (convex) SIP programs. Finally, in the contribution by Sachs, several control problems are described and studied from the viewpoint of SIP, where, in particular, the problems of food sterilization and the flutter of aircraft wings are discussed.

We are very grateful to all authors of this book for their valuable contributions and to the referees of the articles for their qualified reports. We furthermore wish to express our sincere gratitude to John R. Martindale from Kluwer Aca­demic Publishers for offering us the opportunity to edit this volume and for his practical help, encouragement, and understanding support. Finally, we would like to thank Jorg Biesold for the careful preparation of the camera-ready ver­sion of the manuscript.

Preface Xlll

We hope that this volume will give a substantial impetus to the further devel­opment of semi-infinite programming, and we invite the reader to participate in the research and application of this very interesting field.

Cottbus and Erlangen, December 1997

REMBERT REEMTSEN

J AN-J. RUCKMANN

CONTRIBUTERS

i. Kuban Altmel Department of Industrial Engineering, BogazilSi Univer­sity, Istanbul, Thrkey

Stephen Boyd Electrical Engineering Department, Stanford University, Stanford, California, USA

Ian D. Coope Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

Miguel A. Goberna Department of Statistics and Operations Research, University of Alicante, Alicante, Spain

Stephan Garner Department of Mathematics, Technical University of Ber­lin, Berlin, Germany

Rene Henrion Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

Hubertus Th. Jongen Department of Mathematics, RWTH-Aachen, Aa­chen, Germany

Diethard Klatte Institute for Operations Research, University of Zurich, Zurich, Switzerland

Ken O. Kortanek College of Business Administration, University of Iowa, Iowa City, Iowa, USA

Craig T. Lawrence Department of Electrical Engineering and Institute for Systems Research, University of Maryland, College Park, Maryland, USA

Marco A. Lopez Department of Statistics and Operations Research, Uni­versity of Alicante, Alicante, Spain

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Pierre Moulin Beckman Institute, University of Illinois, Urbana, Illinois, USA

Siileyman Ozekici Department of Industrial Engineering, Bogazic;i Univer­sity, Istanbul, 'TUrkey

Alexander W. Potchinkov Institute of Mathematics, Brandenburg Tech­nical University of Cottbus, Cottbus, Germany

Christopher J. Price Department of Mathematics and Statistics, Univer­sity of Canterbury, Christchurch, New Zealand

Rembert Reemtsen Institute of Mathematics, Brandenburg Technical Uni­versity of Cottbus, Cottbus, Germany

Jan-J. Riickmann Institute of Applied Mathematics, University of Erlan­gen-Nuremberg, Erlangen, Germany

Ekkehard W. Sachs Department of Mathematics, University of Thier, Thier, Germany

Alexander Shapiro School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA,

Andre L. Tits Department of Electrical Engineering and Institute for Sys­tems Research, University of Maryland, Maryland, USA

Lieven Vandenberghe Electrical Engineering Department, University of California, Los Angeles, California, USA