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Numerical Methods for Unconstrained Optimization and Nonlinear Equations Numerical Methods for Unconstrained Optimization and Nonlinear Equations SIAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by SIAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in'Chief Robert E. O'Malley, Jr., University of Washington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller, California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimisation and Nonsmooth Analysis George F. Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathemat- ical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematica/ Theory of Reliability Cornelius Lanczos, Linear Differential Operators Richard Bellman, Introduction to Matrix Analysis, Second Edition Beresford N. Parlett, The Symmetric Eigenvalue Problem *First time in print. Classics in Applied Mathematics (continued) Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Peter W. M. John, Statistical Design and Analysis of Experiments Tamer Ba§ar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger Temam, Convex Analysis and Variational Problems Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications F. Natterer, The Mathematics of Computerized Tomography Avinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging R. Wong, Asymptotic Approximations of Integrals O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation David R. Brillinger, Time Series: Data Analysis and Theory Joel N. Franklin, Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems Philip Hartman, Ordinary Differential Equations, Second Edition Michael D. Intriligator, Mathematical Optimization and Economic Theory Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. I: Theory M. Vidyasagar, Nonlinear Systems Analysis, Second Edition Robert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory and Practice Shanti S. Gupta and S. Panchapakesan, Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations Eugene L. Allgower and Kurt Georg, Introduction to Numerical Continuation Methods Heinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value Problems and the Navier- Stofces Equations Copyright © 1996 by the Society for Industrial and Applied Mathematics. This SIAM edition is an unabridged, corrected republication of the work first published by Prentice-Hall, Inc., Englewood Cliffs, NJ, 1983. 1098765 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104- 2688. Library of Congress Cataloging-in-Publication Data Dennis, J. E. ( John E. ) , 1939- Numerical methods for unconstrained optimization and nonlinear equations / J.E. Dennis, Jr., Robert B. Schnabel . p. cm. — ( Classics in applied mathematics ; 16 ) Originally published : Englewood Cliffs, NJ. : Prentice-Hall, ©1983. Includes bibliographical references and indexes. ISBN 0-89871-364-1 ( pbk ) 1. Mathematical optimization. 2. Equations—Numerical solutions. I. Schnabel, Robert B. II. Title III. Series. QA402.5.D44 1996 95-51776 The royalties from the sales of this book are being placed in a fund to help students attend SIAM meetings and other SIAM related activities. This fund is administered by SIAM and qualified individuals are encouraged to write directly to SIAM for guidelines. • is a registered trademark. Numerical Methods for Unconstrained Optimization and Nonlinear Equations J. R Dennis, Jr. Rice University Houston, Texas Robert B. Schnabel University of Colorado Boulder, Colorado siaiiL Society for Industrial and Applied Mathematics Philadelphia To Catherine, Heidi, and Cory Contents PREFACE TO THE CLASSICS EDITION xi PREFACE xiii 1 INTRODUCTION 2 1.1 Problems to be considered 2 1.2 Characteristics of "real-world" problems 5 1.3 Finite-precision arithmetic and measurement of error 10 1.4 Exercises 13 21 NONLINEAR PROBLEMS IN ONE VARIABLE 15 2.1 What is not possible 75 2.2 Newton's method for solving one equation in one unknown 16 2.3 Convergence of sequences of real numbers 19 2.4 Convergence of Newton's method 21 2.5 Globally convergent methods for solving one equation in one unknown 24 2.6 Methods when derivatives are unavailable 27 2.7 Minimization of a function of one variable 32 2.8 Exercises 36 vii viii Contents 3 NUMERICAL LINEAR ALGEBRA BACKGROUND 40 3.1 Vector and matrix norms and orthogonality 41 3.2 Solving systems of linear equations—matrix factorizations 47 3.3 Errors in solving linear systems 51 3.4 Updating matrix factorizations 55 3.5 Eigenvalues and positive definiteness 58 3.6 Linear least squares 60 3.7 Exercises 66 41 MULTIVARIABLE CALCULUS BACKGROUND 69 4.1 Derivatives and multivariable models 69 4.2 Multivariable finite-difference derivatives 77 4.3 Necessary and sufficient conditions for unconstrained minimization 80 4.4 Exercises 83 5 NEWTON'S METHOD FOR NONLINEAR EQUATIONS AND UNCONSTRAINED MINIMIZATION 86 5.1 Newton's method for systems of nonlinear equations 86 5.2 Local convergence of Newton's method 89 5.3 The Kantorovich and contractive mapping theorems 92 5.4 Finite-difference derivative methods for systems of nonlinear equations 94 5.5 Newton's method for unconstrained minimization 99 5.6 Finite-difference derivative methods for unconstrained minimization 103 5.7 Exercises 107 6 GLOBALLY CONVERGENT MODIFICATIONS OF NEWTON'S METHOD 111 6.1 The quasi-Newton framework 112 6.2 Descent directions 113 6.3 Line searches 776 6.3.1 Convergence results for properly chosen steps 120 6.3.2 Step selection by backtracking 126 6.4 The model-trust region approach 729 6.4.1 The locally constrained optimal ("hook") step 134 6.4.2 The double dogleg step 139 6.4.3 Updating the trust region 143 6.5 Global methods for systems of nonlinear equations 147 6.6 Exercises 752 Contents ix 7 STOPPING, SCALING, AND TESTING 155 7.1 Scaling 155 7.2 Stopping criteria 759 7.3 Testing 767 7.4 Exercises 164 8 SECANT METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS 168 8.1 Broyden's method 769 8.2 Local convergence analysis of Broyden's method 174 8.3 Implementation of quasi-Newton algorithms using Broyden's update 186 8.4 Other secant updates for nonlinear equations 189 8.5 Exercises 790 9 SECANT METHODS FOR UNCONSTRAINED MINIMIZATION 194 9.1 The symmetric secant update of Powell 795 9.2 Symmetric positive definite secant updates 198 9.3 Local convergence of positive definite secant methods 203 9.4 Implementation of quasi-Newton algorithms using the positive definite secant update 205 9.5 Another convergence result for the positive definite secant method 270 9.6 Other secant updates for unconstrained minimization 277 9.7 Exercises 272 10 NONLINEAR LEAST SQUARES 218 10.1 The nonlinear least-squares problem 218 10.2 Gauss-Newton-type methods 227 10.3 Full Newton-type methods 228 10.4 Other considerations in solving nonlinear least-squares problems 233 10.5 Exercises 236 11 METHODS FOR PROBLEMS WITH SPECIAL STRUCTURE 239 11.1 The sparse finite-difference Newton method 240
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