Optimization Theory and Applications

by Jochen Werner

Overview

This book is a slightly augmented version of a set of lec- tures on optimization which I held at the University of Got- tingen in the winter semester 1983/84. The lectures were in- tended to give an introduction to the foundations and an im- pression of the applications of optimization theory. Since in- finite 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 com- pletely 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 program- ming and stability theory. I am very grateful to various people for their help in pro- ducing this text. R. Schaback encouraged me to publish my lec- tures and put me in touch with the Vieweg-Verlag. W. BrUbach and O. Herbst proofread the manuscript; the latter also pro- duced 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.

Product Details

ISBN-13:	9783528085940
Publisher:	Vieweg+Teubner Verlag
Publication date:	01/01/1984
Series:	Advanced Lectures in Mathematics
Edition description:	1984
Pages:	233
Product dimensions:	6.69(w) x 9.61(h) x 0.02(d)
Language:	German

Table of Contents

§ 1 Introduction, Examples, Survey.- 1.1 Optimization problems in elementary geometry.- 1.2 Calculus of variations.- 1.3 Approximation problems.- 1.4 Linear programming.- 1.5 Optimal Control.- 1.6 Survey.- 1.7 Literature.- § 2 Linear Programming.- 2.1 Definition and interpretation of the dual program.- 2.2 The FARKAS-Lemma and the Theorem of CARATHEODORY.- 2.3 The strong duality theorem of linear programming.- 2.4 An application: relation between inradius and width of a polyhedron.- 2.5 Literature.- § 3 Convexity in Linear and Normed Linear Spaces.- 3.1 Separating convex sets in linear spaces.- 3.2 Separation of convex sets in normed linear spaces.- 3.3 Convex functions.- 3.4 Literature.- § 4 Convex Optimization Problems.- 4.1 Examples of convex optimization problems.- 4.2 Definition and motivation of the dual program. The weak duality theorem.- 4.3 Strong duality, KUHN-TUCKER saddle point theorem.- 4.4 Quadratic programming.- 4.5 Literature.- § 5 Necessary Optimality Conditions.- 5.1 GATEAUX and FRECHET Differential.- 5.2 The Theorem of LYUSTERNIK.- 5.3 LAGRANGE multipliers. Theorems of KUHN-TUCKER and F. JOHN type.- 5.4 Necessary optimality conditions of first order in the calculus of variations and in optimal control theory.- 5.5 Necessary and sufficient optimality conditions of second order.- 5.6 Literature.- § 6 Existence Theorems for Solutions of Optimization Problems.- 6.1 Functional analytic existence theorems.- 6.2 Existence of optimal controls.- 6.3 Literature.- Symbol Index.