Conjugate gradient algorithms in nonconvex optimization:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2009
|
| Series: | Nonconvex optimization and its applications
89 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 463 - 471 |
| Physical Description: | XXVI, 477 S. graph. Darst. 24 cm |
| ISBN: | 9783540856337 9783540856344 |
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MARC
| LEADER | 00000nam a2200000 cb4500 | ||
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| 100 | 1 | |a Pytlak, Radosław |d 1956- |e Verfasser |0 (DE-588)121118118 |4 aut | |
| 245 | 1 | 0 | |a Conjugate gradient algorithms in nonconvex optimization |c Radosław Pytlak |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2009 | |
| 300 | |a XXVI, 477 S. |b graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Nonconvex optimization and its applications |v 89 | |
| 500 | |a Literaturverz. S. 463 - 471 | ||
| 650 | 4 | |a Algorithms | |
| 650 | 4 | |a Conjugate gradient methods | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 0 | 7 | |a Konjugierte-Gradienten-Methode |0 (DE-588)4255670-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtkonvexe Optimierung |0 (DE-588)4309215-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Nichtkonvexe Optimierung |0 (DE-588)4309215-9 |D s |
| 689 | 0 | 1 | |a Konjugierte-Gradienten-Methode |0 (DE-588)4255670-3 |D s |
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| 830 | 0 | |a Nonconvex optimization and its applications |v 89 |w (DE-604)BV010085908 |9 89 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-017168302 | ||
Record in the Search Index
| _version_ | 1804138689891139584 |
|---|---|
| adam_text | Contents
1
Conjugate
Direction Methods for Quadratic Problems
............. 1
1.1
Introduction
............................................... 1
1.2
Conjugate Direction Methods
................................ 4
1.3
Conjugate Gradient Error Algorithms
.......................... 13
1.4
Conjugate Gradient Residual Algorithms
....................... 21
1.5
The Method of Shortest Residuals
............................. 26
1.6
Rate of Convergence
........................................ 28
1.7
Relations with a Direct Solver for a Set of Linear Equations
....... 38
1.8
Limited Memory
Quasi-Newton
Algorithms
.................... 45
1.9
Reduced-Hessian
Quasi-Newton
Algorithms
.................... 53
1.10
Limited Memory Reduced-Hessian
Quasi-Newton
Algorithms
..... 56
1.11
Conjugate Non-Gradient Algorithm
........................... 59
1.12
Notes
..................................................... 61
2
Conjugate Gradient Methods for Nonconvex Problems
............. 63
2.1
Introduction
............................................... 63
2.2
Line Search Methods
....................................... 65
2.3
General Convergence Results
................................ 70
2.4
Moré-Thuente
Step-Length Selection Algorithm
................ 76
2.5 Hager—
Zhang Step-Length Selection Algorithm
................. 80
2.6
Nonlinear Conjugate Gradient Algorithms
...................... 84
2.7
Global Convergence of the Fletcher-Reeves Algorithm
........... 87
2.8
Other Versions of the Fletcher—Reeves Algorithm
................ 96
2.9
Global Convergence of the
Polak-Rîbière
Algorithm
............. 98
2.10
Hestenes-Stiefel Versions of the Standard Conjugate Gradient
Algorithm
.................................................101
2.11
Notes
.....................................................106
3
Memoryless
Quasi-Newton
Methods
.............................109
3.1
Introduction
...............................................109
Contents
3.2
Conjugate
Gradient
Algorithm as the Memory less
Quasi-Newton
Method
...................................................112
3.3
Beale s Restart Rule
........................................116
3.4
Convergence of the
Shanno
Algorithm
.........................118
3.5
Notes
.....................................................127
Preconditioned Conjugate Gradient Algorithms
...................133
4.1
Introduction
...............................................133
4.2
Rates of Convergence
.......................................134
4.3
Conjugate Gradient Algorithm and the Newton Method
...........139
4.4
Generic Preconditioned Conjugate Gradient Algorithm
...........145
4.5
Quasi-Newton-like Variable Storage Conjugate
Gradient Algorithm
.........................................148
4.6
Scaling the Identity
.........................................155
4.7
Notes
.....................................................158
Limited Memory
Quasi-Newton
Algorithms
.......................159
5.1
Introduction
...............................................159
5.2
Global Convergence of the Limited Memory BFGS Algorithm
.....163
5.3
Compact Representation of the BFGS Approximation of the
Inverse Hessian Matrix
......................................169
5.4
The Compact Representation of the BFGS Approximation of the
Hessian Matrix
.............................................175
5.5
Numerical Experiments
.....................................178
5.6
Notes
.....................................................186
The Method of Shortest Residuals and Nondifferentiable
Optimization
..................................................191
6.1
Introduction
...............................................191
6.2
The Method of Conjugate
Subgradient
by
Lemaréchal
and Wolfe
..192
6.3
Conjugate
Subgradient
Algorithm for Problems with Semismooth
Functions
.................................................204
6.4
Subgradient
Algorithms with Finite Storage
....................210
6.5
Notes
.....................................................215
The Method of Shortest Residuals for Differentiable Problems
......217
7.1
Introduction
...............................................217
7.2
General Algorithm
.........................................219
7.3
Versions of the Method of Shortest Residuals
................... 226
7.4
Polak-Ribière
Version of the Method of Shortest Residuals
.......230
7.5
The Method of Shortest Residuals by Dai and Yuan
..............233
7.6
Global Convergence of the
Lemaréchal-Wolfe
Algorithm Without
Restarts
...................................................241
7.7
A Counter-Example
........................................244
7.8
Numerical Experiments: First Comparisons
.....................250
Contents xv
7.9
Numerical
Experiments:
Comparison to the Memoryless
Quasi-Newton
Method
......................................256
7.10
Numerical Experiments: Comparison to the Limited Memory
Quasi-Newton
Method
......................................257
7.11
Numerical Experiments: Comparison to the Hager-Zhang
Algorithm
.................................................264
7.12
Notes
.....................................................267
8
The Preconditioned Shortest Residuals Algorithm
.................279
8.1
Introduction
...............................................279
8.2
General Preconditioned Method of Shortest Residuals
............281
8.3
Globally Convergent Preconditioned Conjugate Gradient
Algorithm
.................................................286
8.4
Scaling Matrices
...........................................288
8.5
Conjugate Gradient Algorithm with the BFGS Scaling Matrices
... 291
8.6
Numerical Experiments
.....................................293
8.7
Notes
.....................................................297
9
Optimization on a Polyhedron
...................................299
9.1
Introduction
...............................................299
9.2
Exposed Sets
..............................................308
9.3
The Identification of Active Constraints
........................310
9.4
Gradient Projection Method
..................................313
9.5
On the Stopping Criterion
...................................319
9.6
Notes
.....................................................324
10
Conjugate Gradient Algorithms for Problems with Box Constraints
. 327
10.1
Introduction
...............................................327
10.2
General Convergence Theory
.................................328
10.3
The Globally Convergent
Polak-Ribière
Version
of Algorithm
10.1..........................................342
10.4
Convergence Analysis of the Fletcher-Reeves Version
of Algorithm
10.1..........................................350
10.5
Numerical Experiments
.....................................357
10.6
Notes
.....................................................359
11
Preconditioned Conjugate Gradient Algorithms for Problems
with Box Constraints
...........................................371
11.1
Introduction
...............................................371
11.2
Active Constraints Identification by Solving Quadratic Problem
___371
11.3
The Preconditioned Shortest Residuals Algorithm: Outline
........376
11.4
The Preconditioned Shortest Residuals Algorithm:
Global Convergence
........................................384
11.5
The Limited Memory
Quasi-Newton
Method for Problems
with Box Constraints
.......................................386
11.6
Numerical Algebra
.........................................387
xvi Contents
11.7
Algorithm
11.1
and Algorithm
11.2
Applied to Problems
with Strongly Convex Functions
..............................389
11.8
Numerical Experiments
.....................................390
11.9
Notes
.....................................................391
12
Preconditioned Conjugate Gradient Based Reduced-Hessian
Methods
......................................................399
12.1
Introduction
...............................................399
12.2
BFGS Update with Column Scaling
...........................400
12.3
The Preconditioned Method of Shortest Residuals
with Column Scaling
.......................................409
12.4
Reduced-Hessian
Quasi-Newton
Algorithms
....................412
12.5
Limited Memory Reduced-Hessian
Quasi-Newton
Algorithms
.....420
12.6
Preconditioned Conjugate Gradient Based Reduced-Hessian
Algorithm
.................................................424
12.7
Notes
.....................................................428
A Elements of Topology and Analysis
...............................429
A.I The Topology of the Euclidean Space
..........................429
A.2 Continuity and Convexity
....................................432
A.3 Derivatives
................................................435
В
Elements of Linear Algebra
.....................................441
B.I Vector and Matrix Norms
....................................441
B.2 Spectral Decomposition
.....................................442
B.3 Determinant and Trace
......................................447
С
Elements of Numerical Linear Algebra
...........................449
C.I Condition Number and Linear Equations
.......................449
C.2 The
LU
and Cholesky Factorizations
..........................450
C.3 The QR Factorization
.......................................452
C.4 Householder QR
...........................................454
C.5
Givens QR
................................................456
C.6 Gram-Schmidt QR
.........................................459
References
.........................................................463
Index
.............................................................473
|
| any_adam_object | 1 |
| author | Pytlak, Radosław 1956- |
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| author_facet | Pytlak, Radosław 1956- |
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| classification_rvk | SK 870 |
| ctrlnum | (OCoLC)297796432 (DE-599)DNB990020193 |
| dewey-full | 518.1 519.76 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis 519 - Probabilities and applied mathematics |
| dewey-raw | 518.1 519.76 |
| dewey-search | 518.1 519.76 |
| dewey-sort | 3518.1 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035364310 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:32:11Z |
| institution | BVB |
| isbn | 9783540856337 9783540856344 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017168302 |
| oclc_num | 297796432 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-634 DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-634 DE-11 |
| physical | XXVI, 477 S. graph. Darst. 24 cm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Nonconvex optimization and its applications |
| series2 | Nonconvex optimization and its applications |
| spelling | Pytlak, Radosław 1956- Verfasser (DE-588)121118118 aut Conjugate gradient algorithms in nonconvex optimization Radosław Pytlak Berlin [u.a.] Springer 2009 XXVI, 477 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Nonconvex optimization and its applications 89 Literaturverz. S. 463 - 471 Algorithms Conjugate gradient methods Mathematical optimization Konjugierte-Gradienten-Methode (DE-588)4255670-3 gnd rswk-swf Nichtkonvexe Optimierung (DE-588)4309215-9 gnd rswk-swf Nichtkonvexe Optimierung (DE-588)4309215-9 s Konjugierte-Gradienten-Methode (DE-588)4255670-3 s DE-604 Nonconvex optimization and its applications 89 (DE-604)BV010085908 89 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017168302&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pytlak, Radosław 1956- Conjugate gradient algorithms in nonconvex optimization Nonconvex optimization and its applications Algorithms Conjugate gradient methods Mathematical optimization Konjugierte-Gradienten-Methode (DE-588)4255670-3 gnd Nichtkonvexe Optimierung (DE-588)4309215-9 gnd |
| subject_GND | (DE-588)4255670-3 (DE-588)4309215-9 |
| title | Conjugate gradient algorithms in nonconvex optimization |
| title_auth | Conjugate gradient algorithms in nonconvex optimization |
| title_exact_search | Conjugate gradient algorithms in nonconvex optimization |
| title_full | Conjugate gradient algorithms in nonconvex optimization Radosław Pytlak |
| title_fullStr | Conjugate gradient algorithms in nonconvex optimization Radosław Pytlak |
| title_full_unstemmed | Conjugate gradient algorithms in nonconvex optimization Radosław Pytlak |
| title_short | Conjugate gradient algorithms in nonconvex optimization |
| title_sort | conjugate gradient algorithms in nonconvex optimization |
| topic | Algorithms Conjugate gradient methods Mathematical optimization Konjugierte-Gradienten-Methode (DE-588)4255670-3 gnd Nichtkonvexe Optimierung (DE-588)4309215-9 gnd |
| topic_facet | Algorithms Conjugate gradient methods Mathematical optimization Konjugierte-Gradienten-Methode Nichtkonvexe Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017168302&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010085908 |
| work_keys_str_mv | AT pytlakradosław conjugategradientalgorithmsinnonconvexoptimization |