Numerical Optimization

By: Jorge Nocedal, Stephen Wright

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Hardcover | 1 July 2006 | Edition Number 2

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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.

Industry Reviews

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."

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Non-Fiction Mathematics Optimisation Linear Programming Applied Mathematics Business & Management Operational Research

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Published: 1st April 2009

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Preface	p. xvii
Preface to the Second Edition	p. xxi
Introduction	p. 1
Mathematical Formulation	p. 2
Example: A Transportation Problem	p. 4
Continuous versus Discrete Optimization	p. 5
Constrained and Unconstrained Optimization	p. 6
Global and Local Optimization	p. 6
Stochastic and Deterministic Optimization	p. 7
Convexity	p. 7
Optimization Algorithms	p. 8
Notes and References	p. 9
Fundamentals of Unconstrained Optimization	p. 10
What Is a Solution?	p. 12
Recognizing a Local Minimum	p. 14
Nonsmooth Problems	p. 17
Overview of Algorithms	p. 18
Two Strategies: Line Search and Trust Region	p. 19
Search Directions for Line Search Methods	p. 20
Models for Trust-Region Methods	p. 25
Scaling	p. 26
Exercises	p. 27
Line Search Methods	p. 30
Step Length	p. 31
The Wolfe Conditions	p. 33
The Goldstein Conditions	p. 36
Sufficient Decrease and Backtracking	p. 37
Convergence of Line Search Methods	p. 37
Rate of Convergence	p. 41
Convergence Rate of Steepest Descent	p. 42
Newton's Method	p. 44
Quasi-Newton Methods	p. 46
Newton's Method with Hessian Modification	p. 48
Eigenvalue Modification	p. 49
Adding a Multiple of the Identity	p. 51
Modified Cholesky Factorization	p. 52
Modified Symmetric Indefinite Factorization	p. 54
Step-Length Selection Algorithms	p. 56
Interpolation	p. 57
Initial Step Length	p. 59
A Line Search Algorithm for the Wolfe Conditions	p. 60
Notes and References	p. 62
Exercises	p. 63
Trust-Region Methods	p. 66
Outline of the Trust-Region Approach	p. 68
Algorithms Based on the Cauchy Point	p. 71
The Cauchy Point	p. 71
Improving on the Cauchy Point	p. 73
The Dogleg Method	p. 73
Two-Dimensional Subspace Minimization	p. 76
Global Convergence	p. 77
Reduction Obtained by the Cauchy Point	p. 77
Convergence to Stationary Points	p. 79
Iterative Solution of the Subproblem	p. 83
The Hard Case	p. 87
Proof of Theorem 4.1	p. 89
Convergence of Algorithms Based on Nearly Exact Solutions	p. 91
Local Convergence of Trust-Region Newton Methods	p. 92
Other Enhancements	p. 95
Scaling	p. 95
Trust Regions in Other Norms	p. 97
Notes and References	p. 98
Exercises	p. 98
Conjugate Gradient Methods	p. 101
The Linear Conjugate Gradient Method	p. 102
Conjugate Direction Methods	p. 102
Basic Properties of the Conjugate Gradient Method	p. 107
A Practical Form of the Conjugate Gradient Method	p. 111
Rate of Convergence	p. 112
Preconditioning	p. 118
Practical Preconditioners	p. 120
Nonlinear Conjugate Gradient Methods	p. 121
The Fletcher-Reeves Method	p. 121
The Polak-Ribiere Method and Variants	p. 122
Quadratic Termination and Restarts	p. 124
Behavior of the Fletcher-Reeves Method	p. 125
Global Convergence	p. 127
Numerical Performance	p. 131
Notes and References	p. 132
Exercises	p. 133
Quasi-Newton Methods	p. 135
The BFGS Method	p. 136
Properties of the BFGS Method	p. 141
Implementation	p. 142
The SR1 Method	p. 144
Properties of SR1 Updating	p. 147
The Broyden Class	p. 149
Convergence Analysis	p. 153
Global Convergence of the BFGS Method	p. 153
Superlinear Convergence of the BFGS Method	p. 156
Convergence Analysis of the SR1 Method	p. 160
Notes and References	p. 161
Exercises	p. 162
Large-Scale Unconstrained Optimization	p. 164
Inexact Newton Methods	p. 165
Local Convergence of Inexact Newton Methods	p. 166
Line Search Newton-CG Method	p. 168
Trust-Region Newton-CG Method	p. 170
Preconditioning the Trust-Region Newton-CG Method	p. 174
Trust-Region Newton-Lanczos Method	p. 175
Limited-Memory Quasi-Newton Methods	p. 176
Limited-Memory BFGS	p. 177
Relationship with Conjugate Gradient Methods	p. 180
General Limited-Memory Updating	p. 181
Compact Representation of BFGS Updating	p. 181
Unrolling the Update	p. 184
Sparse Quasi-Newton Updates	p. 185
Algorithms for Partially Separable Functions	p. 186
Perspectives and Software	p. 189
Notes and References	p. 190
Exercises	p. 191
Calculating Derivatives	p. 193
Finite-Difference Derivative Approximations	p. 194
Approximating the Gradient	p. 195
Approximating a Sparse Jacobian	p. 197
Approximating the Hessian	p. 201
Approximating a Sparse Hessian	p. 202
Automatic Differentiation	p. 204
An Example	p. 205
The Forward Mode	p. 206
The Reverse Mode	p. 207
Vector Functions and Partial Separability	p. 210
Calculating Jacobians of Vector Functions	p. 212
Calculating Hessians: Forward Mode	p. 213
Calculating Hessians: Reverse Mode	p. 215
Current Limitations	p. 216
Notes and References	p. 217
Exercises	p. 217
Derivative-Free Optimization	p. 220
Finite Differences and Noise	p. 221
Model-Based Methods	p. 223
Interpolation and Polynomial Bases	p. 226
Updating the Interpolation Set	p. 227
A Method Based on Minimum-Change Updating	p. 228
Coordinate and Pattern-Search Methods	p. 229
Coordinate Search Method	p. 230
Pattern-Search Methods	p. 231
A Conjugate-Direction Method	p. 234
Nelder-Mead Method	p. 238
Implicit Filtering	p. 240
Notes and References	p. 242
Exercises	p. 242
Least-Squares Problems	p. 245
Background	p. 247
Linear Least-Squares Problems	p. 250
Algorithms for Nonlinear Least-Squares Problems	p. 254
The Gauss-Newton Method	p. 254
Convergence of the Gauss-Newton Method	p. 255
The Levenberg-Marquardt Method	p. 258
Implementation of the Levenberg-Marquardt Method	p. 259
Convergence of the Levenberg-Marquardt Method	p. 261
Methods for Large-Residual Problems	p. 262
Orthogonal Distance Regression	p. 265
Notes and References	p. 267
Exercises	p. 269
Nonlinear Equations	p. 270
Local Algorithms	p. 274
Newton's Method for Nonlinear Equations	p. 274
Inexact Newton Methods	p. 277
Broyden's Method	p. 279
Tensor Methods	p. 283
Practical Methods	p. 285
Merit Functions	p. 285
Line Search Methods	p. 287
Trust-Region Methods	p. 290
Continuation/Homotopy Methods	p. 296
Motivation	p. 296
Practical Continuation Methods	p. 297
Notes and References	p. 302
Exercises	p. 302
Theory of Constrained Optimization	p. 304
Local and Global Solutions	p. 305
Smoothness	p. 306
Examples	p. 307
A Single Equality Constraint	p. 308
A Single Inequality Constraint	p. 310
Two Inequality Constraints	p. 313
Tangent Cone and Constraint Qualifications	p. 315
First-Order Optimality Conditions	p. 320
First-Order Optimality Conditions: Proof	p. 323
Relating the Tangent Cone and the First-Order Feasible Direction Set	p. 323
A Fundamental Necessary Condition	p. 325
Farkas' Lemma	p. 326
Proof of Theorem 12.1	p. 329
Second-Order Conditions	p. 330
Second-Order Conditions and Projected Hessians	p. 337
Other Constraint Qualifications	p. 338
A Geometric Viewpoint	p. 340
Lagrange Multipliers and Sensitivity	p. 341
Duality	p. 343
Notes and References	p. 349
Exercises	p. 351
Linear Programming: The Simplex Method	p. 355
Linear Programming	p. 356
Optimality and Duality	p. 358
Optimality Conditions	p. 358
The Dual Problem	p. 359
Geometry of the Feasible Set	p. 362
Bases and Basic Feasible Points	p. 362
Vertices of the Feasible Polytope	p. 365
The Simplex Method	p. 366
Outline	p. 366
A Single Step of the Method	p. 370
Linear Algebra in the Simplex Method	p. 372
Other Important Details	p. 375
Pricing and Selection of the Entering Index	p. 375
Starting the Simplex Method	p. 378
Degenerate Steps and Cycling	p. 381
The Dual Simplex Method	p. 382
Presolving	p. 385
Where Does the Simplex Method Fit?	p. 388
Notes and References	p. 389
Exercises	p. 389
Linear Programming: Interior-Point Methods	p. 392
Primal-Dual Methods	p. 393
Outline	p. 393
The Central Path	p. 397
Central Path Neighborhoods and Path-Following Methods	p. 399
Practical Primal-Dual Algorithms	p. 407
Corrector and Centering Steps	p. 407
Step Lengths	p. 409
Starting Point	p. 410
A Practical Algorithm	p. 411
Solving the Linear Systems	p. 411
Other Primal-Dual Algorithms and Extensions	p. 413
Other Path-Following Methods	p. 413
Potential-Reduction Methods	p. 414
Extensions	p. 415
Perspectives and Software	p. 416
Notes and References	p. 417
Exercises	p. 418
Fundamentals of Algorithms for Nonlinear Constrained Optimization	p. 421
Categorizing Optimization Algorithms	p. 422
The Combinatorial Difficulty of Inequality-Constrained Problems	p. 424
Elimination of Variables	p. 426
Simple Elimination using Linear Constraints	p. 428
General Reduction Strategies for Linear Constraints	p. 431
Effect of Inequality Constraints	p. 434
Merit Functions and Filters	p. 435
Merit Functions	p. 435
Filters	p. 437
The Maratos Effect	p. 440
Second-Order Correction and Nonmonotone Techniques	p. 443
Nonmonotone (Watchdog) Strategy	p. 444
Notes and References	p. 446
Exercises	p. 446
Quadratic Programming	p. 448
Equality-Constrained Quadratic Programs	p. 451
Properties of Equality-Constrained QPs	p. 451
Direct Solution of the KKT System	p. 454
Factoring the Full KKT System	p. 454
Schur-Complement Method	p. 455
Null-Space Method	p. 457
Iterative Solution of the KKT System	p. 459
CG Applied to the Reduced System	p. 459
The Projected CG Method	p. 461
Inequality-Constrained Problems	p. 463
Optimality Conditions for Inequality-Constrained Problems	p. 464
Degeneracy	p. 465
Active-Set Methods for Convex QPs	p. 467
Specification of the Active-Set Method for Convex QP	p. 472
Further Remarks on the Active-Set Method	p. 476
Finite Termination of Active-Set Algorithm on Strictly Convex QPs	p. 477
Updating Factorizations	p. 478
Interior-Point Methods	p. 480
Solving the Primal-Dual System	p. 482
Step Length Selection	p. 483
A Practical Primal-Dual Method	p. 484
The Gradient Projection Method	p. 485
Cauchy Point Computation	p. 486
Subspace Minimization	p. 488
Perspectives and Software	p. 490
Notes and References	p. 492
Exercises	p. 492
Penalty and Augmented Lagrangian Methods	p. 497
The Quadratic Penalty Method	p. 498
Motivation	p. 498
Algorithmic Framework	p. 501
Convergence of the Quadratic Penalty Method	p. 502
Ill Conditioning and Reformulations	p. 505
Nonsmooth Penalty Functions	p. 507
A Practical l[subscript 1] Penalty Method	p. 511
A General Class of Nonsmooth Penalty Methods	p. 513
Augmented Lagrangian Method: Equality Constraints	p. 514
Motivation and Algorithmic Framework	p. 514
Properties of the Augmented Lagrangian	p. 517
Practical Augmented Lagrangian Methods	p. 519
Bound-Constrained Formulation	p. 519
Linearly Constrained Formulation	p. 522
Unconstrained Formulation	p. 523
Perspectives and Software	p. 525
Notes and References	p. 526
Exercises	p. 527
Sequential Quadratic Programming	p. 529
Local SQP Method	p. 530
SQP Framework	p. 531
Inequality Constraints	p. 532
Preview of Practical SQP Methods	p. 533
IQP and EQP	p. 533
Enforcing Convergence	p. 534
Algorithmic Development	p. 535
Handling Inconsistent Linearizations	p. 535
Full Quasi-Newton Approximations	p. 536
Reduced-Hessian Quasi-Newton Approximations	p. 538
Merit Functions	p. 540
Second-Order Correction	p. 543
A Practical Line Search SQP Method	p. 545
Trust-Region SQP Methods	p. 546
A Relaxation Method for Equality-Constrained Optimization	p. 547
Sl[subscript 1] QP (Sequential l[subscript 1] Quadratic Programming)	p. 549
Sequential Linear-Quadratic Programming (SLQP)	p. 551
A Technique for Updating the Penalty Parameter	p. 553
Nonlinear Gradient Projection	p. 554
Convergence Analysis	p. 556
Rate of Convergence	p. 557
Perspectives and Software	p. 560
Notes and References	p. 561
Exercises	p. 561
Interior-Point Methods for Nonlinear Programming	p. 563
Two Interpretations	p. 564
A Basic Interior-Point Algorithm	p. 566
Algorithmic Development	p. 569
Primal vs. Primal-Dual System	p. 570
Solving the Primal-Dual System	p. 570
Updating the Barrier Parameter	p. 572
Handling Nonconvexity and Singularity	p. 573
Step Acceptance: Merit Functions and Filters	p. 575
Quasi-Newton Approximations	p. 575
Feasible Interior-Point Methods	p. 576
A Line Search Interior-Point Method	p. 577
A Trust-Region Interior-Point Method	p. 578
An Algorithm for Solving the Barrier Problem	p. 578
Step Computation	p. 580
Lagrange Multipliers Estimates and Step Acceptance	p. 581
Description of a Trust-Region Interior-Point Method	p. 582
The Primal Log-Barrier Method	p. 583
Global Convergence Properties	p. 587
Failure of the Line Search Approach	p. 587
Modified Line Search Methods	p. 589
Global Convergence of the Trust-Region Approach	p. 589
Superlinear Convergence	p. 591
Perspectives and Software	p. 592
Notes and References	p. 593
Exercises	p. 594
Background Material	p. 598
Elements of Linear Algebra	p. 598
Vectors and Matrices	p. 598
Norms	p. 600
Subspaces	p. 602
Eigenvalues, Eigenvectors, and the Singular-Value Decomposition	p. 603
Determinant and Trace	p. 605
Matrix Factorizations: Cholesky, LU, QR	p. 606
Symmetric Indefinite Factorization	p. 610
Sherman-Morrison-Woodbury Formula	p. 612
Interlacing Eigenvalue Theorem	p. 613
Error Analysis and Floating-Point Arithmetic	p. 613
Conditioning and Stability	p. 616
Elements of Analysis, Geometry, Topology	p. 617
Sequences	p. 617
Rates of Convergence	p. 619
Topology of the Euclidean Space R[superscript n]	p. 620
Convex Sets in R[superscript n]	p. 621
Continuity and Limits	p. 623
Derivatives	p. 625
Directional Derivatives	p. 628
Mean Value Theorem	p. 629
Implicit Function Theorem	p. 630
Order Notation	p. 631
Root-Finding for Scalar Equations	p. 633
A Regularization Procedure	p. 635
References	p. 637
Index	p. 653
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Numerical Optimization

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