Introduction to the Theory of Nonlinear Optimization Third Editi....pdf

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Introduction to the Theory of Nonhnear Optimization

Johannes Jahn Introduction to the Theory of NonHnear Optimization Third Edition With 31 Figures Sprin g er

Prof. Dr. Johannes Jahn Universitat Erlangen-Niirnberg Institut fur Angewandte Mathematik Martensstr. 3 91058 Erlangen Germany jahn@am.uni-erlangen.de Library of Congress Control Number: 2006938674 ISBN 978-3-540-49378-5 Springer Berlin Heidelberg New York ISBN 978-3-540-61407-4 Second Edition Springer Berlin Heidelberg New York 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, reuse of illustrations, recitation, broad casting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. Springer is part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 1994,1996,2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production: LE-TgX Jelonek, Schmidt & Vockler GbR, Leipzig Cover-design: Erich Kirchner, Heidelberg SPIN 11932048 42/3100YL - 5 4 3 2 10 Printed on acid-free paper

To Claudia and Martin

Preface This book presents an application-oriented introduction to the the ory of nonhnear optimization. It describes basic notions and concep tions of optimization in the setting of normed or even Banach spaces. Various theorems are appHed to problems in related mathematical areas. For instance, the Euler-Lagrange equation in the calculus of variations, the generahzed Kolmogorov condition and the alternation theorem in approximation theory as well as the Pontryagin maximum principle in optimal control theory are derived from general results of optimization. Because of the introductory character of this text it is not intended to give a complete description of all approaches in optimization. For instance, investigations on conjugate duality, sensitivity, stability, re cession cones and other concepts are not included in the book. The bibliography gives a survey of books in the area of nonlinear optimization and related areas like approximation theory and optimal control theory. Important papers are cited as footnotes in the text. This third edition is an enlarged and revised version containing an additional chapter on extended semidefinite optimization and an updated bibliography. I am grateful to S. GeuB, S. Gmeiner, S. Keck, Prof. Dr. E.W. Sachs and H. Winkler for their support, and I am especially indebted to D.G. Cunningham, Dr. G. Eichfelder, Dr. F. Hettlich, Dr. J. Klose, Prof. Dr. E.W. Sachs, Dr. T. Staib and Dr. M. Stingl for fruitful discussions. Erlangen, September 2006 Johannes Jahn

Contents Preface 1 Introduction and Problem Formulation 2 Existence Theorems for Minimal Points 2.1 Problem Formulation 2.2 Existence Theorems 2.3 Set of Minimal Points 2.4 Application to Approximation Problems 2.5 Application to Optimal Control Problems Exercises 3 Generalized Derivatives 3.1 Directional Derivative 3.2 Gateaux and Frechet Derivatives 3.3 Subdifferential 3.4 Quasidifferential 3.5 Clarke Derivative Exercises 4 Tangent Cones 4.1 Definition and Properties 4.2 Optimality Conditions 4.3 A Lyusternik Theorem Exercises 5 Generalized Lagrange Multiplier Rule 5.1 Problem Formulation vii 1 7 7 8 18 19 23 29 31 31 37 49 57 67 75 79 79 88 95 103 105 105

X Contents 5.2 Necessary Optimality Conditions 5.3 Sufficient Optimality Conditions 5.4 Application to Optimal Control Problems Exercises 6 Duality 6.1 Problem Formulation 6.2 Duality Theorems 6.3 Saddle Point Theorems 6.4 Linear Problems 6.5 Application to Approximation Problems Exercises 108 126 136 156 159 159 164 168 172 175 184 7 Application to Extended Semidefinite Optimization 187 187 202 207 210 7.1 Lowner Ordering Cone and Extensions 7.2 Optimality Conditions 7.3 Duality Exercises 8 Direct Treatment of Special Optimization Problems 213 213 221 238 8.1 Linear Quadratic Optimal Control Problems 8.2 Time Minimal Control Problems Exercises A Weak Convergence B Reflexivity of Banach Spaces C Hahn-Banach Theorem D Partially Ordered Linear Spaces Bibliography Answers to the Exercises Index 241 243 245 249 253 275 289

Chapter 1 Introduction and Problem Formulation In optimization one investigates problems of the determination of a minimal point of a functional on a nonempty subset of a real linear space. To be more specific this means: Let X be a real linear space, let S' be a nonempty subset of X, and let / : iS —> R be a given functional. We ask for the minimal points of / on S. An element X E S is called a minimal point offonS if f{x) < f{x) for all xeS. The set S is also called constraint set^ and the functional / is called objective functional In order to introduce optimization we present various typical op timization problems from Applied Mathematics. First we discuss a design problem from structural engineering. Example 1.1. As a simple example consider the design of a beam with a rectangular cross-section and a given length I (see Fig. 1.1 and 1.2). The height xi and the width X2 have to be determined. The design variables Xi and X2 have to be chosen in an area which makes sense in practice. A certain stress condition must be satisfied, i.e. the arising stresses cannot exceed a feasible stress. This leads to the inequality 2000 < x\x2. (1.1)

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