Optimization Theory and Applications

ISBN-13: 9783528085940 / Angielski / Miękka / 1984 / 234 str.

Jochen Werner

Opis

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.

Spis treści

§ 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.