《Numerical Optimization》——非线性最优化经典著作

非线性最优化

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"Numerical Optimization非线性最优化" 《Numerical Optimization》是由Jorge Nocedal和Stephen J. Wright合著的一本专业书籍，它深入探讨了非线性最优化这一关键的数学和计算领域。该书由Springer出版社出版，属于Springer Series in Operations Research系列，该系列由Peter Glynn和Stephen M. Robinson担任编辑。非线性最优化是优化理论的一个分支，主要研究如何在非线性函数的约束条件下寻找最大值或最小值。在实际问题中，如工程设计、经济模型、机器学习和数据分析等领域，非线性优化方法被广泛应用于解决复杂问题。本书包含85个插图，帮助读者直观理解复杂的概念和算法。作者Jorge Nocedal来自西北大学电子与计算机工程系和数学与计算机科学分部，而Stephen J. Wright则来自阿贡国家实验室和威斯康星大学麦迪逊分校的工业工程和运营管理系。他们的专业背景确保了书中内容的权威性和实用性。书中的内容可能涵盖了以下知识点： 1. **优化问题的基本概念**：定义非线性优化问题，包括目标函数和约束条件。 2. **梯度法**：介绍梯度下降法及其变种，讨论局部最优解和全局最优解的概念。 3. **牛顿法和拟牛顿法**：详述牛顿法的基本思想，以及有限内存BFGS和DFP等拟牛顿法，解释其优点和适用情况。 4. **线性代数和矩阵理论**：在非线性优化中线性代数的重要性，如矩阵的性质和特征值分析。 5. **迭代过程和收敛性**：讨论迭代算法的收敛性条件，以及如何评估优化算法的效率。 6. **约束处理**：处理等式和不等式约束的方法，如拉格朗日乘子法和惩罚函数法。 7. **全局优化方法**：针对全局最优解的算法，如分支定界、模拟退火和遗传算法。 8. **实际应用案例**：通过实例展示非线性优化在实际问题中的应用和解决方案。 9. **软件工具**：介绍常用的非线性优化软件库，如MATLAB的fmincon、Python的scipy.optimize等。 10. **数值稳定性和算法实现**：讨论数值稳定性问题，以及如何在实际编程中避免误差积累。《Numerical Optimization》不仅对理论进行了深入探讨，还提供了丰富的实例和算法实现细节，适合研究生、研究人员和工程师作为参考书目。书中包含的文献引用也便于读者进一步研究相关主题。

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xvi Contents

Practical Continuation Methods ...................... 306

NotesandReferences ............................... 310

Exercises ...................................... 311

12 Theory of Constrained O ptimization 314

LocalandGlobalSolutions ........................ 316

Smoothness ................................ 317

12.1 Examples .................................. 319

A Single Equality Constraint ........................ 319

A Single Inequality Constraint ....................... 321

Two Inequality Constraints ........................ 324

12.2 First-Order Optimality Conditions .................... 327

Statement of First-Order Necessary Conditions .............. 327

Sensitivity ................................. 330

12.3 Derivation of the First-Order Conditions ................. 331

Feasible Sequences ............................. 331

Characterizing Limiting Directions: Constraint Qualiﬁcations ...... 336

Introducing Lagrange Multipliers ..................... 339

Proof of Theorem 12.1 ........................... 341

12.4 Second-Order Conditions ......................... 342

Second-Order Conditions and Projected Hessians ............ 348

ConvexPrograms ............................. 349

12.5 Other Constraint Qualiﬁcations ...................... 350

12.6 A Geometric Viewpoint .......................... 353

NotesandReferences ............................... 356

Exercises ...................................... 357

13 Linear Programming: The Simplex Method 360

Linear Programming ............................ 362

13.1 Optimality and Duality .......................... 364

Optimality Conditions ........................... 364

TheDualProblem............................. 365

13.2 Geometry of the Feasible Set ........................ 368

Basic Feasible Points ............................ 368

Vert ices of the Feasible Polytope ...................... 370

13.3 The Simplex Method ............................ 372

Outline of the Method ........................... 372

Finite Termination of the Simplex Method ................ 374

ASingleStepoftheMethod........................ 376

13.4 Linear Algebra in the Simplex Method .................. 377

13.5 Other (Important) Details ......................... 381

Pricing and Selection of the Entering Index ................ 381

Contents xvii

Starting the Simplex Method ....................... 384

Degenerate Steps and Cycling ....................... 387

13.6 Where Does the Simplex Method Fit? ................... 389

NotesandReferences ............................... 390

Exercises ...................................... 391

14 Linear Programming: Interior-Point Methods 392

14.1 Primal–Dual Methods ........................... 394

Outline ................................... 394

TheCentralPath.............................. 397

A Primal–Dual Framework ........................ 399

Path-Following Methods .......................... 400

14.2 A Practical Primal–Dual Algorithm .................... 402

Solving the Linear Systems ......................... 406

14.3 Other Primal–Dual Algorithms and Extensions .............. 407

Other Path-Following Methods ...................... 407

Potential-Reduction Methods ....................... 407

Extensions ................................. 408

14.4 Analysis of Algorithm 14.2 ......................... 409

NotesandReferences ............................... 414

Exercises ...................................... 415

15 Fundamentals of Algorithms for Nonlinear Constrained Optimization 418

InitialStudyofaProblem ......................... 420

15.1 Categorizing Optimization Algorithms .................. 422

15.2 Elimination of Variables .......................... 424

Simple Elimination for Linear Constraints ................ 426

General Reduction Strategies for Linear Constraints ........... 429

The Effect of Inequality Constraints .................... 431

15.3 Measuring Progress: Merit Functions ................... 432

NotesandReferences ............................... 436

Exercises ...................................... 436

16 Quadratic Programming 438

An Example: Portfolio Optimization ................... 440

16.1 Equality–Constrained Quadratic Programs ................ 441

Properties of Equality-Constrained QPs .................. 442

16.2 Solving the KKT System .......................... 445

DirectSolutionoftheKKTSystem .................... 446

Range-Space Method ........................... 447

Null-Space Method ............................ 448

AMethodBasedonConjugacy ...................... 450

xviii Contents

16.3 Inequality-Constrained Problems ..................... 451

Optimality Conditions for Inequality-Constrained Problems ....... 452

Degeneracy................................. 453

16.4 Active-Set Methods for Convex QP .................... 455

Speciﬁcation of the Active-Set Method for Convex QP .......... 460

An Example ................................ 461

FurtherRemarksontheActive-SetMethod................ 463

Finite Termination of the Convex QP Algorithm ............. 464

Updating Factorizations .......................... 465

16.5 Active-Set Methods for Indeﬁnite QP ................... 468

Illustration................................. 470

Choice of Starting Point .......................... 472

FailureoftheActive-SetMethod ..................... 473

Detecting Indeﬁniteness Using the LBL

Factorization ......... 473

16.6 The Gradient–Projection Method ..................... 474

CauchyPointComputation ........................ 475

Subspace Minimization .......................... 478

16.7 Interior-Point Methods .......................... 479

Extensions and Comparison with Active-Set Methods .......... 482

16.8 Duality ................................... 482

NotesandReferences ............................... 483

Exercises ...................................... 484

17 Penalty, Barrier, and Augmented L agrangian Methods 488

17.1 The Quadratic Penalty Method ...................... 490

Motivation................................. 490

Algorithmic Framework .......................... 492

Convergence of the Quadratic Penalty Function ............. 493

17.2 The Logarithmic Barrier Method ..................... 498

Properties of Logarithmic Barr ier Functions ............... 498

Algorithms Based on the Log-Barrier Function .............. 503

Properties of the Log-Barrier Function and Framework 17.2 ....... 505

Handling Equality Constraints ...................... 507

Relationship to Primal–Dual Methods .................. 508

17.3 Exact Penalty Functions .......................... 510

17.4 Augmented Lagrangian Method ...................... 511

Motivation and Algorithm Framework .................. 512

Extension to Inequality Constraints .................... 514

PropertiesoftheAugmentedLagrangian ................. 517

Practical Implementation ......................... 520

17.5 Sequential Linearly Constrained Methods ................. 522

NotesandReferences ............................... 523

Contents xix

Exercises ...................................... 524

18 Sequential Quadratic Programming 526

18.1 Local SQP Method ............................. 528

SQPFramework.............................. 529

Inequality Constraints ........................... 531

IQPvs.EQP ................................ 531

18.2 Preview of Practical SQP Methods ..................... 532

18.3 Step Computation ............................. 534

Equality Constraints ............................ 534

Inequality Constraints ........................... 536

18.4 The Hessian of the Quadratic Model ................... 537

Full Quasi-Newton Approximations .................... 538

Hessian of Augmented Lagrangian .................... 539

Reduced-Hessian Approximations ..................... 540

18.5 Merit Functions and Descent ....................... 542

18.6 A Line Search SQP Method ........................ 545

18.7 Reduced-Hessian SQP Methods ...................... 546

Some Properties of Reduced-Hessian Methods .............. 547

Update Criteria for Reduced-Hessian Updating .............. 548

Changes of Bases .............................. 549

A Practical Reduced-Hessian Method ................... 550

18.8 Trust-Region SQP Me thods ........................ 551

Approach I: Shifting the Constraints ................... 553

Approach II: Two Elliptical Constraints .................. 554

Approach III: S

QP (Sequential 

Quadratic Programming) ...... 555

18.9 A Practical Trust-Region SQP Algorithm ................. 558

18.10 Rate of Convergence ............................ 561

Convergence Rate of Reduced-Hessian Methods ............. 563

18.11 The Maratos Effect ............................. 565

Second-Order Correction ......................... 568

Watchdog(Nonmonotone)Strategy.................... 569

NotesandReferences ............................... 571

Exercises ...................................... 572

A Background Material 574

A.1 Elements of Analysis, Geometry, Topology ................ 575

Topology of the Euclidean Space IR

.................... 575

Continuity and Limits ........................... 578

Derivatives................................. 579

DirectionalDerivatives .......................... 581

MeanValueTheorem ........................... 582

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