Numerical optimization:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | Englisch |
| Veröffentlicht: |
New York, NY
Springer
[2006]
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Springer series in operations research and financial engineering
|
| Schlagwörter: | |
| Links: | https://doi.org/10.1007/978-0-387-40065-5 https://doi.org/10.1007/978-0-387-40065-5 https://doi.org/10.1007/978-0-387-40065-5 https://doi.org/10.1007/978-0-387-40065-5 https://doi.org/10.1007/978-0-387-40065-5 https://doi.org/10.1007/978-0-387-40065-5 https://doi.org/10.1007/978-0-387-40065-5 https://doi.org/10.1007/978-0-387-40065-5 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015516939&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | 1 Online-Ressource (xxii, 664 Seiten) Illustrationen, Diagramme |
| ISBN: | 9780387400655 |
| DOI: | 10.1007/978-0-387-40065-5 |
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Datensatz im Suchindex
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|---|---|
| _version_ | 1821930751058247681 |
| adam_text | Contents
Preface xvii
Preface to the Second Edition xxi
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
Stochastic and Deterministic Optimization 7
Convexity 7
Optimization Algorithms 8
Notes and References 9
2 Fundamentals of Unconstrained Optimization 10
2.1 What Is a Solution? 12
viii Contents
Recognizing a Local Minimum 14
Nonsmooth Problems 17
2.2 Overview of Algorithms 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 Identity 51
Modified Cholesky Factorization 52
Modified Symmetric Indefinite Factorization 54
3.5 Step Length Selection Algorithms 56
Interpolation 57
Initial Step Length 59
A Line Search Algorithm 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 Algorithms Based on the Cauchy Point 71
The Cauchy Point 71
Improving on the Cauchy Point 73
The Dogleg Method 73
Two Dimensional Subspace Minimization 76
4.2 Global Convergence 77
Reduction Obtained by the Cauchy Point 77
Convergence to Stationary Points 79
4.3 Iterative Solution of the Subproblem 83
Contents ix
The Hard Case 87
Proof of Theorem 4.1 89
Convergence of Algorithms Based on Nearly Exact Solutions 91
4.4 Local Convergence of Trust Region Newton Methods 92
4.5 Other Enhancements 95
Scaling 95
Trust Regions in Other 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 the Conjugate Gradient Method 107
A Practical Form of the Conjugate Gradient Method Ill
Rate of Convergence 112
Preconditioning 118
Practical Preconditioners 120
5.2 Nonlinear Conjugate Gradient Methods 121
The Fletcher Reeves Method 121
The Polak Ribiere Method and Variants 122
Quadratic Termination and Restarts 124
Behavior of the Fletcher Reeves Method 125
Global Convergence 127
Numerical Performance 131
Notes and References 132
Exercises 133
6 Quasi Newton Methods 135
6.1 The BFGS Method 136
Properties oftheBFGS Method 141
Implementation 142
6.2 The SRI Method 144
Properties of SRI Updating 147
6.3 The Broyden Class 149
6.4 Convergence Analysis 153
Global Convergence oftheBFGS Method 153
Superlinear Convergence of the BFGS Method 156
Convergence Analysis of the SRI Method 160
Notes and References 161
Exercises 162
x Contents
7 Large Scale Unconstrained Optimization 164
7.1 Inexact 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 Limited Memory Updating 181
Compact Representation of BFGS Updating 181
Unrolling the Update 184
7.3 Sparse Quasi Newton Updates 185
7.4 Algorithms for Partially Separable Functions 186
7.5 Perspectives and Software 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
Approximating the Hessian 201
Approximating a Sparse Hessian 202
8.2 Automatic Differentiation 204
An Example 205
The Forward Mode 206
The Reverse Mode 207
Vector Functions and Partial Separability 210
Calculating Jacobians of Vector Functions 212
Calculating Hessians: Forward Mode 213
Calculating Hessians: Reverse Mode 215
Current Limitations 216
Notes and References 217
Exercises 217
9 Derivative Free Optimization 220
9.1 Finite Differences and Noise 221
9.2 Model Based Methods 223
Interpolation and Polynomial Bases 226
Updating the Interpolation Set 227
Contents xi
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 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
The Levenberg Marquardt Method 258
Implementation oftheLevenberg Marquardt Method 259
Convergence of the Levenberg Marquardt Method 261
Methods for Large Residual Problems 262
10.4 Orthogonal Distance Regression 265
Notes and References 267
Exercises 269
11 Nonlinear Equations 270
11.1 Local Algorithms 274
Newton s Method for Nonlinear Equations 274
Inexact Newton Methods 277
Broyden s Method 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
xii Contents
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 of Theorem 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 Duality 343
Notes and References 349
Exercises 351
13 Linear Programming: The Simplex Method 355
Linear Programming 356
13.1 Optimality and Duality 358
Optimality Conditions 358
The Dual Problem 359
13.2 Geometry of the Feasible Set 362
Bases and Basic Feasible Points 362
Vertices of the Feasible Polytope 365
13.3 The Simplex Method 366
Outline 366
A Single Step of the Method 370
13.4 Linear Algebra in the Simplex Method 372
13.5 Other Important Details 375
Pricing and Selection of the Entering Index 375
Starting the Simplex Method 378
Degenerate Steps and Cycling 381
13.6 The Dual Simplex Method 382
13.7 Presolving 385
13.8 Where Does the Simplex Method Fit? 388
Notes and References 389
Exercises 389
Contents xiii
14 Linear Programming: Interior Point Methods 392
14.1 Primal Dual Methods 393
Outline 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 Practical Algorithm 411
Solving the Linear Systems 411
14.3 Other Primal Dual Algorithms and Extensions 413
Other Path Following Methods 413
Potential Reduction Methods 414
Extensions 415
14.4 Perspectives and Software 416
Notes and References 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 Elimination of Variables 426
Simple Elimination using Linear Constraints 428
General Reduction Strategies for Linear Constraints 431
Effect of Inequality Constraints 434
15.4 Merit Functions and Filters 435
Merit Functions 435
Filters 437
15.5 The Maratos Effect 440
15.6 Second Order Correction and Nonmonotone Techniques 443
Nonmonotone (Watchdog) Strategy 444
Notes and References 446
Exercises 446
16 Quadratic Programming 448
16.1 Equality Constrained Quadratic Programs 451
Properties of Equality Constrained QPs 451
16.2 Direct Solution oftheKKT System 454
Factoring the Full KKT System 454
Schur Complement Method 455
Null Space Method 457
xiv Contents
16.3 Iterative Solution of the KKT System 459
CG Applied to the Reduced System 459
The Projected CG 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 Algorithm on Strictly Convex QPs . . . 477
Updating Factorizations 478
16.6 Interior Point Methods 480
Solving the Primal Dual System 482
Step Length Selection 483
A Practical Primal Dual Method 484
16.7 The Gradient Projection Method 485
Cauchy Point Computation 486
Subspace Minimization 488
16.8 Perspectives and Software 490
Notes and References 492
Exercises 492
17 Penalty and Augmented Lagrangian Methods 497
17.1 The Quadratic Penalty Method 498
Motivation 498
Algorithmic Framework 501
Convergence of the Quadratic Penalty Method 502
111 Conditioning and Reformulations 505
17.2 Nonsmooth Penalty Functions 507
A Practical lx Penalty Method 511
A General Class of Nonsmooth Penalty Methods 513
17.3 Augmented Lagrangian Method: Equality Constraints 514
Motivation and Algorithmic Framework 514
Properties of the Augmented Lagrangian 517
17.4 Practical Augmented Lagrangian Methods 519
Bound Constrained Formulation 519
Linearly Constrained Formulation 522
Unconstrained Formulation 523
17.5 Perspectives and Software 525
Notes and References 526
Exercises 527
Contents xv
18 Sequential Quadratic Programming 529
18.1 Local SQP Method 530
SQP Framework 531
Inequality Constraints 532
18.2 Preview of Practical SQP Methods 533
IQPandEQP 533
Enforcing Convergence 534
18.3 Algorithmic Development 535
Handling Inconsistent Linearizations 535
Full Quasi Newton Approximations 536
Reduced Hessian Quasi Newton Approximations 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
S£iQP (Sequential lY Quadratic Programming) 549
Sequential Linear Quadratic Programming (SLQP) 551
A Technique for Updating the Penalty Parameter 553
18.6 Nonlinear 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 Algorithm 566
19.3 Algorithmic Development 569
Primal vs. Primal Dual System 570
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 Algorithm for Solving the Barrier Problem 578
Step Computation 580
Lagrange Multipliers Estimates and Step Acceptance 581
xvi Contents
Description of a Trust Region Interior Point Method 582
19.6 The Primal Log Barrier Method 583
19.7 Global Convergence Properties 587
Failure of the Line Search Approach 587
Modified Line Search Methods 589
Global Convergence of the Trust Region Approach 589
19.8 Superlinear Convergence 591
19.9 Perspectives and Software 592
Notes and References 593
Exercises 594
A Background Material 598
A.1 Elements of Linear Algebra 598
Vectors and Matrices 598
Norms 600
Subspaces 602
Eigenvalues, Eigenvectors, and the Singular Value Decomposition .... 603
Determinant and Trace 605
Matrix Factorizations: Cholesky, LU, QR 606
Symmetric 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 the Euclidean Space R 620
Convex Sets in R 621
Continuity and Limits 623
Derivatives 625
Directional Derivatives 628
Mean Value Theorem 629
Implicit Function Theorem 630
Order Notation 631
Root Finding for Scalar Equations 633
B A Regularization Procedure 635
References 637
Index 653
|
| any_adam_object | 1 |
| author | Nocedal, Jorge Wright, Stephen J. 1960- |
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| dewey-full | 519.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.6 |
| dewey-search | 519.6 |
| dewey-sort | 3519.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| doi_str_mv | 10.1007/978-0-387-40065-5 |
| edition | Second edition |
| format | Electronic eBook |
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| id | DE-604.BV022307057 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:52:24Z |
| institution | BVB |
| isbn | 9780387400655 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015516939 |
| oclc_num | 255028705 |
| open_access_boolean | |
| owner | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 DE-19 DE-BY-UBM DE-11 |
| owner_facet | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 DE-19 DE-BY-UBM DE-11 |
| physical | 1 Online-Ressource (xxii, 664 Seiten) Illustrationen, Diagramme |
| psigel | ZDB-2-SMA |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in operations research and financial engineering |
| spellingShingle | Nocedal, Jorge Wright, Stephen J. 1960- Numerical optimization Numerisches Verfahren (DE-588)4128130-5 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4043664-0 |
| title | Numerical optimization |
| title_auth | Numerical optimization |
| title_exact_search | Numerical optimization |
| title_full | Numerical optimization Jorge Nocedal ; Stephen J. Wright |
| title_fullStr | Numerical optimization Jorge Nocedal ; Stephen J. Wright |
| title_full_unstemmed | Numerical optimization Jorge Nocedal ; Stephen J. Wright |
| title_short | Numerical optimization |
| title_sort | numerical optimization |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Numerisches Verfahren Optimierung |
| url | https://doi.org/10.1007/978-0-387-40065-5 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015516939&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT nocedaljorge numericaloptimization AT wrightstephenj numericaloptimization |