Lectures on Mathematical Theory of Extremum Problems
The author of this book, Igor' Vladimirovich Girsanov, was one of the first mathematicians to study general extremum problems and to realize the feasibility and desirability of a unified theory of extremal problems, based on a functional analytic approac
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67 I. V. Girsanov
Lectures on Mathematical Theory of Extremum Problems
Spri nger-Verlag Berlin· Heidelberg· New York 1972
Advisory Board H. Albach· A. V. Balakrishnan' F. Ferschl . R. E. Kalman' W. Krelle' G. Seegmiiller N. Wirth Igor Vladimirovich Girsanovt Edited by Prof. B. T. Poljak Moscow State University Computer Center Moscow V-234/USSR Translated from the Russian by D. Louvish Israel Program for Scientific Translations Kiryat Moshe P. O. Box 7145 ] erusalem/Israel
AMS Subject Classifications (1970): 46N05, 49B30, 49B40, 52A40
ISBN -13:978-3-540-05857-1
e- ISBN -13:978-3-642-80684-1
DO I: 10.1007/978-3-642-80684-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 72-80360.
Extremal problems are now playing an ever-increasing role in applications of mathematics.
It has been discovered that, notwithstanding the great diversity of
these problems, they can be attacked by a unified functional-analytic approach, first suggested by A. Ya. Dubovitskii and A. A. Milyutin.
The book is devoted to an
exposition of this approach and its application to the analysis of specific extremal problems.
All requisite material from functional analysis is first presented, and
a general scheme for derivation of optimum conditions is then described.
Using
this scheme, necessary extremum conditions are then derived for a series of problems - ranging from Pontryagin's maximum principle in optimal control theory to duality theorems in linear programming. The book should be of interest not only to mathematicians, but also to those working in other fields involving optimization problems.
TABLE OF CONTENTS
Editor's preface
1
Lecture 1,
Introduction
2
Lecture 2,
Topological linear spaces, convex sets, weak topologies
11
Lecture 3,
Hahn-Banach Theorem
21
Lecture 4,
Supporting hyperplanes and extremal points
25
Lecture 5,
Cones, dual cones
30
Lecture 6,
Necessary extremum conditions (Euler-Lagrange equation)
38
Lecture 7,
Directions of decrease
43
Lecture 8,
Feasible directions
58
Lecture 9,
Tangent directions
61
Lecture 10,
Calculation of dual cones
69
Lecture 11,
Lagrange multipliers and the Kuhn-Tucker Theorem
78
Lecture 12,
Problem of optimal control.
Local maximum principle
83
Lecture 13,
Problem of optimal control,
Maximum principle
93
Lecture 14,
Problem of optimal control.
Constraints on phase coordinates,
minimax problem
105
Lecture 15,
Sufficient extremum conditions
114
Lecture 16,
Sufficient extremum eonditions,
Examples
121
Suggestions for further r
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