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数值最优化（第二版） 
定 价：198 元 丛书名：国外数学名著系列（影印版）
9 7 6 8 0 7 5 0 5 3 1 0 1 读者对象:本书适用于工程、运筹学、数学、计算机科学以及商务方面的研究生 
Contents
Preface prefcetothe Second Edition 1 Introduction 1 Mathematical Formulation 2 Example:A Transportation Problem 4 Continuous versus Discrete Optimization 5 Constrained and Unconstrained Optimization 6 Global and Local Optimization 6 Stocbastic and Deterministic Optimization 7 Convexity 7 Optimization Algorithms 8 Notes and References 9 2 Fundamentals of Unconstrained Optimization 10 2.1 What ls a Solution? 12 Recognizing a Local Minimum 14 Nonsmooth Problems 17 2.2 Overview of A1gorithms 18 Two Strategies:Line Search and Trust Region 19 Search Directions for Line Search Methods 20 Models for Trust-Region Methods 25 Scaling 26 Exercises 27 3 Line Search Methods 30 3.1 Step Length 31 The Wolfe Conditions 33 The Goldstein Conditions 36 Sufficient Decrease and Backtracking 37 3.2 Convergence of Line Search Methods 37 3.3 Rate of Convergence 41 Convergence Rate of Steepest Descent 42 Newton's Method 44 Quasi-Newton Methods 46 3.4 Newton's Method with Hessian Modification 48 Eigenvalue Modification 49 Adding a Multiple of the ldentity 51 Modified Cholesky Factorization 52 Modified Symmetric Indefinite Factorization 54 3.5 Step-Length Selection Algorithms 56 lnterpolation 57 lnitial Step Length 59 A Line Search A1gorithm for the Wolfe Conditions 60 Notes and References 62 Exercises 63 4 Trust-Region Methods 66 Outline of the Trust-Region Approach 68 4.1 A1gorithms Based on the Cauchy Point 71 The Cauchy Point 71 lmpro时ng on the Cauchy Point 73 The Dogleg Method 73 Two-Dinlensional Subspace Mininlization 76 4.2 Global Convergence 77 Reduction Obtained by the Cauchy Point 77 Convergence to Stationary Points 79 4.3 lterative Solution of the Subproblem 83 The Hard Case 87 Proof of Theorem 4.1 89 Convergence of Algorithms Based on Nearly Exact Solutions 91 4.4 Local Convergence ofTrust-Region Newton Methods 92 4.5 0ther Enhancements 95 Scaling 95 Trust Regions in 0ther Norms 97 Notes and References 98 Exercises 98 5 Conjugate Gradient Methods 101 5.1 The linear Conjugate Gradient Method 102 Conjugate Direction Methods 102 Basic Properties of thee Conjugate Gradient Method 107 A Practical Form of the Conjugate Gradient Method 111 Rate of Convergence 112 Preconditioning 118 Practical Preconditioners 120 5.2 Nonlinear Conjugate Gradient Methods 121 The Fletcher-Reeves Method 121 The Polak-Ribière Method and Variants 122 Quadratic Termination and Restarts 124 Behavior of the Fletcher-Reeves Method 125 Global Convergence 127 Numerical Performance 131 Notes and Reference 132 Exercises 133 6 Quasi-Newton Methods 135 6.1 The BFGS Method 136 Properties ofthe BFGS Method 141 Implementation 142 6.2 The SR1 Method 144 Properties of SR1 Updating 147 6.3 The Broyden Class 149 6.4 Convergence Analysis 153 Global Convergence of the BFGS Method 153 Superlinear Convergence of the BFGS Method 156 Convergence Analysis of the SR1 Method 160 Notes and References 161 Exercises 162 7 Large-Scale Unconstrained optimization 164 7.1 lnexact Newton Methods 165 Local Convergence of Inexact Newton Methods 166 Line Search Newton-CG Method 168 Trust-Region Newton-CG Method 170 Preconditioning the Trust-Region Newton-CG Method 174 Trust-Region Newton-Lanczos Method 175 7.2 Limited-Memory Quasi-Newton Methods 176 Limited-Memory BFGS 177 Relationship with Conjugate Gradient Methods 180 General Lirnited:d-Memory Updatiug 181 Compact Representation of BFGS Updating 181 Unrolling the Update 184 7.3 Sparse Quasi-Newton Updates 185 7.4 Algorithms for Partially Separable Fnnctions 186 7.5 Perspectives and Sotrware 189 Notes and References 190 Exercises 191 8 Calculating Derivatives 193 8.1 Finite-Difference Derivative Approximations 194 Approximating the Gradient 195 Approximating a Sparse Jacobian 197 Approximatiug the Hessian 201 Approximatiug a Sparse Hessian 202 8.2 Automatic Differentiation 204 Au Example 205 The Forward Mode 206 The Reverse Mode 207 Vector Fnnctions and Partial Separablity 210 Calculating Jacobians ofVector Funlctions 212 Calculating Hessians:Forward Mode 213 Calculating Hessians:Reverse Mode 215 Current Lirnitations 216 Notess and References 217 Exercises 217 9 Derivatve-Free Optiimization 220 9.1 Finite Differences and Noise 221 9.2 Model-Based Methods 223 Interpolation aod Polyoomial Bases 226 Updating the Interpolation Set 227 A Method Based on Minimum-Change Updating 228 9.3 Coordinate and Pattern-Search Methods 229 Coordinate Search Method 230 Pattern-Search Methods 231 9.4 A Conjugate-Direction Method 234 9.5 Nelder-Mead Method 238 9.6 Implicit Filtering 240 Notes and References 242 Exercises 242 10 Least-Sqnares Problems 245 10.1 Background 247 10.2 Linear Least-Squares Problems 250 10.3 Algorithms for Nonlinear Least-Squares Problems 254 The Gauss-Newton Method 254 Convergence of the Gauss Newton Method 255 The Levenberg-Marquardt Method 258 Implementation of the Levenberg-Marquardt Method 259 Convergence of the Levenberg-Marquardt Method 261 Methods for Large-Residual Problems262 10.4 Orthogonal Distance Regression 265 Nootes and References 267 Exerclses 269 11 Nonlinear Equations 270 11.1 Local A1gorithms 274 Newton's Method for Nonlinear Equations 274 Inexact Newton Methods 277 Broyden's Methods 279 Tensor Methods 283 11.2 Practical Methods 285 Merit Functions 285 Line Search Methods 287 Trust-Region Methods 290 11.3 Continuation/Homotopy Methods 296 Motivation 296 Practical Continuation Methods 297 Notes and References 302 Exercises 302 12 Theory of Constrained Optimization 304 Local and Global Solutions 305 Smoothness 306 12.1 Examples 307 A Single Equality Constraint 308 A Single Inequality Constraint 310 Two Inequality Constraints 313 12.2 Tangent Cone and Constraint Qualifications 315 12.3 First-Order Optimality Conditions 320 12.4 First-Order Optimality Conditions:Proof 323 Relating the Tangent Cone and the First-Order Feasible Direction Set 323 A Fundamental Necessary Condition 325 Farkas' Lemma 326 Proof ofTbeorem 12.1 329 12.5 Second-Order Conditions 330 Second-Order Conditions and projected Hessians 337 12.6 Other Constraint Qualifications 338 12.7 A Geometric Viewpoint 340 12.8 Lagrange Multipliers and Sensitivity 341 12.9 Duality343 Notes and References 349 Exercises 351 13 Linear Programming:Tbe Sirnplex Method 355 Linear Programming 356 13.1 Optimality and Duality 358 Optimality Conditions 358 Tbe Dual Problem 359 13.2 Geometry of the Feasible Set 362 Bases and Basic Feasible Points 362 A Single Step of the Feasible Polytope 365 13.3 The Sirnplex Metbod 366 Outline 366 A Single Step of the Metbod 370 13.4 Linear Algebra in the Sirnplex Metbod372 13.5 Other Important Detaills 375 Pricing and Selection of the Entering Index 375 Starting the Sirnplex Method378 Degenerate Steps and Cycling 381 13.6 Tbe Dual Sirnplex Method 382 13.7 Presolving 385 13.8 Where Does the Sirnplex Metbod Fit1 388 Notes and References 389 Exfercises 389 14 Linear Programming:lnterior-Point Methods 392 14.1 Primal-Dual Methods 393 Outlioe 393 The Central Path 397 Central Path Neighborhoods and path-Following Methods 399 14.2 Practical Primal-Dual Algorithms 407 Corrector and Centering Steps 407 Step Lengths 409 Starting Point 410 A Practiica1 Algorithm 411 Solving Linear Systems 411 14.3 Other Primal-Dual Algorithms and Extensions 413 0ther Parimal-Followmg Methods 413 Potential-Reduction Metheods 414 Extenlsions 415 14.4 Perspectives and Software 416 Notes and References 417 Exercises 418 15 Fundamentals of A1gorithms for Nonlinear Constrained Optization 421 15.1 Categorizing Optimization Algorithms 422 15.2 The Combmatorial Difficulty of Inequality Constrained Problems 424 15.3 Elimiuation of Variables 426 Simple Elimination usmg Lmear Constraints 428 General Reduction Strategies for Lmear Constraints 431 Effect of lnequality Constraints 434 15.4 Merit Functions and Filtes 435 Merit Functions 435 Filters 437 15.5 The Maratos Effect 440 15.6 Second-Order Correction and Nonmonotone Tecbniques 443 Nonmonotone (Watcbdog) Strategy 444 Notes and References 446 Exercises 446 16 Quadratic Programs 448 16.1 Equality-Constrained Quadratic Programs 451 Properties of Equality-Constrained QPs 451 16.2 Direct Solution of the KKT System 454 Factormg 也e Full KKT System 454 Scbur-Complement Method 455 Null-Space Method 457 16.3 Iterative Solution of the KKT System 459 CG Applied to the Reduced System 459 The ProjectedCG Method 461 16.4 Inequality-Constrained Problems 463 Optimality Conditions for Inequality-Constrained Problems 464 Degeneracy 465 16.5 Active-Set Methods for Convex QPs 467 Specification of the Active-Set Method for Convex QP 472 Further Remarks on the Active-Set Method 476 Finite Termination of Active-Set A1gorithm on Strictly Convex QPs 477 Updating Factorizations 478 16.6 Interior-Point Methods 480 Solving the PrinIal-Dual System 482 Step Length Selection 483 A Practical PrinIal-Dual Method 484 16.7 The Gradient Projection Method 485 Caucby Point Computation 486 Subspace Mininimization 488 16.8 Perspectives and Software 490 Notes and References 492 Exercises 492 17 Penalty and Angmented Lagrangian Methods 497 17.1 Tbe Quadratic penalty Method 498 Motivation 498 Algorithmic Framework 501 Convergence of the Quadratic Penalty Method 502 Ⅲ Conditioning and Reformulations 505 17.2 Nonsmooth Peualty Functions 507 A Practical e1 Penalty Method 511 A General Class ofNonsmooth Penalty Methods 513 17.3 Augmented Lagrangian Method:Equality Constraints 514 Motivation and A1gorithmic Framework 514 Properties of the Augmented Lagrangian 517 17.4 Practical Augmented Lagrangian Methods 519 Bound-Constrained Formulation 519 Linearly Constrained Formulation 522 Unconstrainde Formulation 523 17.5 Perspectives and Software 525 Notes and References 526 Exercises 527 18 Sequential Quadratic Programming 529 18.1 Local SQP Method 530 SQP Framework 531 Inequality Constraints 532 18.2 Preview ofPractical SQP Methods 533 IQP and EQP 533 Enforcing Convergence 534 18.3 Algorithmic Development 535 Handling Inconsistent Linearizations 535 FuIl Quasi-Newton Approxirnations 536 Reduced-Hessian Quasi-Newton Approxirnations 538 Merit Functions 540 Second-Order Correction 543 18.4 A Practical Line Search SQP Method 545 18.5 Trust-Region SQP Methods 546 A Relaxation Method for Equality-Constrained Optimization 547 St1QP(Sequential t1 Quadratic Programming) 549 Sequential Linear-Quadratic Programming (SLQP) 551 Aτèchnique for Updating the Penalty Parameter 553 18.6 Nonlinque Gradient Projection 554 18.7 Convergence Analysis 556 Rate of Convergence 557 18.8 Perspectives and Software 560 Notes and References 561 Exercises 561 19 Interior-Point Methods for Nonlinear Programming 563 19.1 Two InterPretations 564 19.2 A Basic Interior-Point A1gorithm 566 19.3 A1gorithmic Development 569 Primal vs.Primal-Dual System570 Solving the Primal-Dual System 570 Updating the Barrier Parameter 572 Handling Nonconvexity and Singularity 573 Step Acceptance:Merit Functions and Filters 575 Quasi-Newton Approximations 575 Feasible Interior-Point Methods 576 19.4 A Line Search Interior-Point Method 577 19.5 A Trust-Region Interior-Point Method 578 An A1gorithm for Solving the Barrier Problem 578 Step Computation 580 Lagrange MuItipliers Estimates and Step Acceptance 581 Description of a Trust-Region Interior-Point Method 582 19.6 The Primal Log-Barrier Method 583 19.7 Global Convergence Propertiles 587 Failure of tbe Line Search Approach 587 Modified Line Search Metbods 589 Global Convergence of the Trust-Region Approach 589 19.8 Superlinear Convergence 591 19.9 Perspectives and Sofware 592 Notes and References 593 Exercises 594 A Background Material 598 A.l Elements of Linear A1gebra 598 Vectors and Matriices 598 Norms 600 Subspaces 602 Eigenva1ues, Eigenvectors，and the Singular-Value Decomposition 603 Determinant and Trace 605 Matrix Factorizations:Cholesky,LU，QR 606 Synunetric Indefinite Factorization 610 Sherman-Morrison-Woodbury Formula 612 Interlacing Eigenvalue Theorem 613 Error Analysis and Floating-Point Arithmetic 613 Conditioning and Stability 616 A.2 Elements of Analysis,Geometry,Topology 617 Sequences 617 Rates of Convergence 619 Topology of tbe Euclideean Space Rn 620 Convex Sets in Rn 621 Continuity and Limits 623 Derivatives 625 Directional Derivatives 628 Mean Value Theorern 629 Implicit Function Theorem 630 Order Notation 631 Root-Finding for Scalar Equations 633 B A Reaularization Procedure 635 References 637 Index 653
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