Nonsmooth Optimization : Analysis and Algorithms with Applications to Optimal Control

Analysis and Algorithms with Applications to Optimal Control

By: Marko M Makela, Pekka Neittaanmaki

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Hardcover | 1 May 1992

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268 Pages

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This book is a self-contained elementary study for nonsmooth analysis and optimization, and their use in solution of nonsmooth optimal control problems. The first part of the book is concerned with nonsmooth differential calculus containing necessary tools for nonsmooth optimization. The second part is devoted to the methods of nonsmooth optimization and their development. A proximal bundle method for nonsmooth nonconvex optimization subject to nonsmooth constraints is constructed. In the last part nonsmooth optimization is applied to problems arising from optimal control of systems covered by partial differential equations. Several practical problems, like process control and optimal shape design problems are considered.

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Non-Fiction Mathematics Applied Mathematics Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory Optimisation Game Theory Calculus & Mathematical Analysis Differential Calculus & Equations Probability & Statistics

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Preface
List of Symbols
Nonsmooth Analysis
Introduction	p. 3
Notations	p. 5
Basic Results	p. 6
Convex Analysis	p. 8
Functions, Derivatives and Subgradients	p. 8
Sets, Tangents and Normals	p. 17
Links between Geometry and Differentiation	p. 22
Nonsmooth Differential Theory	p. 29
Generalization of Derivative	p. 29
Subdifferential Calculus	p. 36
[Epsilon]-subdifferentials	p. 52
Convex Case	p. 52
Nonconvex Case	p. 55
Generalized Jacobians	p. 57
Nonsmooth Geometry	p. 59
Generalization of Tangents and Normals	p. 59
Epigraphs and Level Sets of Nonsmooth Functions	p. 64
Nonsmooth Optimization Theory	p. 70
Optimality Conditions	p. 70
Linearization of Unconstrained Problem	p. 75
Linearization of Constrained Problem	p. 83
Nonsmooth Optimization
Introduction	p. 93
A Survey of Bundle Methods	p. 97
Preliminaries	p. 97
Smooth Minimization Methods	p. 97
Subgradient methods	p. 99
Lemarechal's [Epsilon]-steepest Descent Method	p. 101
Theoretical Foundation	p. 101
Implementation	p. 102
Kiwiel's Generalized Cutting Plane Method	p. 104
The Cutting Plane Method	p. 104
A Generalization of the Cutting Plane Method	p. 106
Proximal Bundle and Bundle Trust Region Methods	p. 107
The Tilted Proximal Bundle Method	p. 109
Proximal Bundle Method for Nonconvex Constrained Optimization	p. 112
Direction Finding	p. 113
Derivation of the Direction Finding Problem	p. 113
Subgradient Aggregation	p. 117
Nonconvexity	p. 122
Line Search	p. 126
Stopping Criterion	p. 130
Weight Updating	p. 132
Algorithm	p. 135
Numerical Experiments	p. 138
Test Examples	p. 138
Numerical Results	p. 142
Nonsmooth Optimal Control
Introduction	p. 147
Preliminaries	p. 149
Basics for Sensitivity Analysis	p. 149
Preliminaries from Functional Analysis	p. 150
Distributed Parameter Control Problems	p. 154
Introduction	p. 154
Setting of the Problem. Existence of the Solution	p. 154
Practical Model Problems	p. 156
Axially Loaded Rod with Stress Constraints	p. 157
Clambed Beam with Displacement Constraints	p. 165
Clambed Beam with Obstacle	p. 172
Optimal Shape Design	p. 183
Introduction	p. 183
Setting of the Abstract Problem	p. 186
Optimal Shape Design with Unilateral Boundary Value Problem	p. 191
Setting of the Problem	p. 192
Discretization of the Problem	p. 195
Design Sensitivity Analysis	p. 199
Numerical Results	p. 204
Design of Optimal Covering	p. 210
Setting of the Problem	p. 211
Discretization of the Problem	p. 214
Design Sensitivity Analysis	p. 216
Numerical Results	p. 218
Boundary Control for Stefan Type Problems	p. 223
Introduction	p. 223
Setting of the Abstract Problem	p. 223
Simulation of the Continuous Casting Process	p. 225
Setting of the Problem	p. 225
Penalization of the State Constraints	p. 228
Discretization of the Problem	p. 228
Sensitivity Analysis	p. 231
Numerical Results	p. 233
References	p. 239
Index	p. 251
Table of Contents provided by Blackwell. All Rights Reserved.

Nonsmooth Optimization : Analysis and Algorithms with Applications to Optimal Control

Analysis and Algorithms with Applications to Optimal Control

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Hardcover

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