Introduction to the Theory of Nonlinear Optimization

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Springer Science & Business Media, Jun 30, 2007 - Mathematics - 292 pages
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This book serves as an introductory text to optimization theory in normed spaces and covers all areas of nonlinear optimization. It presents fundamentals with particular emphasis on the application to problems in the calculus of variations, approximation and optimal control theory. The reader is expected to have a basic knowledge of linear functional analysis.

 

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Contents

Introduction and Problem Formulation
1
Existence Theorems for Minimal Points
6
22 Existence Theorems
8
23 Set of Minimal Points
18
24 Application to Approximation Problems
19
25 Application to Optimal Control Problems
23
Exercises
29
Generalized Derivatives
31
Duality
159
62 Duality Theorems
164
63 Saddle Point Theorems
168
64 Linear Problems
172
65 Application to Approximation Problems
175
Exercises
184
Application to Extended Semidefinite Optimization
186
72 Optimality Conditions
202

32 Gâteaux and Fréchet Derivatives
37
33 Subdifferential
49
34 Quasidifferential
57
35 Clarke Derivative
67
Exercises
75
Tangent Cones
79
42 Optimality Conditions
88
43 A Lyusternik Theorem
95
Exercises
103
Generalized Lagrange Multiplier Rule
105
52 Necessary Optimality Conditions
108
53 Sufficient Optimality Conditions
126
54 Application to Optimal Control Problems
136
Exercises
156
73 Duality
207
Exercises
210
Direct Treatment of Special Optimization Problems
213
82 Time Minimal Control Problems
221
Exercises
238
Weak Convergence
241
Reflexivity of Banach Spaces
243
HahnBanach Theorem
245
Partially Ordered Linear Spaces
249
Bibliography
252
Answers to the Exercises
275
Index
289
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