Optimization theory and methods: nonlinear programming
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| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
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New York, NY
Springer
2006
|
| Schriftenreihe: | Springer optimization and its applications
1 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XII, 687 S. graph. Darst. |
| ISBN: | 0387249753 9780387249759 |
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MARC
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| 084 | |a MAT 916f |2 stub | ||
| 100 | 1 | |a Sun, Wenyu |e Verfasser |0 (DE-588)135560020 |4 aut | |
| 245 | 1 | 0 | |a Optimization theory and methods |b nonlinear programming |c Wenyu Sun ; Ya-Xiang Yuan |
| 264 | 1 | |a New York, NY |b Springer |c 2006 | |
| 300 | |a XII, 687 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Springer optimization and its applications |v 1 | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 4 | |a Nonlinear programming | |
| 650 | 0 | 7 | |a Lineare Optimierung |0 (DE-588)4035816-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtlineare Optimierung |0 (DE-588)4128192-5 |2 gnd |9 rswk-swf |
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| 689 | 0 | 0 | |a Nichtlineare Optimierung |0 (DE-588)4128192-5 |D s |
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| 689 | 1 | 0 | |a Lineare Optimierung |0 (DE-588)4035816-1 |D s |
| 689 | 1 | |8 1\p |5 DE-604 | |
| 700 | 1 | |a Yuan, Ya-xiang |e Verfasser |0 (DE-588)1043054243 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 0-387-24976-1 |
| 830 | 0 | |a Springer optimization and its applications |v 1 |w (DE-604)BV021746093 |9 1 | |
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Datensatz im Suchindex
| _version_ | 1804135580080013312 |
|---|---|
| adam_text | Contents
Preface
χι
1
Introduction
1
1.1
Introduction
............................ 1
1.2
Mathematics Foundations
.................... 2
1.2.1
Norm
...........................
З
1.2.2
Inverse and Generalized Inverse of a Matrix
...... 9
1.2.3
Properties of Eigenvalues
................ 12
1.2.4
Rank-One Update
.................... 17
1.2.5
Function and Differential
................ 22
1.3
Convex Sets and Convex Functions
............... 31
1.3.1
Convex Sets
........................ 32
1.3.2
Convex Functions
.................... 36
1.3.3
Separation and Support of Convex Sets
........ 50
1.4
Optimality Conditions for Unconstrained Case
........ 57
1.5
Structure of Optimization Methods
............... 63
Exercises
................................ 68
2
Line Search
71
2.1
Introduction
............................ 71
2.2
Convergence Theory for Exact Line Search
.......... 74
2.3
Section Methods
......................... 84
2.3.1
The Golden Section Method
............... 84
2.3.2
The Fibonacci Method
.................. 87
2.4
Interpolation Method
....................... 89
2.4.1
Quadratic Interpolation Methods
............ 89
2.4.2
Cubic Interpolation Method
............... 98
2.5
Inexact Line Search Techniques
................. 102
vi
CONTENTS
2.5.1
Armijo
and Goldstein Rule
............... 103
2.5.2
Wolfe-Powell Rule
.................... 104
2.5.3
Goldstein Algorithm and Wolfe-Powell Algorithm
. . 106
2.5.4
Backtracking Line Search
................ 108
2.5.5
Convergence Theorems of Inexact Line Search
.... 109
Exercises
................................ 116
3
Newton s Methods
119
3.1
The Steepest Descent Method
.................. 119
3.1.1
The Steepest Descent Method
.............. 119
3.1.2
Convergence of the Steepest Descent Method
..... 120
3.1.3
Barzilai and Borwein Gradient Method
........ 126
3.1.4
Appendix: Kantorovich Inequality
........... 129
3.2
Newton s Method
......................... 130
3.3
Modified Newton s Method
................... 135
3.4
Finite-Difference Newton s Method
............... 140
3.5
Negative Curvature Direction Method
............. 147
3.5.1
Gill-Murray Stable Newton s Method
......... 148
3.5.2
Fiacco-McCormick Method
............... 151
3.5.3
Fletcher-Freeman Method
................ 152
3.5.4
Second-Order Step Rules
................ 155
3.6
Inexact Newton s Method
.................... 163
Exercises
................................ 172
4
Conjugate Gradient Method
175
4.1
Conjugate Direction Methods
.................. 175
4.2
Conjugate Gradient Method
................... 178
4.2.1
Conjugate Gradient Method
............... 178
4.2.2
Beale s Three-Term Conjugate Gradient Method
... 185
4.2.3
Preconditioned Conjugate Gradient Method
...... 188
4.3
Convergence of Conjugate Gradient Methods
......... 191
4.3.1
Global Convergence of Conjugate Gradient Methods
. 191
4.3.2
Convergence Rate of Conjugate Gradient Methods
. . 198
Exercises
................................ 200
5
Quasi-Newton
Methods
203
5.1
Quasi-Newton
Methods
..................... 203
5.1.1
Quasi-Newton
Equation
................. 204
CONTENTS
vii
5.1.2 Symmetrie
Rank-One
(SRI) Update.......... 207
5.1.3 DFP Update....................... 210
5.1.4 BFGS Update and PSB Update ............ 217
5.1.5
The Least Change Secant
Update............ 223
5.2
The Broyden Class
........................ 225
5.3 Global
Convergence of
Quasi-Newton
Methods
........ 231
5.3.1
Global Convergence under Exact Line Search
..... 232
5.3.2
Global Convergence under Inexact Line Search
.... 238
5.4
Local Convergence of
Quasi-Newton
Methods
......... 240
5.4.1 Superlinear
Convergence of General
Quasi-Newton
Meth¬
ods
............................. 241
5.4.2
Linear Convergence of General
Quasi-Newton
Methods
250
5.4.3
Local Convergence of Broyden s Rank-One Update
. . 255
5.4.4
Local and Linear Convergence of DFP Method
.... 258
5.4.5 Superlinear
Convergence of BFGS Method
....... 261
5.4.6
Superlinear Convergence of DFP Method
....... 265
5.4.7
Local Convergence of Broyden s Class Methods
.... 271
5.5
Self-Scaling Variable Metric (SSVM) Methods
......... 273
5.5.1
Motivation to SSVM Method
.............. 273
5.5.2
Self-Scaling Variable Metric (SSVM) Method
..... 277
5.5.3
Choices of the Scaling Factor
.............. 279
5.6
Sparse
Quasi-Newton
Methods
................. 282
5.7
Limited Memory BFGS Method
................. 292
Exercises
................................ 301
6
Trust-Region and Conic Model Methods
303
6.1
Trust-Region Methods
...................... 303
6.1.1
Trust-Region Methods
.................. 303
6.1.2
Convergence of Trust-Region Methods
......... 308
6.1.3
Solving A Trust-Region Subproblem
.......... 316
6.2
Conic Model and
Collinear
Scaling Algorithm
......... 324
6.2.1
Conic Model
....................... 324
6.2.2
Generalized
Quasi-Newton
Equation
.......... 326
6.2.3
Updates that Preserve Past Information
........ 330
6.2.4
Collinear
Scaling BFGS Algorithm
........... 334
6.3
Tensor Methods
.......................... 337
6.3.1
Tensor Method for Nonlinear Equations
........ 337
6.3.2
Tensor Methods for Unconstrained Optimization
... 341
viii CONTENTS
Exercises
................................ 349
7
Nonlinear Least-Squares Problems
353
7.1
Introduction
............................ 353
7.2
Gauss-Newton Method
...................... 355
7.3
Levenberg-Marquardt Method
.................. 362
7.3.1
Motivation and Properties
................ 362
7.3.2
Convergence of Levenberg-Marquardt Method
..... 367
7.4
Implementation of L-M Method
................. 372
7.5
Quasi-Newton
Method
...................... 379
Exercises
................................ 382
8
Theory of Constrained Optimization
385
8.1
Constrained Optimization Problems
.............. 385
8.2
First-Order Optimality Conditions
............... 388
8.3
Second-Order Optimality Conditions
.............. 401
8.4
Duality
.............................. 406
Exercises
................................ 409
9
Quadratic Programming
411
9.1
Optimality for Quadratic Programming
............ 411
9.2
Duality for Quadratic Programming
.............. 413
9.3
Equality-Constrained Quadratic Programming
........ 419
9.4
Active Set Methods
....................... 427
9.5
Dual Method
........................... 435
9.6
Interior Ellipsoid Method
.................... 441
9.7
Primal-Dual Interior-Point Methods
.............. 445
Exercises
................................ 451
10
Penalty Function Methods
455
10.1
Penalty Function
......................... 455
10.2
The Simple Penalty Function Method
............. 461
10.3
Interior Point Penalty Functions
................ 466
10.4
Augmented Lagrangian Method
................. 474
10.5
Smooth Exact Penalty Functions
................ 480
10.6
Nonsmooth Exact Penalty Functions
.............. 482
Exercises
................................ 490
CONTENTS ix
11
Feasible Direction Methods
493
11.1
Feasible Point Methods
..................... 493
11.2
Generalized Elimination
..................... 502
11.3
Generalized Reduced Gradient Method
............. 509
11.4
Projected Gradient Method
................... 512
11.5
Linearly Constrained Problems
................. 515
Exercises
................................ 520
12
Sequential Quadratic Programming
523
12.1
Lagrange-Newton Method
.................... 523
12.2
Wilson-Han-Powell Method
................... 530
12.3 Superlinear
Convergence of SQP Step
............. 537
12.4
Maratos Effect
.......................... 541
12.5
Watchdog Technique
....................... 543
12.6
Second-Order Correction Step
.................. 545
12.7
Smooth Exact Penalty Functions
................ 550
12.8
Reduced Hessian Matrix Method
................ 554
Exercises
................................ 558
13
TR Methods for Constrained Problems
561
13.1
Introduction
............................ 561
13.2
Linear Constraints
........................ 563
13.3
Trust-Region Subproblems
.................... 568
13.4
Null Space Method
........................ 571
13.5
CDT Subproblem
......................... 580
13.6
Powell-Yuan Algorithm
..................... 585
Exercises
................................ 594
14
Nonsmooth Optimization
597
14.1
Generalized Gradients
..............·....... 597
14.2
Nonsmooth Optimization Problem
............... 607
14.3
The
Subgradient
Method
.................... 609
14.4
Cutting Plane Method
...................... 615
14.5
The Bundle Methods
....................... 617
14.6
Composite Nonsmooth Function
................ 620
14.7
Trust Region Method for Composite Problems
........ 623
14.8
Nonsmooth Newton s Method
.................. 628
Exercises
................................ 634
CONTENTS
Appendix: Test
Functions
637
§1. Test
Functions for Unconstrained Optimization Problems
637
§2.
Test Functions for Constrained Optimization Problems
. 638
Bibliography
649
Index
682
|
| adam_txt |
Contents
Preface
χι
1
Introduction
1
1.1
Introduction
. 1
1.2
Mathematics Foundations
. 2
1.2.1
Norm
.
З
1.2.2
Inverse and Generalized Inverse of a Matrix
. 9
1.2.3
Properties of Eigenvalues
. 12
1.2.4
Rank-One Update
. 17
1.2.5
Function and Differential
. 22
1.3
Convex Sets and Convex Functions
. 31
1.3.1
Convex Sets
. 32
1.3.2
Convex Functions
. 36
1.3.3
Separation and Support of Convex Sets
. 50
1.4
Optimality Conditions for Unconstrained Case
. 57
1.5
Structure of Optimization Methods
. 63
Exercises
. 68
2
Line Search
71
2.1
Introduction
. 71
2.2
Convergence Theory for Exact Line Search
. 74
2.3
Section Methods
. 84
2.3.1
The Golden Section Method
. 84
2.3.2
The Fibonacci Method
. 87
2.4
Interpolation Method
. 89
2.4.1
Quadratic Interpolation Methods
. 89
2.4.2
Cubic Interpolation Method
. 98
2.5
Inexact Line Search Techniques
. 102
vi
CONTENTS
2.5.1
Armijo
and Goldstein Rule
. 103
2.5.2
Wolfe-Powell Rule
. 104
2.5.3
Goldstein Algorithm and Wolfe-Powell Algorithm
. . 106
2.5.4
Backtracking Line Search
. 108
2.5.5
Convergence Theorems of Inexact Line Search
. 109
Exercises
. 116
3
Newton's Methods
119
3.1
The Steepest Descent Method
. 119
3.1.1
The Steepest Descent Method
. 119
3.1.2
Convergence of the Steepest Descent Method
. 120
3.1.3
Barzilai and Borwein Gradient Method
. 126
3.1.4
Appendix: Kantorovich Inequality
. 129
3.2
Newton's Method
. 130
3.3
Modified Newton's Method
. 135
3.4
Finite-Difference Newton's Method
. 140
3.5
Negative Curvature Direction Method
. 147
3.5.1
Gill-Murray Stable Newton's Method
. 148
3.5.2
Fiacco-McCormick Method
. 151
3.5.3
Fletcher-Freeman Method
. 152
3.5.4
Second-Order Step Rules
. 155
3.6
Inexact Newton's Method
. 163
Exercises
. 172
4
Conjugate Gradient Method
175
4.1
Conjugate Direction Methods
. 175
4.2
Conjugate Gradient Method
. 178
4.2.1
Conjugate Gradient Method
. 178
4.2.2
Beale's Three-Term Conjugate Gradient Method
. 185
4.2.3
Preconditioned Conjugate Gradient Method
. 188
4.3
Convergence of Conjugate Gradient Methods
. 191
4.3.1
Global Convergence of Conjugate Gradient Methods
. 191
4.3.2
Convergence Rate of Conjugate Gradient Methods
. . 198
Exercises
. 200
5
Quasi-Newton
Methods
203
5.1
Quasi-Newton
Methods
. 203
5.1.1
Quasi-Newton
Equation
. 204
CONTENTS
vii
5.1.2 Symmetrie
Rank-One
(SRI) Update. 207
5.1.3 DFP Update. 210
5.1.4 BFGS Update and PSB Update . 217
5.1.5
The Least Change Secant
Update. 223
5.2
The Broyden Class
. 225
5.3 Global
Convergence of
Quasi-Newton
Methods
. 231
5.3.1
Global Convergence under Exact Line Search
. 232
5.3.2
Global Convergence under Inexact Line Search
. 238
5.4
Local Convergence of
Quasi-Newton
Methods
. 240
5.4.1 Superlinear
Convergence of General
Quasi-Newton
Meth¬
ods
. 241
5.4.2
Linear Convergence of General
Quasi-Newton
Methods
250
5.4.3
Local Convergence of Broyden's Rank-One Update
. . 255
5.4.4
Local and Linear Convergence of DFP Method
. 258
5.4.5 Superlinear
Convergence of BFGS Method
. 261
5.4.6
Superlinear Convergence of DFP Method
. 265
5.4.7
Local Convergence of Broyden's Class Methods
. 271
5.5
Self-Scaling Variable Metric (SSVM) Methods
. 273
5.5.1
Motivation to SSVM Method
. 273
5.5.2
Self-Scaling Variable Metric (SSVM) Method
. 277
5.5.3
Choices of the Scaling Factor
. 279
5.6
Sparse
Quasi-Newton
Methods
. 282
5.7
Limited Memory BFGS Method
. 292
Exercises
. 301
6
Trust-Region and Conic Model Methods
303
6.1
Trust-Region Methods
. 303
6.1.1
Trust-Region Methods
. 303
6.1.2
Convergence of Trust-Region Methods
. 308
6.1.3
Solving A Trust-Region Subproblem
. 316
6.2
Conic Model and
Collinear
Scaling Algorithm
. 324
6.2.1
Conic Model
. 324
6.2.2
Generalized
Quasi-Newton
Equation
. 326
6.2.3
Updates that Preserve Past Information
. 330
6.2.4
Collinear
Scaling BFGS Algorithm
. 334
6.3
Tensor Methods
. 337
6.3.1
Tensor Method for Nonlinear Equations
. 337
6.3.2
Tensor Methods for Unconstrained Optimization
. 341
viii CONTENTS
Exercises
. 349
7
Nonlinear Least-Squares Problems
353
7.1
Introduction
. 353
7.2
Gauss-Newton Method
. 355
7.3
Levenberg-Marquardt Method
. 362
7.3.1
Motivation and Properties
. 362
7.3.2
Convergence of Levenberg-Marquardt Method
. 367
7.4
Implementation of L-M Method
. 372
7.5
Quasi-Newton
Method
. 379
Exercises
. 382
8
Theory of Constrained Optimization
385
8.1
Constrained Optimization Problems
. 385
8.2
First-Order Optimality Conditions
. 388
8.3
Second-Order Optimality Conditions
. 401
8.4
Duality
. 406
Exercises
. 409
9
Quadratic Programming
411
9.1
Optimality for Quadratic Programming
. 411
9.2
Duality for Quadratic Programming
. 413
9.3
Equality-Constrained Quadratic Programming
. 419
9.4
Active Set Methods
. 427
9.5
Dual Method
. 435
9.6
Interior Ellipsoid Method
. 441
9.7
Primal-Dual Interior-Point Methods
. 445
Exercises
. 451
10
Penalty Function Methods
455
10.1
Penalty Function
. 455
10.2
The Simple Penalty Function Method
. 461
10.3
Interior Point Penalty Functions
. 466
10.4
Augmented Lagrangian Method
. 474
10.5
Smooth Exact Penalty Functions
. 480
10.6
Nonsmooth Exact Penalty Functions
. 482
Exercises
. 490
CONTENTS ix
11
Feasible Direction Methods
493
11.1
Feasible Point Methods
. 493
11.2
Generalized Elimination
. 502
11.3
Generalized Reduced Gradient Method
. 509
11.4
Projected Gradient Method
. 512
11.5
Linearly Constrained Problems
. 515
Exercises
. 520
12
Sequential Quadratic Programming
523
12.1
Lagrange-Newton Method
. 523
12.2
Wilson-Han-Powell Method
. 530
12.3 Superlinear
Convergence of SQP Step
. 537
12.4
Maratos Effect
. 541
12.5
Watchdog Technique
. 543
12.6
Second-Order Correction Step
. 545
12.7
Smooth Exact Penalty Functions
. 550
12.8
Reduced Hessian Matrix Method
. 554
Exercises
. 558
13
TR Methods for Constrained Problems
561
13.1
Introduction
. 561
13.2
Linear Constraints
. 563
13.3
Trust-Region Subproblems
. 568
13.4
Null Space Method
. 571
13.5
CDT Subproblem
. 580
13.6
Powell-Yuan Algorithm
. 585
Exercises
. 594
14
Nonsmooth Optimization
597
14.1
Generalized Gradients
.·. 597
14.2
Nonsmooth Optimization Problem
. 607
14.3
The
Subgradient
Method
. 609
14.4
Cutting Plane Method
. 615
14.5
The Bundle Methods
. 617
14.6
Composite Nonsmooth Function
. 620
14.7
Trust Region Method for Composite Problems
. 623
14.8
Nonsmooth Newton's Method
. 628
Exercises
. 634
CONTENTS
Appendix: Test
Functions
637
§1. Test
Functions for Unconstrained Optimization Problems
637
§2.
Test Functions for Constrained Optimization Problems
. 638
Bibliography
649
Index
682 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Sun, Wenyu Yuan, Ya-xiang |
| author_GND | (DE-588)135560020 (DE-588)1043054243 |
| author_facet | Sun, Wenyu Yuan, Ya-xiang |
| author_role | aut aut |
| author_sort | Sun, Wenyu |
| author_variant | w s ws y x y yxy |
| building | Verbundindex |
| bvnumber | BV021732644 |
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| callnumber-subject | T - General Technology |
| classification_rvk | SK 870 |
| classification_tum | MAT 916f |
| ctrlnum | (OCoLC)60311926 (DE-599)BVBBV021732644 |
| dewey-full | 519.7/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.7/6 |
| dewey-search | 519.7/6 |
| dewey-sort | 3519.7 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV021732644 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:26:53Z |
| indexdate | 2024-07-09T20:42:46Z |
| institution | BVB |
| isbn | 0387249753 9780387249759 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014946130 |
| oclc_num | 60311926 |
| open_access_boolean | |
| owner | DE-20 DE-824 DE-703 DE-706 DE-91G DE-BY-TUM DE-29T DE-11 DE-634 DE-19 DE-BY-UBM DE-188 |
| owner_facet | DE-20 DE-824 DE-703 DE-706 DE-91G DE-BY-TUM DE-29T DE-11 DE-634 DE-19 DE-BY-UBM DE-188 |
| physical | XII, 687 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series | Springer optimization and its applications |
| series2 | Springer optimization and its applications |
| spelling | Sun, Wenyu Verfasser (DE-588)135560020 aut Optimization theory and methods nonlinear programming Wenyu Sun ; Ya-Xiang Yuan New York, NY Springer 2006 XII, 687 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer optimization and its applications 1 Mathematical optimization Nonlinear programming Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Nichtlineare Optimierung (DE-588)4128192-5 s DE-604 Lineare Optimierung (DE-588)4035816-1 s 1\p DE-604 Yuan, Ya-xiang Verfasser (DE-588)1043054243 aut Erscheint auch als Online-Ausgabe 0-387-24976-1 Springer optimization and its applications 1 (DE-604)BV021746093 1 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2739404&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014946130&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 | Sun, Wenyu Yuan, Ya-xiang Optimization theory and methods nonlinear programming Springer optimization and its applications Mathematical optimization Nonlinear programming Lineare Optimierung (DE-588)4035816-1 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| subject_GND | (DE-588)4035816-1 (DE-588)4128192-5 (DE-588)4123623-3 |
| title | Optimization theory and methods nonlinear programming |
| title_auth | Optimization theory and methods nonlinear programming |
| title_exact_search | Optimization theory and methods nonlinear programming |
| title_exact_search_txtP | Optimization theory and methods nonlinear programming |
| title_full | Optimization theory and methods nonlinear programming Wenyu Sun ; Ya-Xiang Yuan |
| title_fullStr | Optimization theory and methods nonlinear programming Wenyu Sun ; Ya-Xiang Yuan |
| title_full_unstemmed | Optimization theory and methods nonlinear programming Wenyu Sun ; Ya-Xiang Yuan |
| title_short | Optimization theory and methods |
| title_sort | optimization theory and methods nonlinear programming |
| title_sub | nonlinear programming |
| topic | Mathematical optimization Nonlinear programming Lineare Optimierung (DE-588)4035816-1 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| topic_facet | Mathematical optimization Nonlinear programming Lineare Optimierung Nichtlineare Optimierung Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2739404&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014946130&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV021746093 |
| work_keys_str_mv | AT sunwenyu optimizationtheoryandmethodsnonlinearprogramming AT yuanyaxiang optimizationtheoryandmethodsnonlinearprogramming |