Numerical Methods for Unconstrained Optimization and Nonlinear Equations
Published on 31. December 1996
Book
Paperback/Softback
396 pages
978-0-89871-364-0 (ISBN)
Description
This book has become the standard for a complete, state-of-the-art description of the methods for unconstrained optimization and systems of nonlinear equations. Originally published in 1983, it provides information needed to understand both the theory and the practice of these methods and provides pseudocode for the problems. The algorithms covered are all based on Newton's method or ""quasi-Newton"" methods, and the heart of the book is the material on computational methods for multidimensional unconstrained optimization and nonlinear equation problems. The republication of this book by SIAM is driven by a continuing demand for specific and sound advice on how to solve real problems.
The level of presentation is consistent throughout, with a good mix of examples and theory, making it a valuable text at both the graduate and undergraduate level. It has been praised as excellent for courses with approximately the same name as the book title and would also be useful as a supplemental text for a nonlinear programming or a numerical analysis course. Many exercises are provided to illustrate and develop the ideas in the text. A large appendix provides a mechanism for class projects and a reference for readers who want the details of the algorithms. Practitioners may use this book for self-study and reference.
For complete understanding, readers should have a background in calculus and linear algebra. The book does contain background material in multivariable calculus and numerical linear algebra.
The level of presentation is consistent throughout, with a good mix of examples and theory, making it a valuable text at both the graduate and undergraduate level. It has been praised as excellent for courses with approximately the same name as the book title and would also be useful as a supplemental text for a nonlinear programming or a numerical analysis course. Many exercises are provided to illustrate and develop the ideas in the text. A large appendix provides a mechanism for class projects and a reference for readers who want the details of the algorithms. Practitioners may use this book for self-study and reference.
For complete understanding, readers should have a background in calculus and linear algebra. The book does contain background material in multivariable calculus and numerical linear algebra.
More details
Series
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
College/higher education
Product notice
Paperback (trade)
Dimensions
Height: 233 mm
Width: 150 mm
Thickness: 23 mm
Weight
522 gr
ISBN-13
978-0-89871-364-0 (9780898713640)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Content
Preface to the Classics edition
Preface
Chapter 1: Introduction. Problems to be considered
Characteristics of "real-world" problems
Finite-precision arithmetic and measurement of error
Exercises
Chapter 2: Nonlinear Problems in One Variable. What is not possible
Newton's method for solving one equation in one unknown
Convergence of sequences of real numbers
Convergence of Newton's method
Globally convergent methods for solving one equation in one unknown
Methods when derivatives are unavailable
Minimization of a function of one variable
Exercises
Chapter 3: Numerical Linear Algebra Background. Vector and matrix norms and orthogonality
Solving systems of linear equations-matrix factorizations
Errors in solving linear systems
Updating matrix factorizations
Eigenvalues and positive definiteness
Linear least squares
Exercises
Chapter 4: Multivariable Calculus Background
Derivatives and multivariable models
Multivariable finite-difference derivatives
Necessary and sufficient conditions for unconstrained minimization
Exercises
Chapter 5: Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations
Local convergence of Newton's method
The Kantorovich and contractive mapping theorems
Finite-difference derivative methods for systems of nonlinear equations
Newton's method for unconstrained minimization
Finite-difference derivative methods for unconstrained minimization
Exercises
Chapter 6: Globally Convergent Modifications of Newton's Method. The quasi-Newton framework
Descent directions
Line searches
The model-trust region approach
Global methods for systems of nonlinear equations
Exercises
Chapter 7: Stopping, Scaling, and Testing. Scaling
Stopping criteria
Testing
Exercises
Chapter 8: Secant Methods for Systems of Nonlinear Equations. Broyden's method
Local convergence analysis of Broyden's method
Implementation of quasi-Newton algorithms using Broyden's update
Other secant updates for nonlinear equations
Exercises
Chapter 9: Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell
Symmetric positive definite secant updates
Local convergence of positive definite secant methods
Implementation of quasi-Newton algorithms using the positive definite secant update
Another convergence result for the positive definite secant method
Other secant updates for unconstrained minimization
Exercises
Chapter 10: Nonlinear Least Squares. The nonlinear least-squares problem
Gauss-Newton-type methods
Full Newton-type methods
Other considerations in solving nonlinear least-squares problems
Exercises
Chapter 11: Methods for Problems with Special Structure. The sparse finite-difference Newton method
Sparse secant methods
Deriving least-change secant updates
Analyzing least-change secant methods
Exercises
Appendix A: A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel)
Appendix B: Test Problems (by Robert Schnabel)
References
Author Index
Subject Index.
Preface
Chapter 1: Introduction. Problems to be considered
Characteristics of "real-world" problems
Finite-precision arithmetic and measurement of error
Exercises
Chapter 2: Nonlinear Problems in One Variable. What is not possible
Newton's method for solving one equation in one unknown
Convergence of sequences of real numbers
Convergence of Newton's method
Globally convergent methods for solving one equation in one unknown
Methods when derivatives are unavailable
Minimization of a function of one variable
Exercises
Chapter 3: Numerical Linear Algebra Background. Vector and matrix norms and orthogonality
Solving systems of linear equations-matrix factorizations
Errors in solving linear systems
Updating matrix factorizations
Eigenvalues and positive definiteness
Linear least squares
Exercises
Chapter 4: Multivariable Calculus Background
Derivatives and multivariable models
Multivariable finite-difference derivatives
Necessary and sufficient conditions for unconstrained minimization
Exercises
Chapter 5: Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations
Local convergence of Newton's method
The Kantorovich and contractive mapping theorems
Finite-difference derivative methods for systems of nonlinear equations
Newton's method for unconstrained minimization
Finite-difference derivative methods for unconstrained minimization
Exercises
Chapter 6: Globally Convergent Modifications of Newton's Method. The quasi-Newton framework
Descent directions
Line searches
The model-trust region approach
Global methods for systems of nonlinear equations
Exercises
Chapter 7: Stopping, Scaling, and Testing. Scaling
Stopping criteria
Testing
Exercises
Chapter 8: Secant Methods for Systems of Nonlinear Equations. Broyden's method
Local convergence analysis of Broyden's method
Implementation of quasi-Newton algorithms using Broyden's update
Other secant updates for nonlinear equations
Exercises
Chapter 9: Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell
Symmetric positive definite secant updates
Local convergence of positive definite secant methods
Implementation of quasi-Newton algorithms using the positive definite secant update
Another convergence result for the positive definite secant method
Other secant updates for unconstrained minimization
Exercises
Chapter 10: Nonlinear Least Squares. The nonlinear least-squares problem
Gauss-Newton-type methods
Full Newton-type methods
Other considerations in solving nonlinear least-squares problems
Exercises
Chapter 11: Methods for Problems with Special Structure. The sparse finite-difference Newton method
Sparse secant methods
Deriving least-change secant updates
Analyzing least-change secant methods
Exercises
Appendix A: A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel)
Appendix B: Test Problems (by Robert Schnabel)
References
Author Index
Subject Index.