Optimality Conditions : Abnormal and Degenerate Problems - Aram Arutyunov

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Optimality Conditions

Abnormal and Degenerate Problems

By: Aram Arutyunov

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Hardcover | 31 October 2000

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312 Pages

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24.13 x 16.51 x 1.91

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This book is devoted to one of the main questions of the theory of extremal prob lems, namely, to necessary and sufficient extremality conditions. It is intended mostly for mathematicians and also for all those who are interested in optimiza tion problems. The book may be useful for advanced students, post-graduated students, and researchers. The book consists of four chapters. In Chap. 1 we study the abstract minimization problem with constraints, which is often called the mathemati cal programming problem. Chapter 2 is devoted to one of the most important classes of extremal problems, the optimal control problem. In the third chapter we study one of the main objects of the calculus of variations, the integral quadratic form. In the concluding, fourth, chapter we study local properties of smooth nonlinear mappings in a neighborhood of an abnormal point. The problems which are studied in this book (of course, in addition to their extremal nature) are united by our main interest being in the study of the so called abnormal or degenerate problems. This is the main distinction of the present book from a large number of books devoted to theory of extremal problems, among which there are many excellent textbooks, and books such as, e.g., [13, 38, 59, 78, 82, 86, 101, 112, 119], to mention a few.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Optimisation Game Theory Medicine Medicine in General Computing & I.T.Computer Programming & Software Development Applied Mathematics Linear Programming

Preface	p. vii
Extremal Problems with Constraints	p. 1
Extremal Problems with Constraints. Normal and Abnormal Points	p. 1
Elementary Statements (Finite-Dimensional Case)	p. 8
Certain Notation and Concepts	p. 11
Statement of the First- and Second-Order Conditions	p. 15
Lower Estimates for Upper Topological Limits of Sequences of Subspaces	p. 22
Proof of Theorems 4.1, 4.2, and 4.3	p. 23
Proof of Theorem 5.1	p. 31
Sufficient Second-Order Conditions	p. 36
Interconnection of Necessary and Sufficient Second-Order Conditions. 2-Normal Mappings	p. 41
Properties of 2-Normal Mappings	p. 44
Lagrange-Avakov Function and Necessary Extremality Conditions	p. 53
Theorem on the Tangent Cone. Tuples	p. 60
Proof of Theorem 11.1	p. 69
Higher-Order Necessary Conditions	p. 71
Sufficient Conditions for Abnormal Problems. Higher-Order Sufficient Conditions	p. 74
Proof of Theorems 15.1 and 15.2	p. 78
Optimal Control Problem. Pontryagin Maximum Principle	p. 89
Statement of the Problem	p. 89
Basic Assumptions and Notation	p. 92
Pontryagin Maximum Principle for the Simplest Problem	p. 98
Statement of the Pontryagin Maximum Principle. State Constraints and the Degeneration Phenomenon	p. 107
Linear-Convex Problems	p. 115
Proof of the Weakened Maximum Principle for a Linear-Convex Problem Without State Constraints	p. 123
Proof of the Maximum Principle in a Linear-Convex Problem with State Constraints	p. 133
Proof of the Pontryagin Maximum Principle. Finite-Dimensional Approximation Method	p. 141
Penalty Method. Necessary Conditions in the [mu]-Problem	p. 147
Completing the Proof of the Weakened Maximum Principle	p. 155
v-Problem and Completing the Proof of the Maximum Principle	p. 164
A Little More About the Nondegeneracy of the Maximum Principle	p. 167
Relaxations and Perturbations of Optimal Control Problems	p. 174
Degenerate Quadratic forms of the Calculus of Variations	p. 181
Statement of the Problem	p. 181
Constructions and the Notation. The spaces W[superscript n subscript A,B] [theta subscript 1, theta subscript 2] and W[superscript n] [theta subscript 1, theta subscript 2]	p. 183
Statement of the Main Results	p. 188
Discussion of the Main Results. Examples	p. 196
Proof of Theorem 3.1	p. 198
Proof of Theorem 3.2	p. 208
Proof of Theorem 3.4	p. 221
Necessary and Sufficient Conditions for a Local Minimum in Degenerate Problems of the Calculus of Variations	p. 242
Study of Mappings in a Neighborhood of an Abnormal Point	p. 245
Implicit Function Theorem and Abnormal Points	p. 245
Discussion and Auxiliary Results	p. 249
Proof of the Inverse and Implicit Function Theorems	p. 258
On the Existence of Regular Zeros for a Quadratic Mapping	p. 265
Level Set of a Smooth Mapping in a Neighborhood of an Abnormal Point	p. 268
Criterion for the Strong 2-Regularity of Quadratic Mappings	p. 279
References	p. 287
Index	p. 293
List of Notation	p. 297
Table of Contents provided by Syndetics. All Rights Reserved.

Optimality Conditions

Abnormal and Degenerate Problems

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Hardcover

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