Bertsekas / Rheinboldt Constrained Optimization and Lagrange Multiplier Methods

1. Auflage 2014
ISBN: 978-1-4832-6047-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 412 Seiten, Web PDF

Computer Science and Applied Mathematics: Constrained Optimization and Lagrange Multiplier Methods focuses on the advancements in the applications of the Lagrange multiplier methods for constrained minimization. The publication first offers information on the method of multipliers for equality constrained problems and the method of multipliers for inequality constrained and nondifferentiable optimization problems. Discussions focus on approximation procedures for nondifferentiable and ill-conditioned optimization problems; asymptotically exact minimization in the methods of multipliers; duality framework for the method of multipliers; and the quadratic penalty function method. The text then examines exact penalty methods, including nondifferentiable exact penalty functions; linearization algorithms based on nondifferentiable exact penalty functions; differentiable exact penalty functions; and local and global convergence of Lagrangian methods. The book ponders on the nonquadratic penalty functions of convex programming. Topics include large scale separable integer programming problems and the exponential method of multipliers; classes of penalty functions and corresponding methods of multipliers; and convergence analysis of multiplier methods. The text is a valuable reference for mathematicians and researchers interested in the Lagrange multiplier methods.

Autoren/Hrsg.

Bertsekas, Dimitri P.

Rheinboldt, Werner

Inhaltsverzeichnis

1;Front Cover;1
2;Constrained Optimization and Lagrange Multiplier Methods;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;12
7;Chapter 1. Introduction;16
7.1;1.1 General Remarks;16
7.2;1.2 Notation and Mathematical Background;21
7.3;1.3 Unconstrained Minimization;33
7.4;1.4 Constrained Minimization;81
7.5;1.5 Algorithms for Minimization Subject to Simple Constraints;91
7.6;1.6 Notes and Sources;108
8;Chapter 2. The Method of Multipliers for Equality Constrained Problems;110
8.1;2.1 The Quadratic Penalty Function Method;111
8.2;2.2 The Original Method of Multipliers;119
8.3;2.3 Duality Framework for the Method of Multipliers;140
8.4;2.4 Multiplier Methods with Partial Elimination of Constraints;156
8.5;2.5 Asymptotically Exact Minimization in Methods of Multipliers;162
8.6;2.6 Primal-Dual Methods Not Utilizing a Penalty Function;168
8.7;2.7 Notes and Sources;171
9;Chapter 3. The Method of Multipliers for Inequality Constrained and Nondifferentiable Optimization Problems;173
9.1;3.1 One-Sided Inequality Constraints;173
9.2;3.2 Two-Sided Inequality Constraints;179
9.3;3.3 Approximation Procedures for Nondifferentiable and Ill-Conditioned Optimization Problems;182
9.4;3.4 Notes and Sources;193
10;Chapter 4. Exact Penalty Methods and Lagrangian Methods;194
10.1;4.1 Nondifferentiable Exact Penalty Functions;195
10.2;4.2 Linearization Algorithms Based on Nondifferentiable Exact Penalty Functions;211
10.3;4.3 Differentiable Exact Penalty Functions;221
10.4;4.4 Lagrangian Methods—Local Convergence;246
10.5;4.5 Lagrangian Methods—Global Convergence;272
10.6;4.6 Notes and Sources;312
11;Chapter 5. Nonquadratic Penalty Functions — Convex Programming;317
11.1;5.1 Classes of Penalty Functions and Corresponding Methods of Multipliers;317
11.2;5.2 Convex Programming and Duality;330
11.3;5.3 Convergence Analysis of Multiplier Methods;341
11.4;5.4 Rate of Convergence Analysis;356
11.5;5.5 Conditions for Penalty Methods to Be Exact;374
11.6;5.6 Large Scale Separable Integer Programming Problems and the Exponential Method of Multipliers;379
11.7;5.7 Notes and Sources;395
12;References;398
13;Index;408