无约束最优化与非线性方程的数值方法

PREFACE TO THE CLASSICS EDITIONPREFACE1 INTRODUCTION 1.1 Problems to be considered 1.2 Characteristics of"real-world" problems 1.3 Finite-precision arithmetic and measurement of error 1.4 Exercises2 NONLINEAR PROBLEMS IN ONE VARIABLE 2.1 What is not possible 2.2 Newton's method for solving one equation in one unknown 2.3 Convergence of sequences of real numbers 2.4 Convergence of Newton's method 2.5 Globally convergent methods for solving one equation in one unknown 2.6 Methods when derivatives are unavailable 2.7 Minimization of a function of one variable 2.8 Exercises3 NUMERICAL LINEAR ALGEBRA BACKGROUND 3.1 Vector and matrix norms and orthogonality 3.2 Solving systems of linear equations--matrix factorizations 3.3 Errors in solving linear systems 3.4 Updating matrix factorizations 3.5 Eigenvalues and positive definiteness 3.6 Linear least squares 3.7 Exercises4 MULTIVARIABLE CALCULUS BACKGROUND 4.1 Derivatives and multivariable models 4.2 Multivariable finite-difference derivatives 4.3 Necessary and sufficient conditions for unconstrained minimization 4.4 Exercises 5 NEWTON'S METHOD FOR NONLINEAR EQUATIONS AND UNCONSTRAINED MINIMIZATION 5.1 Newton's method for systems of nonlinear equations 5.2 Local convergence of Newton's method 5.3 The Kantorovich and contractive mapping theorems 5.4 Finite-difference derivative methods for systems of nonlinear equations 5.5 Newton's method for unconstrained minimization 5.6 Finite-difference derivative methods for unconstrained minimization 5.7 Exercises6 GLOBALLY CONVERGENT MODIFICATIONS OF NEWTON'S METHOD 6.1 The quasi-Newton framework 6.2 Descent directions 6.3 Line searches 6.3.1 Convergence results for properly chosen steps 6.3.2 Step selection by backtracking 6.4 The model-trust region approach 6.4.1 The locally constrained optimal ("hook") step 6.4.2 The double dogleg step 6.4.3 Updating the trust region 6.5 Global methods for systems of nonlinear equations 6.6 Exercises7 STOPPING, SCALING, AND TESTING 7.1 Scaling 7.2 Stopping criteria 7.3 Testing 7.4 Exercises8 SECANT METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS 8.1 Broyden's method 8.2 Local convergence analysis of Broyden's method 8.3 Implementation of quasi-Newton algorithms using Broyden's update 8.4 Other secant updates for nonlinear equations 8.5 Exercises9 SECANT METHODS FOR UNCONSTRAINED MINIMIZATION 9.1 The symmetric secant update of Powell 9.2 Symmetric positive definite secant updates 9.3 Local convergence of positive definite secant methods 9.4 Implementation of quasi-Newton algorithms using the positive definite secant update 9.5 Another convergence result for the positive definite secant method 9.6 Other secant updates for unconstrained minimization 9.7 Exercises10 NONLINEAR LEAST SQUARES 10.1 The nonlinear least-squares problem 10.2 Gauss-Newton-type methods 10.3 Full Newton-type methods 10.4 Other considerations in solving nonlinear least-squares problems 10.5 Exercises11 METHODS FOR PROBLEMS WITH SPECIAL STRUCTURE 11.1 The sparse finite-difference Newton method 11.2 Sparse secant methods 11.3 Deriving least-change secant updates 11.4 Analyzing least-change secant methods 11.5 ExercisesA APPENDIX: A MODULAR SYSTEM OF ALGORITHMS FOR UNCONSTRAINED MINIMIZATION AND NONLINEAR EQUATIONS (by Robert Schnabel)B APPENDIX: TEST PROBLEMS (by Robert SchnabeI) REFERENCES AUTHOR INDEX SUBJECT INDEX