## Optimization Theory and ApplicationsThis 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. |

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### Contents

INTRODUCTION EXAMPLES SURVEY | 1 |

NECESS ARY OPTIMALITY CONDITIONS | 5 |

CONVEXITY IN LINEAR AND NORMED LINEAR SPACES | 56 |

Copyright | |

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affine manifold approximation problem arbitrary assume b-Ax Banach space bºy calculus of variations closed hyperplane compact constraint qualification contradiction converges convex cone convex functions convex program convex sets define Definition differentiable dual program equations Example FARKAS-Lemma finite dimensional follows halfspace hyperplane H icr epi f Lagrange multiplier Lemma linear program linear subspace LYUSTERNIK Maximize Minimize cºx Minimize f necessary optimality conditions nonnegative halfspace normal form normed linear space optimal control optimization problem production plan program dual proof prove quadratic program Remark resp separation theorem solution span strong duality theorem subset Suppose given triangle weak duality theorem weakly sequentially x e IR x"ox