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Optimization Theory and Methods

 Ebook
Sofort lieferbar | Lieferzeit:3-5 Tage I
ISBN-13:
9780387249766
Einband:
Ebook
Seiten:
688
Autor:
Wenyu Sun
Serie:
1, Springer Optimization and Its Applications
eBook Typ:
PDF
eBook Format:
EPUB
Kopierschutz:
1 - PDF Watermark
Sprache:
Englisch
Beschreibung:

"This book, a result of the author's teaching and research experience in various universities and institutes over the past ten years, can be used as a textbook for an optimization course for graduates and senior undergraduates. It systematically describes optimization theory and several powerful methods, including recent results. For most methods, the authors discuss an idea's motivation, study the derivation, establish the global and local convergence, describe algorithmic steps, and discuss the numerical performance. The book deals with both theory and algorithms of optimization concurrently. It also contains an extensive bibliography. Finally, apart from its use for teaching, Optimization Theory and Methods will be very beneficial as a research reference. TOC:Preface.- Introduction.- Line Search.- Newton's Methods.- Conjugate Gradient Method.- Quasi-Newton Methods.- Trust-Region and Conic Model Methods.- Nonlinear Least-Squares Problems.- Theory of Constrained Optimization.- Quadratic Programming.- Penalty Function Methods.- Feasible Direction Methods.- Sequential Quadratic Programming.- TR Methods for Constrained Problems.- Nonsmooth Optimization.- Appendix: Test Functions.- Bibliography.- Index."
Preface

1 Introduction
1.1 Introduction
1.2 Mathematics Foundations
1.2.1 Norm
1.2.2 Inverse and Generalized Inverse of a Matrix
1.2.3 Properties of Eigenvalues
1.2.4 Rank-One Update
1.2.5 Function and Differential
1.3 Convex Sets and Convex Functions
1.3.1 Convex Sets
1.3.2 Convex Functions
1.3.3 Separation and Support of Convex Sets
1.4 Optimality Conditions for Unconstrained Case
1.5 Structure of Optimization Methods
Exercises

2 Line Search
2.1 Introduction
2.2 Convergence Theory for Exact Line Search
2.3 Section Methods
2.3.1 The Golden Section Method
2.3.2 The Fibonacci Method
2.4 Interpolation Method
2.4.1 Quadratic Interpolation Methods
2.4.2 Cubic Interpolation Method
2.5 Inexact Line Search Techniques
2.5.1 Armijo and Goldstein Rule
2.5.2 Wolfe-Powell Rule
2.5.3 Goldstein Algorithm and Wolfe-Powell Algorithm
2.5.4 Backtracking Line Search
2.5.5 Convergence Theorems of Inexact Line Search
Exercises

3 Newton's Methods
3.1 The Steepest Descent Method
3.1.1 The Steepest Descent Method
3.1.2 Convergence of the Steepest Descent Method
3.1.3 Barzilai and Borwein Gradient Method
3.1.4 Appendix: Kantorovich Inequality
3.2 Newton's Method
3.3 Modified Newton's Method
3.4 Finite-Difference Newton's Method
3.5 Negative Curvature Direction Method
3.5.1 Gill-Murray Stable Newton's Method
3.5.2 Fiacco-McCormick Method
3.5.3 Fletcher-Freeman Method
3.5.4 Second-Order Step Rules
3.6 Inexact Newton's Method
Exercises

4 Conjugate Gradient Method
4.1 Conjugate Direction Methods
4.2 Conjugate Gradient Method
4.2.1 Conjugate Gradient Method
4.2.2 Beale's Three-Term Conjugate Gradient Method
4.2.3 Preconditioned Conjugate Gradient Method
4.3 Convergence of Conjugate Gradient Methods
4.3.1 Global Convergence of Conjugate Gradient Methods
4.3.2 Convergence Rate of Conjugate Gradient Methods
Exercises

5 Quasi-Newton Methods
5.1 Quasi-Newton Methods
5.1.1 Quasi-Newton Equation
5.1.2 Symmetric Rank-One (SR1) Update
5.1.3 DFP Update
5.1.4 BFGS Update and PSB Update
5.1.5 The Least Change Secant Update
5.2 The Broyden Class
5.3 Global Convergence of Quasi-Newton Methods
5.3.1 Global Convergence under Exact Line Search
5.3.2 Global Convergence under Inexact Line Search
5.4 Local Convergence of Quasi-Newton Methods
5.4.1 Superlinear Convergence of General Quasi-Newton Methods
5.4.2 Linear Convergence of General Quasi-Newton Methods
5.4.3 Local Convergence of Broyden's Rank-One Update
5.4.4 Local and Linear Convergence of DFP Method
5.4.5 Superlinear Convergence of BFGS Method
5.4.6 Superlinear Convergence of DFP Method
5.4.7 Local Convergence of Broyden's Class Methods
5.5 Self-Scaling Variable Metric (SSVM) Methods
5.5.1 Motivation to SSVM Method
5.5.2 Self-Scaling Variable Metric (SSVM) Method
5.5.3 Choices of the Scaling Factor
5.6 Sparse Quasi-Newton Methods
5.7 Limited Memory BFGS Method
Exercises

6 Trust-Region and Conic Model Methods
6.1 Trust-Region Methods
6.1.1 Trust-Region Methods
6.1.2 Convergence of Trust-Region Methods
6.1.3 Solving A Trust-Region Subproblem
6.2 Conic Model and Collinear Scaling Algorithm
6.2.1 Conic Model
6.2.2 Generalized Quasi-Newton Equation
6.2.3 Updates that Preserve Past Information
6.2.4 Collinear Scaling BFGS Algorithm
6.3 Tensor Methods
6.3.1 Tensor Method for Nonlinear Equations
6.3.2 Tensor Methods for Unconstrained Optimization
Exercises

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