Trust-region methods:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Philadelphia [u.a.]
SIAM u.a.
2000
|
| Schriftenreihe: | MPS, SIAM series on optimization
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIX, 959 S. graph. Darst. |
| ISBN: | 0898714605 |
Internformat
MARC
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| 100 | 1 | |a Conn, Andrew R. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Trust-region methods |c Andrew R. Conn ; Nicholas I. M. Gould ; Philippe L. Toint |
| 264 | 1 | |a Philadelphia [u.a.] |b SIAM u.a. |c 2000 | |
| 300 | |a XIX, 959 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a MPS, SIAM series on optimization | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 0 | 7 | |a Trust-Region-Algorithmus |0 (DE-588)4311403-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lineare Optimierung |0 (DE-588)4035816-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Optimierung |0 (DE-588)4043664-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Trust-Region-Algorithmus |0 (DE-588)4311403-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Lineare Optimierung |0 (DE-588)4035816-1 |D s |
| 689 | 1 | |5 DE-604 | |
| 689 | 2 | 0 | |a Optimierung |0 (DE-588)4043664-0 |D s |
| 689 | 2 | |8 1\p |5 DE-604 | |
| 700 | 1 | |a Gould, Nicholas I. M. |e Verfasser |4 aut | |
| 700 | 1 | |a Toint, Philippe L. |e Verfasser |0 (DE-588)113099193 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009176821&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-009176821 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
Datensatz im Suchindex
| _version_ | 1804128236278382592 |
|---|---|
| adam_text | Contents
Preface
xv
1
Introduction
1
1.1
What Is a Trust-Region Algorithm?
................... 1
1.2
Historical Development of Trust-Region Methods
........... 8
1.3
Some Applications of Trust-Region Algorithms
............. 9
1 Preliminaries
13
2
Basic Concepts
15
2.1
Basic Notation
............................... 15
2.2
Eigenvalues and Eigenvectors
...................... 16
2.3
Vector and Matrix Norms
........................ 20
2.3.1
Vector and Functional Norms
.................. 21
2.3.2
Matrix Norms
........................... 22
3
Basic Analysis and Optimality Conditions
25
3.1
Basic Real Analysis
............................ 25
3.1.1
Basic Topology
.......................... 25
3.1.2
Derivatives and Taylor s Theorem
................ 27
3.1.3
Convexity
............................. 30
3.1.4
Nonsmooth Functions
....................... 33
3.1.5
Geometry
............................. 34
3.2
Optimality Conditions
.......................... 37
3.2.1
Differentiable Unconstrained Problems
............. 38
3.2.2
Differentiable Constrained Problems
.............. 39
3.2.3
Convex Programming
....................... 44
3.2.4
Nonsmooth Problems
....................... 47
3.3
Sequences and Basic Iterations
...................... 50
3.3.1
Sequences
............................. 50
3.3.2
Newton s Method
......................... 51
3.3.3
Forcing Functions
......................... 53
4
Basic
Linear
Algebra
55
4.1
Linear
Equations
in Optimization
.................... 55
4.1.1
Structure and Sparsity
...................... 56
4.1.2
Structured Systems
........................ 57
4.2
Special Matrices
.............................. 57
4.2.1
Diagonal Matrices
......................... 58
4.2.2
Triangular Matrices
........................ 58
4.2.3
Orthonormal
Matrices
...................... 58
4.2.4
Tridiagonal and Band Matrices
................. 59
4.3
Direct Methods for Solving Linear Systems
............... 60
4.3.1
Stability in the Face of Rounding Errors
............ 60
4.3.2
Sparse Systems
.......................... 61
4.3.3
Symmetric Positive Definite Matrices
.............. 63
4.3.4
Symmetric Indefinite Matrices
.................. 65
4.3.5
Frontal and
Multifrontal
Methods
................ 66
4.3.6
Matrix Modification
....................... 67
4.4
Least-Squares Problems and Projections
................ 68
4.4.1
Least-Squares Problems
..................... 68
4.4.2
Projections, Range-, and Null-Space Bases
........... 71
5
Krylov Subspace Methods
75
5.1
The Conjugate Gradient Method
.................... 75
5.1.1
Minimization in a Subspace
................... 76
5.1.2
Conjugate Directions
....................... 77
5.1.3
Generating Conjugate Directions
................ 78
5.1.4
Conjugate Gradients
....................... 82
5.1.5
Convergence of the Conjugate Gradient Method
........ 83
5.1.6
Preconditioned Conjugate Gradients
.............. 86
5.2
The Lanczos Method
........................... 91
5.2.1
Computing an
Orthonormal
Basis for the Krylov Space
.... 92
5.2.2
Relationship with the Conjugate Direction Method
...... 95
5.2.3
Finding Conjugate Directions from an
Orthonormal
Krylov
Basis
................................ 96
5.2.4
Approximation of Critical Points within the Subspace
..... 99
5.2.5
Rayleigh-Ritz Approximations to Eigenpairs
.......... 101
5.2.6
Preconditioned Lanczos
...................... 102
5.3
Linear Least-Squares Problems
..................... 106
5.4
Problems with Constraints
........................ 108
5.4.1
Projected Preconditioned Conjugate Gradients
........ 109
II Trust-Region Methods for Unconstrained Optimization
ИЗ
6
Global Convergence of the Basic Algorithm
115
6.1
The Basic Trust-Region Algorithm
................... 115
6.2
Assumptions
................................ 121
6.2.1
Assumptions on the Problem
.................. 121
6.2.2
Assumptions on the Algorithm
................. 122
6.3
The Cauchy Point and the Model Decrease
............... 123
6.3.1
The Cauchy Arc
.......................... 123
6.3.2
The Cauchy Point for Quadratic Models
............ 124
6.3.3
The Approximate Cauchy Point
................. 128
6.3.4
The Final Condition on the Model Decrease
.......... 130
6.4
Convergence to First-Order Critical Points
............... 133
6.5
Second-Order Convex Models
...................... 139
6.6
The
Eigenpoint
and Second-Order Nonconvex Models
......... 147
6.6.1
Exploitation of Negative Curvature by Minimization
..... 148
6.6.2
Exploitation of Negative Curvature by a Linesearch
...... 150
6.6.3
Convergence Theorems
...................... 153
6.7
Trust-Region Scaling
........................... 162
6.7.1
Geometry and Scaling
...................... 162
6.7.2
Uniformly Equivalent Norms Again
............... 166
7
The Trust-Region Subproblem
169
7.1
The Solution of Trust-Region Subproblems
............... 169
7.2
Characterization of the ^-Norm Model Minimizer
........... 171
7.3
Finding the ¿2-Norm Model Minimizer
................. 176
7.3.1
Finding the ^-Norm Model Minimizer
............. 176
7.3.2
Finding the Root of ||s(A)||2
-
Δ
= 0.............. 181
7.3.3
Newton s Method and the Secular Equation
.......... 182
7.3.4
Safeguarding Newton s Method
................. 185
7.3.5
Updating the Intervals of Uncertainty
.............. 189
7.3.6
Finding
λ
in the Interval of Uncertainty
............ 189
7.3.7
Finding Good Lower Bounds on
-λι
.............. 190
7.3.8
Initial Values
........................... 192
7.3.9
The Complete Algorithm
..................... 193
7.3.10
Termination
............................ 194
7.3.11
Enhancements
........................... 197
7.4
The Scaled ^-Norm Problem
...................... 200
7.5
Approximating the Model Minimizer
.................. 201
7.5.1
The Truncated Conjugate Gradient Method
.......... 202
7.5.2
How Good Is the Steihaug-Toint Point?
............ 208
7.5.3
Dogleg and Double-Dogleg Paths
................ 218
7.5.4
The Truncated Lanczos Approach
................ 221
7.5.5 Computing
the
Eigenpoint.................... 230
7.5.6
Eigenvalue-Based Approaches
.................. 231
7.6
Projection-Based Approaches
...................... 235
7.7
Norms that Reflect the Underlying Geometry
............. 236
7.7.1
The Ideal Trust Region
...................... 236
7.7.2
The Absolute-Value Trust Region
................ 237
7.7.3
Solving Diagonal and Block Diagonal Trust-Region Subprob-
lems
................................ 238
7.7.4
Coping with Singularity
..................... 239
7.8
The
¿oo-Norm
Problem
.......................... 243
8
Further Convergence Theory Issues
249
8.1
Convergence of General Measures
.................... 249
8.1.1
First-Order and Criticality Measures
.............. 249
8.1.2
First-Order Convergence Theory
................ 251
8.1.3
Second-Order Convergence Theory
............... 255
8.1.4
Convergence in Dual Norms
................... 259
8.1.5
A More General Cauchy Point
.................. 265
8.1.6
The Scaled Cauchy Point
..................... 269
8.2
Weakly Equivalent Norms
........................ 272
8.3
Minimization in Infinite-Dimensional Spaces
.............. 274
8.3.1
Hubert Spaces
.......................... . 275
8.3.2
Banach Spaces
........................... 277
8.4
Using Approximate Derivatives
..................... 280
8.4.1
Concepts and Assumptions
.................... 280
8.4.2
Global Convergence
........................ 285
8.4.3
Finite-Difference Approximations to Derivatives
........ 297
8.5
Composite Problems and Models
.................... 302
9
Conditional Models
307
9.1
Motivation and Formal Description
................... 307
9.2
Convergence to First-Order Critical Points
............... 313
9.3
Convergence to Second-Order Critical Points
.............. 319
9.4
Conditional Models and Derivative-Free Optimization
......... 322
9.4.1
Derivative-Free Minimization: Why and How?
......... 322
9.4.2
Basic Concepts in Multivariate Interpolation
.......... 324
9.4.3
The Interpolation Error
..................... 330
9.5
Conditional Models and Models with Memory
............ . 336
9.5.1
Adding Memory to the Model of the Objective Function
. . . 336
9.5.2
The Effect of Memory on the Modelling Error
......... 338
10
Algorithmic Extensions
347
10.1
A Nonmonotone
Trust-Region Algorithm
................ 347
10.1.1
The
Nonmonotone
Algorithm
.................. 348
10.1.2
Convergence Theory Revisited
.................. 350
10.1.3
The Reference Iteration and Other Practicalities
........ 355
10.2
Structured Trust Regions
......................... 359
10.2.1
Motivation and Basic Concepts
................. 359
10.2.2
A Trust-Region Algorithm Using Problem Structure
..... 364
10.2.3
Convergence Theory
....................... 370
10.3
Trust Regions and Linesearches
..................... 376
10.3.1
Linesearch Algorithms as Trust-Region Methods
........ 376
10.3.2
Backtracking at Unsuccessful Iterations
............. 380
10.3.3
Which Steps Are Gradient Related?
............... 383
10.4
Other Measures of Model Accuracy
................... 387
10.4.1
Magical Steps
........................... 387
10.4.2
Correction Steps
......................... 391
10.4.3
Modified Models
......................... 392
10.5
Alternative Trust-Region Management
................. 394
10.5.1
Internal Doubling
......................... 394
10.5.2
Basing the Radius Update on the Steplength
.......... 396
10.6
Problems with Dynamic Accuracy
.................... 399
10.6.1
An Algorithm Using Dynamic Accuracy
............ 400
10.6.2
Conditional Models and Dynamic Accuracy
.......... 406
11
Nonsmooth Problems
407
11.1
Algorithms for Nonsmooth Optimization
................ 408
11.2
Convergence to a First-Order Critical Point
.............. 414
11.3
Variations on This Theme
........................ 423
11.3.1
A Nonmonotonic Variant
..................... 423
11.3.2
Correction Steps
......................... 424
11.4
Computing the Generalized Gradient
.................. 426
11.5
Suitable Models
.............................. 434
III Trust-Region Methods for Constrained Optimization
with Convex Constraints
439
12
Projection Methods for Convex Constraints
441
12.1
Simple Feasible Domains and Their Geometry
............. 441
12.1.1
Simple Bounds on the Variables
................. 442
12.1.2
Other Simple Domains
...................... 443
12.1.3
The Projected-Gradient Path
.................. 444
12.1.4
A New Criticality Measure
.................... 448
12.2
An Algorithm for Problems with Convex Constraints
......... 451
12.2.1
The Generalized Cauchy Point
.................. 453
12.2.2
Convergence to First-Order Critical Points
........... 458
12.3
Active Constraints Identification
..................... 460
12.3.1
Further Assumptions
....................... 460
12.3.2
The Geometry of the Set of Limit Points
............ 461
12.3.3
The Identification of Active Constraints
............ 467
12.4
Convergence to Second-Order Critical Points
.............. 474
12.4.1
The Role of the Lagrangian
................... 474
12.4.2
Convex Models
.......................... 476
12.4.3
Nonconvex Models
........................ 481
13
Barrier Methods for Inequality Constraints
491
13.1
Convex Constraints and Barriers
.................... 491
13.2
A Trust-Region Method for Barrier Functions
............. 498
13.3
Constrained Cauchy and
Eigenpoints.................. 504
13.4
The Primal Log-Barrier Algorithm
................... 511
13.5
Reciprocal Barriers
............................ 517
13.6
A Primal-Dual Algorithm
........................ 518
13.6.1
The Algorithm
.......................... 518
13.6.2
Convergence Properties
...................... 522
13.6.3
Updating the Vector of Dual Variables
............. 525
13.7
Scaling of Barrier Methods
........................ 527
13.7.1
Reintroducing Iteration-Dependent Norms
........... 527
13.7.2
Scaling of the Inner Iterations
.................. 528
13.7.3
Scaling of the Outer Iterations
.................. 530
13.8
Upper and Lower Bounds on the Variables
............... 535
13.9
Barrier Methods for General Constraints
................ 536
13.10
Adding Linear Equality Constraints
................... 542
13.11
The Affme-Scaling Method
........................ 550
13.12
The Method of Coleman and Li
..................... 554
13.12.1
The Algorithm
.......................... 554
13.12.2
Convergence Theory
....................... 559
IV Trust-Region Methods for General Constrained Opti¬
mization and Systems of Nonlinear Equations
571
14
Penalty-Function Methods
573
14.1
Penalty Functions and Constrained Optimization
........... 573
14.2
Smooth Penalty Functions
........................ 575
14.3
Quadratic Penalty-Function Methods
.................. 582
14.4
Augmented Lagrangian Function Methods
............... 593
14.5
Nonsmooth Exact Penalty Functions
.................. 610
14.6
Smooth Exact Penalty Functions
.................... 616
15
Sequential Quadratic Programming Methods
623
15.1
Introduction
................................ 623
15.2
What Is Sequential Quadratic Programming?
............. 624
15.2.1
Methods for Problems with Equality Constraints
....... 624
15.2.2
Methods for Inequality Constraints
............... 631
15.2.3
Quadratic Programming
..................... 633
15.3
Merit Functions and SQP Methods
................... 636
15.3.1
The Augmented Lagrangian Penalty Function
......... 637
15.3.2
Nonsmooth Exact Penalty Functions
.............. 637
15.3.3
Smooth Exact Penalty Functions
................ 655
15.4
Composite-Step Trust-Region SQP Methods
.............. 657
15.4.1
Vardi-Like Approaches
...................... 658
15.4.2
Byrd-Omojokun-like Approaches
................ 694
15.4.3
Celis-Dennis-Tapia-like Approaches
............... 711
15.4.4
Inequality Constraints
...................... 717
15.5
The Filter Method
............................ 721
15.5.1
A Composite-Step Approximate SQP Framework
....... 721
15.5.2
The Notion of a Filter
...................... 725
15.5.3
An SQP Filter Algorithm
.................... 727
15.5.4
Convergence to First-Order Critical Points
........... 730
15.5.5
An Alternative Step Strategy
.................. 742
15.6
Nonquadratic Models
........................... 745
15.7
Concluding Remarks
........................... 746
16
Nonlinear Equations and Nonlinear Fitting
749
16.1
Nonlinear Equations, Nonlinear Least Squares
............. 749
16.2
Nonlinear Equations in Other Norms
.................. 759
16.2.1
Global Convergence
........................ 761
16.2.2
Asymptotic Convergence of the Basic Method
......... 761
16.2.3
Second-Order Corrections
.................... 766
16.3
Complementarity Problems
........................ 770
16.3.1
The Problem and an Associated Merit Function
........ 770
16.3.2
Applying the Basic Nonsmooth Algorithm
........... 772
16.3.3
Regular Complementarity Problems and Their Solutions
. . . 774
V Final Considerations
779
17
Practicalities
781
17.1
Choosing the Algorithmic Parameters
.................. 781
17.2
Choosing the Initial Trust-Region Radius
................ 784
17.3 Computing
the Generalized Cauchy Point
............... 789
17.4
Other Numerical Considerations
..................... 792
17.4.1
Computing the Model Decrease
................. 792
17.4.2
Computing the Value of pk
.................... 793
17.4.3
Stopping Conditions
....................... 794
17
AA Noise and/or Expensive Function Evaluations
......... 798
17.5
Software
.................................. 799
Afterword
801
Appendix: A Summary of Assumptions
803
Annotated Bibliography
813
Subject and Notation Index
935
Author Index
951
|
| any_adam_object | 1 |
| author | Conn, Andrew R. Gould, Nicholas I. M. Toint, Philippe L. |
| author_GND | (DE-588)113099193 |
| author_facet | Conn, Andrew R. Gould, Nicholas I. M. Toint, Philippe L. |
| author_role | aut aut aut |
| author_sort | Conn, Andrew R. |
| author_variant | a r c ar arc n i m g nim nimg p l t pl plt |
| building | Verbundindex |
| bvnumber | BV013444088 |
| callnumber-first | Q - Science |
| callnumber-label | QA402 |
| callnumber-raw | QA402.5 |
| callnumber-search | QA402.5 |
| callnumber-sort | QA 3402.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 870 |
| classification_tum | MAT 913f |
| ctrlnum | (OCoLC)85688875 (DE-599)BVBBV013444088 |
| dewey-full | 519.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.3 |
| dewey-search | 519.3 |
| dewey-sort | 3519.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV013444088 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:46:02Z |
| institution | BVB |
| isbn | 0898714605 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009176821 |
| oclc_num | 85688875 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-20 DE-355 DE-BY-UBR DE-703 DE-29T DE-384 DE-83 DE-11 DE-188 DE-739 |
| owner_facet | DE-91G DE-BY-TUM DE-20 DE-355 DE-BY-UBR DE-703 DE-29T DE-384 DE-83 DE-11 DE-188 DE-739 |
| physical | XIX, 959 S. graph. Darst. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | SIAM u.a. |
| record_format | marc |
| series2 | MPS, SIAM series on optimization |
| spelling | Conn, Andrew R. Verfasser aut Trust-region methods Andrew R. Conn ; Nicholas I. M. Gould ; Philippe L. Toint Philadelphia [u.a.] SIAM u.a. 2000 XIX, 959 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier MPS, SIAM series on optimization Mathematical optimization Trust-Region-Algorithmus (DE-588)4311403-9 gnd rswk-swf Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Trust-Region-Algorithmus (DE-588)4311403-9 s DE-604 Lineare Optimierung (DE-588)4035816-1 s Optimierung (DE-588)4043664-0 s 1\p DE-604 Gould, Nicholas I. M. Verfasser aut Toint, Philippe L. Verfasser (DE-588)113099193 aut Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009176821&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Conn, Andrew R. Gould, Nicholas I. M. Toint, Philippe L. Trust-region methods Mathematical optimization Trust-Region-Algorithmus (DE-588)4311403-9 gnd Lineare Optimierung (DE-588)4035816-1 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4311403-9 (DE-588)4035816-1 (DE-588)4043664-0 |
| title | Trust-region methods |
| title_auth | Trust-region methods |
| title_exact_search | Trust-region methods |
| title_full | Trust-region methods Andrew R. Conn ; Nicholas I. M. Gould ; Philippe L. Toint |
| title_fullStr | Trust-region methods Andrew R. Conn ; Nicholas I. M. Gould ; Philippe L. Toint |
| title_full_unstemmed | Trust-region methods Andrew R. Conn ; Nicholas I. M. Gould ; Philippe L. Toint |
| title_short | Trust-region methods |
| title_sort | trust region methods |
| topic | Mathematical optimization Trust-Region-Algorithmus (DE-588)4311403-9 gnd Lineare Optimierung (DE-588)4035816-1 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Mathematical optimization Trust-Region-Algorithmus Lineare Optimierung Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009176821&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT connandrewr trustregionmethods AT gouldnicholasim trustregionmethods AT tointphilippel trustregionmethods |