Ebook: Connectedness and Necessary Conditions for an Extremum

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 ap­ proach 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 lin­ ear topological space. We also show an application of our result to some problems of nonlinear programming and optimal control.

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This monograph is the first book in the study of necessary conditions of an extremum in which topological connectedness plays a major role. Many new and original results are presented here. The synthesis of the well-known Dybrovitskii-Milyutin approach, based on functional analysis, and topological methods permits the derivation of the so-called alternative conditions of an extremum: 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, the present approach gives a transparent answer to the question why the Kuhn-Tucker theorem establishes the restrictions on the signs of the Lagrange multipliers for the inequality constraints but why this theorem does not establish any analogous restrictions on the multipliers for the equality constraints.
Examples from mathematical economics illustrate the alternative conditions of any extremum. Parallels are drawn between these examples and the problems of static equilibrium in classical mechanics.
Audience: This volume will be of use to mathematicians and graduate students interested in the areas of optimization, optimal control and mathematical economics.

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This monograph is the first book in the study of necessary conditions of an extremum in which topological connectedness plays a major role. Many new and original results are presented here. The synthesis of the well-known Dybrovitskii-Milyutin approach, based on functional analysis, and topological methods permits the derivation of the so-called alternative conditions of an extremum: 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, the present approach gives a transparent answer to the question why the Kuhn-Tucker theorem establishes the restrictions on the signs of the Lagrange multipliers for the inequality constraints but why this theorem does not establish any analogous restrictions on the multipliers for the equality constraints.
Examples from mathematical economics illustrate the alternative conditions of any extremum. Parallels are drawn between these examples and the problems of static equilibrium in classical mechanics.
Audience: This volume will be of use to mathematicians and graduate students interested in the areas of optimization, optimal control and mathematical economics.
Content:
Front Matter....Pages i-xi
Preliminaries....Pages 1-20
Alternative Conditions for an Extremum of the First Order....Pages 21-59
Alternative Conditions for an Extremum in Nonlinear Programming....Pages 60-102
Alternative Conditions for an Extremum in Optimal Control Problems....Pages 103-156
Necessary Conditions for an Extremum in a Measure Space....Pages 157-188
Back Matter....Pages 189-203

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This monograph is the first book in the study of necessary conditions of an extremum in which topological connectedness plays a major role. Many new and original results are presented here. The synthesis of the well-known Dybrovitskii-Milyutin approach, based on functional analysis, and topological methods permits the derivation of the so-called alternative conditions of an extremum: 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, the present approach gives a transparent answer to the question why the Kuhn-Tucker theorem establishes the restrictions on the signs of the Lagrange multipliers for the inequality constraints but why this theorem does not establish any analogous restrictions on the multipliers for the equality constraints.
Examples from mathematical economics illustrate the alternative conditions of any extremum. Parallels are drawn between these examples and the problems of static equilibrium in classical mechanics.
Audience: This volume will be of use to mathematicians and graduate students interested in the areas of optimization, optimal control and mathematical economics.
Content:
Front Matter....Pages i-xi
Preliminaries....Pages 1-20
Alternative Conditions for an Extremum of the First Order....Pages 21-59
Alternative Conditions for an Extremum in Nonlinear Programming....Pages 60-102
Alternative Conditions for an Extremum in Optimal Control Problems....Pages 103-156
Necessary Conditions for an Extremum in a Measure Space....Pages 157-188
Back Matter....Pages 189-203
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