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Numerical Methods for Unconstrained Optimization and Nonlinear Equations

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by J. E. Dennis, Jr. and Robert B. Schnabel


1996 / xvi + 378 pages / Softcover / ISBN: 978-0-898713-64-0 / List Price $72.00 / SIAM Member Price $50.40 / Order Code CL16

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.

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 uknown; 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.

Royalties from the sale of this book are contributed to the SIAM Student Travel fund.

ISBN: 9780898713640

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