Well-Posed Optimization Problems
This book presents in a unified way the mathematical theory of well-posedness in optimization. The basic concepts of well-posedness and the links among them are studied, in particular Hadamard and Tykhonov well-posedness. Abstract optimization problems as
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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A. L. Dontchev T. Zolezzi
Well-Posed Optimization Problems
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Authors Asen L. Dontchev Bulgarian Academy of Sciences Institute of Mathematics Acad. G. Bonchev 8 1113 Sofia, Bulgaria Tullio Zolezzi Universita di Genova Dipartimento di Matematica Via L.B. Alberti 4 16132 Genova, Italy
Mathematics Subject Classification (1991): Primary: 49-02, 49K40 Secondary: 41A50, 49Jl5, 90C31 ISBN 3-540-56737-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56737-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany
Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper
To our parents, and To Dora, Mira, Kiril, Orietta and Guido
PREFACE
This book aims to present, in a unified way, some basic aspects of the mathematical theory of well-posedness in scalar optimization. The first fundamental concept in this area. is inspired by the classical idea of J. Hadamard, which goes back to the beginning of this century. It requires existence and uniqueness of the optimal solution together with continuous dependence on the problem's data. In the early sixties A. Tykhonov introduced another concept of well-posedness imposing convergence of every minimizing sequence to the unique minimum point. Its relevance to (and motivation from) the approximate (numerical) solution of optimization problems is clear. In the book we study both the Tykhonov and the Hadamard concepts of well-posedness, the links between them and also some extensions (e.g. relaxing the uniqueness). Both the pure and the applied sides of our topic are presented. The first four chapters are devoted to abstract optimization problems. Applications to optimal control, cs'culus of variations and mathematical programming are the subject matter of the remaining five chapters. Chapter I contains the basic facts about Tykhonov well-posedness and its generalizations. The main metric, topological and differential characterizations are discussed. The Tykhonov regularization method is outlined. Chapter II is the key chapter (as we see from its introduction) because it is devoted to a basic issue: the relationships between Tykhonov and Hadamard well-posedness. We emphasize the fundamental links between the two conc
