【下载】Numerical Optimization.2nd~Jorge Nocedal.Stephen Wright.2006

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Numerical Optimization (Springer Series in Operations Research and Financial Engineering) [Hardcover]
Jorge Nocedal (Author), Stephen Wright (Author)

Editorial Reviews

Review

MMOR Mathematical Methods of Operations Research, 2001: "The books looks very suitable to be used in an graduate-level course in optimization for students in mathematics, operations research, engineering, and others. Moreover, it seems to be very helpful to do some self-studies in optimization, to complete own knowledge and can be a source of new ideas.... I recommend this excellent book to everyone who is interested in optimization problems."

Product Description

Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems.

For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

There is a selected solutions manual for instructors for the new edition.

Product Details Hardcover: 664 pages Publisher: Springer; 2nd edition (July 27, 2006) Language: English ISBN-10: 0387303030 ISBN-13: 978-0387303031

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关键词：Optimization Numerical Stephen Nocedal numeric Optimization Stephen Wright Jorge Nocedal

Contents

Preface xvii

Preface to the Second Edition xxi

1 Introduction 1

MathematicalFormulation 2

Example:ATransportationProblem 4

ContinuousversusDiscreteOptimization 5

ConstrainedandUnconstrainedOptimization 6

GlobalandLocalOptimization 6

Stochastic and Deterministic Optimization 7

Convexity 7

Optimization Algorithms 8

NotesandReferences 9

2 Fundamentals of Unconstrained Optimization 10

2.1 What IsaSolution? 12

Recognizing a Local Minimum 14

NonsmoothProblems 17

2.2 Overview of Algorithms 18

TwoStrategies:LineSearchandTrustRegion 19

SearchDirections forLineSearchMethods 20

Models for Trust-Region Methods 25

Scaling 26

Exercises 27

3 Line SearchMethods 30

3.1 StepLength 31

TheWolfe Conditions 33

The Goldstein Conditions 36

Sufficient Decrease and Backtracking 37

3.2 ConvergenceofLineSearchMethods 37

3.3 RateofConvergence 41

ConvergenceRateofSteepestDescent 42

Newton’sMethod 44

Quasi-NewtonMethods 46

3.4 Newton’s Method with Hessian Modification 48

EigenvalueModification 49

Adding a Multiple of the Identity 51

Modified Cholesky Factorization 52

ModifiedSymmetricIndefiniteFactorization 54

3.5 Step-Length Selection Algorithms 56

Interpolation 57

InitialStepLength 59

A Line Search Algorithm for theWolfe Conditions 60

NotesandReferences 62

Exercises 63

4 Trust-RegionMethods 66

Outline of the Trust-Region Approach 68

4.1 Algorithms Based on the Cauchy Point 71

TheCauchyPoint 71

ImprovingontheCauchyPoint 73

TheDoglegMethod 73

Two-Dimensional Subspace Minimization 76

4.2 GlobalConvergence 77

ReductionObtainedbytheCauchyPoint 77

ConvergencetoStationaryPoints 79

4.3 IterativeSolutionof theSubproblem 83

TheHardCase 87

ProofofTheorem4.1 89

Convergence of Algorithms Based on Nearly Exact Solutions 91

4.4 Local Convergence of Trust-Region Newton Methods 92

4.5 OtherEnhancements 95

Scaling 95

TrustRegions inOtherNorms 97

NotesandReferences 98

Exercises 98

5 Conjugate GradientMethods 101

5.1 TheLinearConjugateGradientMethod 102

ConjugateDirectionMethods 102

BasicPropertiesof theConjugateGradientMethod 107

APracticalFormof theConjugateGradientMethod 111

RateofConvergence 112

Preconditioning 118

Practical Preconditioners 120

5.2 NonlinearConjugateGradientMethods 121

TheFletcher–ReevesMethod 121

The Polak–Ribi`ereMethodandVariants 122

Quadratic Termination and Restarts 124

Behaviorof theFletcher–ReevesMethod 125

GlobalConvergence 127

NumericalPerformance 131

NotesandReferences 132

Exercises 133

6 Quasi-NewtonMethods 135

6.1 TheBFGSMethod 136

Propertiesof theBFGSMethod 141

Implementation 142

6.2 TheSR1Method 144

PropertiesofSR1Updating 147

6.3 TheBroydenClass 149

6.4 ConvergenceAnalysis 153

GlobalConvergenceof theBFGSMethod 153

SuperlinearConvergenceof theBFGSMethod 156

ConvergenceAnalysisof theSR1Method 160

NotesandReferences 161

Exercises 162

7 Large-Scale Unconstrained Optimization 164

7.1 InexactNewtonMethods 165

LocalConvergenceof InexactNewtonMethods 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-MemoryQuasi-NewtonMethods 176

Limited-MemoryBFGS 177

RelationshipwithConjugateGradientMethods 180

GeneralLimited-MemoryUpdating 181

CompactRepresentationofBFGSUpdating 181

UnrollingtheUpdate 184

7.3 SparseQuasi-NewtonUpdates 185

7.4 Algorithms for Partially Separable Functions 186

7.5 PerspectivesandSoftware 189

NotesandReferences 190

Exercises 191

8 Calculating Derivatives 193

8.1 Finite-Difference Derivative Approximations 194

ApproximatingtheGradient 195

ApproximatingaSparseJacobian 197

Approximating the Hessian 201

Approximating a Sparse Hessian 202

8.2 AutomaticDifferentiation 204

AnExample 205

TheForwardMode 206

TheReverseMode 207

VectorFunctionsandPartialSeparability 210

CalculatingJacobiansofVectorFunctions 212

Calculating Hessians: Forward Mode 213

Calculating Hessians: Reverse Mode 215

CurrentLimitations 216

NotesandReferences 217

Exercises 217

9 Derivative-Free Optimization 220

9.1 Finite Differences and Noise 221

9.2 Model-BasedMethods 223

InterpolationandPolynomialBases 226

UpdatingtheInterpolationSet 227

A Method Based on Minimum-Change Updating 228

9.3 Coordinate and Pattern-Search Methods 229

Coordinate Search Method 230

Pattern-SearchMethods 231

9.4 AConjugate-DirectionMethod 234

9.5 Nelder–MeadMethod 238

9.6 ImplicitFiltering 240

NotesandReferences 242

Exercises 242

0 Least-Squares 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

TheLevenberg–MarquardtMethod 258

Implementationof theLevenberg–MarquardtMethod 259

Convergenceof theLevenberg–MarquardtMethod 261

Methods forLarge-ResidualProblems 262

10.4 Orthogonal Distance Regression 265

NotesandReferences 267

Exercises 269

11 Nonlinear Equations 270

11.1 Local Algorithms 274

Newton’sMethodforNonlinearEquations 274

InexactNewtonMethods 277

Broyden’sMethod 279

TensorMethods 283

11.2 PracticalMethods 285

MeritFunctions 285

LineSearchMethods 287

Trust-Region Methods 290

11.3 Continuation/HomotopyMethods 296

Motivation 296

PracticalContinuationMethods 297

NotesandReferences 302

Exercises 302

12 Theory of Constrained Optimization 304

LocalandGlobalSolutions 305

Smoothness 306

12.1 Examples 307

ASingleEqualityConstraint 308

ASingleInequalityConstraint 310

TwoInequalityConstraints 313

12.2 TangentConeandConstraintQualifications 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

ProofofTheorem12.1 329

12.5 Second-Order Conditions 330

Second-Order Conditions and Projected Hessians 337

12.6 OtherConstraintQualifications 338

12.7 AGeometricViewpoint 340

12.8 LagrangeMultipliersandSensitivity 341

12.9 Duality 343

NotesandReferences 349

Exercises 351

13 Linear Programming: The SimplexMethod 355

LinearProgramming 356

13.1 OptimalityandDuality 358

Optimality Conditions 358

TheDualProblem 359

13.2 Geometryof theFeasibleSet 362

BasesandBasicFeasiblePoints 362

Verticesof theFeasiblePolytope 365

13.3 TheSimplexMethod 366

Outline 366

ASingleStepof theMethod 370

13.4 LinearAlgebraintheSimplexMethod 372

13.5 Other ImportantDetails 375

PricingandSelectionof theEnteringIndex 375

StartingtheSimplexMethod 378

DegenerateStepsandCycling 381

13.6 TheDualSimplexMethod 382

13.7 Presolving 385

13.8 WhereDoes theSimplexMethodFit? 388

NotesandReferences 389

Exercises 389

14 Linear Programming: Interior-PointMethods 392

14.1 Primal-DualMethods 393

Outline 393

TheCentralPath 397

Central Path Neighborhoods and Path-Following Methods 399

14.2 Practical Primal-Dual Algorithms 407

CorrectorandCenteringSteps 407

Step Lengths 409

StartingPoint 410

APracticalAlgorithm 411

SolvingtheLinearSystems 411

14.3 Other Primal-Dual Algorithms and Extensions 413

Other Path-Following Methods 413

Potential-ReductionMethods 414

Extensions 415

14.4 PerspectivesandSoftware 416

NotesandReferences 417

Exercises 418

15 Fundamentals of Algorithms for Nonlinear Constrained Optimization 421

15.1 Categorizing Optimization Algorithms 422

15.2 The Combinatorial Difficulty of Inequality-Constrained Problems 424

15.3 EliminationofVariables 426

SimpleEliminationusingLinearConstraints 428

GeneralReductionStrategies forLinearConstraints 431

Effectof InequalityConstraints 434

15.4 MeritFunctionsandFilters 435

MeritFunctions 435

Filters 437

15.5 TheMaratosEffect 440

15.6 Second-OrderCorrectionandNonmonotoneTechniques 443

Nonmonotone(Watchdog)Strategy 444

NotesandReferences 446

Exercises 446

16 Quadratic Programming 448

16.1 Equality-ConstrainedQuadraticPrograms 451

PropertiesofEquality-ConstrainedQPs 451

16.2 DirectSolutionof theKKTSystem 454

FactoringtheFullKKTSystem 454

Schur-ComplementMethod 455

Null-Space Method 457

16.3 IterativeSolutionof theKKTSystem 459

CGAppliedtotheReducedSystem 459

TheProjectedCGMethod 461

16.4 Inequality-ConstrainedProblems 463

Optimality Conditions for Inequality-Constrained Problems 464

Degeneracy 465

16.5 Active-SetMethods forConvexQPs 467

Specificationof theActive-SetMethodforConvexQP 472

FurtherRemarksontheActive-SetMethod 476

Finite Termination of Active-Set Algorithm on Strictly Convex QPs 477

UpdatingFactorizations 478

16.6 Interior-PointMethods 480

SolvingthePrimal-DualSystem 482

StepLengthSelection 483

APracticalPrimal-DualMethod 484

16.7 TheGradientProjectionMethod 485

CauchyPointComputation 486

Subspace Minimization 488

16.8 PerspectivesandSoftware 490

NotesandReferences 492

Exercises 492

17 Penalty and Augmented LagrangianMethods 497

17.1 TheQuadraticPenaltyMethod 498

Motivation 498

Algorithmic Framework 501

Convergenceof theQuadraticPenaltyMethod 502

Ill Conditioning and Reformulations 505

17.2 NonsmoothPenaltyFunctions 507

A Practical _1 PenaltyMethod 511

AGeneralClassofNonsmoothPenaltyMethods 513

17.3 AugmentedLagrangianMethod:EqualityConstraints 514

Motivation and Algorithmic Framework 514

Propertiesof theAugmentedLagrangian 517

17.4 PracticalAugmentedLagrangianMethods 519

Bound-ConstrainedFormulation 519

LinearlyConstrainedFormulation 522

UnconstrainedFormulation 523

17.5 PerspectivesandSoftware 525

NotesandReferences 526

Exercises 527

18 Sequential Quadratic Programming 529

18.1 LocalSQPMethod 530

SQPFramework 531

InequalityConstraints 532

18.2 PreviewofPracticalSQPMethods 533

IQPandEQP 533

EnforcingConvergence 534

18.3 Algorithmic Development 535

HandlingInconsistentLinearizations 535

FullQuasi-NewtonApproximations 536

Reduced-Hessian Quasi-Newton Approximations 538

MeritFunctions 540

Second-OrderCorrection 543

18.4 APracticalLineSearchSQPMethod 545

18.5 Trust-Region SQP Methods 546

A Relaxation Method for Equality-Constrained Optimization 547

S_1QP (Sequential _1QuadraticProgramming) 549

SequentialLinear-QuadraticProgramming(SLQP) 551

ATechniqueforUpdatingthePenaltyParameter 553

18.6 NonlinearGradientProjection 554

18.7 ConvergenceAnalysis 556

RateofConvergence 557

18.8 PerspectivesandSoftware 560

NotesandReferences 561

Exercises 561

19 Interior-PointMethods for Nonlinear Programming 563

19.1 TwoInterpretations 564

19.2 ABasicInterior-PointAlgorithm 566

19.3 Algorithmic Development 569

Primalvs.Primal-DualSystem 570

SolvingthePrimal-DualSystem 570

UpdatingtheBarrierParameter 572

Handling Nonconvexity and Singularity 573

StepAcceptance:MeritFunctionsandFilters 575

Quasi-NewtonApproximations 575

FeasibleInterior-PointMethods 576

19.4 ALineSearchInterior-PointMethod 577

19.5 A Trust-Region Interior-Point Method 578

AnAlgorithmforSolvingtheBarrierProblem 578

StepComputation 580

LagrangeMultipliersEstimatesandStepAcceptance 581

Description of a Trust-Region Interior-Point Method 582

19.6 ThePrimalLog-BarrierMethod 583

19.7 GlobalConvergenceProperties 587

Failureof theLineSearchApproach 587

ModifiedLineSearchMethods 589

Global Convergence of the Trust-Region Approach 589

19.8 SuperlinearConvergence 591

19.9 PerspectivesandSoftware 592

NotesandReferences 593

Exercises 594

A BackgroundMaterial 598

A.1 ElementsofLinearAlgebra 598

VectorsandMatrices 598

Norms 600

Subspaces 602

Eigenvalues, Eigenvectors, and the Singular-Value Decomposition 603

DeterminantandTrace 605

Matrix Factorizations: Cholesky, LU, QR 606

SymmetricIndefiniteFactorization 610

Sherman–Morrison–WoodburyFormula 612

InterlacingEigenvalueTheorem 613

Error Analysis and Floating-Point Arithmetic 613

Conditioning and Stability 616

A.2 ElementsofAnalysis,Geometry,Topology 617

Sequences 617

RatesofConvergence 619

Topology of the Euclidean Space IRn 620

Convex Sets in IRn 621

ContinuityandLimits 623

Derivatives 625

DirectionalDerivatives 628

MeanValueTheorem 629

ImplicitFunctionTheorem 630

OrderNotation 631

Root-Finding for Scalar Equations 633

B A Regularization Procedure 635

References 637

Index 653

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