Connectedness and Necessary Conditions for an Extremum

by Alexey Abramov

Overview

The present book is the outcome of efforts to introduce topological connectedness as one of the basic tools for the study of necessary conditions for an extremum. Apparently this monograph is the first book in the theory of maxima and minima where topological connectedness is used so widely for this purpose. Its application permits us to obtain new results in this sphere and to consider the classical results from a nonstandard point of view. Regarding the style of the present book it should be remarked that it is comparatively elementary. The author has made constant efforts to make the book as self-contained as possible. Certainly, familiarity with the basic facts of topology, functional analysis, and the theory of optimization is assumed. The book is written for applied mathematicians and graduate students interested in the theory of optimization and its applications. We present the synthesis of the well known Dybovitskii'-Milyutin approach for the study of necessary conditions for an extremum, based on functional analysis, and topological methods. This synthesis allows us to show that in some cases we have the following important result: if the Euler equation has no non trivial solution at a point of an extremum, then some inclusion is valid for the functionals belonging to the dual space. This general result is obtained for an optimization problem considered in a linear topological space. We also show an application of our result to some problems of nonlinear programming and optimal control.

Product Details

ISBN-13:	9780792349105
Publisher:	Springer Netherlands
Publication date:	03/31/1998
Series:	Mathematics and Its Applications , #431
Edition description:	1998
Pages:	204
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

0. Preliminaries.- 1. Alternative conditions for an extremum of the first order.- 2. Alternative conditions for an extremum in nonlinear programming.- 3. Alternative conditions for an extremum in optimal control problems.- 4. Necessary conditions for an extremum in a measure space.- List of notation.

Editorial Reviews

For mathematicians and graduate students interested in optimization, optimal control, and mathematical economics, investigates the necessary conditions of an extremum in which topological connectedness plays a major role. Presenting new and original results, synthesizes the Dybrovitskii-Milyutin approach based on functional analysis with topological methods to derive the supposed alternative conditions of an extremum, that is, if the Euler equation has the trivial solution only at an extreme point, then some inclusion is valid for the functionals belonging to the dual space. Also explains why the Kuhn- Tucker theorem establishes the restrictions on the signs of the Lagrange multipliers for the inequality restraint but no analogous restrictions on the multipliers for the equality constraints. Translated and revised from the 1996 Russian edition published by Computing Centre RAS Publications, Moscow. Annotation c. by Book News, Inc., Portland, Or.

Booknews