Optimization Theory and Applications
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Advanced Lectures
in Mathematics Edited by Gerd Fischer
Jochen Werner Optimization. Theory and Applications Manfred Denker Nonparametric Statistics
Jochen Werner
Optimization Theory and Applications
M
Friedr. Vieweg & Sohn
Braunschweig/Wiesbaden
CIP-Kurztlteleufnehme der Deutschen Bibliothek Werner, Jochen: Optimizetion - theory end epplications I Jochen Werner. -,Breunschweig; Wiesbeden: Vieweg, 1984. (Advences lectures in metherneticsl ISBN-13: 978-3-528'()8594'() e-ISBN-13:978-3-322-84035-6 DOl: 10.1007/978-3-322-84035-6
1984 All rights reserved © Friedr. Vieweg & Sohn Verlegsgesellscheft mbH. Breunschweig 1984 No pert of this publicetion mey be reproduced, stored in e retrievel system or trensmitted in eny form or by eny meens, electronic, mechenicel, photocopying, recording or otherwise, without prior permission of the copyright holder. Produced,by IVD, Welluf b. Wiesbeden
ISBN-13:97S-3-52S-GS594-0
v
PREFACE This book is a slightly augmented version of a set of lectures on optimization which I held at the University of Gottingen in the winter semester 1983/84. The lectures were intended to give an introduction to the foundations and an impression of the applications of optimization theory. Since infinite dimensional problems were also to be treated and one could only assume a minimal knowledge of functional analysis, the necessary tools from functional analysis were almost completely developed during the course of the semester. The most important aspects of the course are the duality theory for convex programming and necessary optimality conditions for nonlinear optimization problems; here we strive to make the geometric background particularly clear. For lack of time and space we were not able to go into several important problems in optimization - e.g. vector optimization, geometric programming and stability theory. I am very grateful to various people for their help in producing this text. R. Schaback encouraged me to publish my lectures and put me in touch with the Vieweg-Verlag. W. BrUbach and O. Herbst proofread the manuscript; the latter also produced the drawings and assembled the index. I am indebted to W. LUck for valuable suggestions for improvement. I am also particularly grateful to R. Switzer, who translated the German text into English. Finally I wish to thank Frau P. Trapp for her
Gare and patience in typing the final version.
Gottingen, June 1984
Jochen Werner
VI
CONTENTS § 1
INTRODUCTION, EXAMPLES, SURVEY 1.1 1.2 1.3 1.4 1.5 1.6 1.7
§ 2
2.2 2.3 2.4 2.5
14 19 24 27 28
Definition and interpretation of the dual program The FARKAS-Lemma and the Theorem of CARATHEODORY The strong duality theorem of linear programming An application: relation between inradius and width of a polyhedron Literature
,30 37 44 50 55
CONVEXITY 'IN LINEAR AND NORMED LINEAR SPACES 3.1 3.2
3.3 3.4
§ 4
10
LINEAR PROGRAMMING 2.1
§ 3
Optimization problems in elementary g-eometry Calculus of variations Approximation problems Linear programming Optimal Control Survey Literature
Separating conv
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