Optimization over integers:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Belmont, Mass.
Dynamic Ideas
2005
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 602 S. graph. Darst. |
| ISBN: | 0975914626 9780975914625 |
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| 100 | 1 | |a Bertsimas, Dimitris |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Optimization over integers |c Dimitris Bertsimas ; Robert Weismantel |
| 264 | 1 | |a Belmont, Mass. |b Dynamic Ideas |c 2005 | |
| 300 | |a X, 602 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 7 | |a Discrete programmering |2 gtt | |
| 650 | 7 | |a Optimaliseren |2 gtt | |
| 650 | 4 | |a Algorithms | |
| 650 | 4 | |a Integer programming | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 0 | 7 | |a Ganzzahlige Optimierung |0 (DE-588)4155950-2 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Weismantel, Robert |e Verfasser |4 aut | |
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Datensatz im Suchindex
| _version_ | 1804136176103194624 |
|---|---|
| adam_text | Titel: Optimization over integers
Autor: Bertsimas, Dimitris
Jahr: 2005
Contents
1 Formulations 1
1.1 Modeling techniques...................... 2
1.2 Guidelines for strong formulations .............. 8
1.3 Modeling with exponentially many constraints ....... 13
1.4 Modeling with exponentially many variables......... 23
1.5 Summary............................ 25
1.6 Exercises ............................ 25
1.7 Notes and sources ....................... 33
2 Methods to enhance formulations 35
2.1 Methods to generate valid inequalities............ 36
2.2 Methods to generate facet defining inequalities....... 46
2.3 Valid inequalities in independence systems.......... 52
2.4 On the strength of valid inequalities............. 56
2.5 Nonlinear formulations..................... 61
2.6 Summary............................ 69
2.7 Exercises ............................ 69
2.8 Notes and sources ....................... 76
3 Ideal formulations 79
3.1 Total unimodularity...................... 80
3.2 Dual methods.......................... 87
3.3 Randomized rounding methods................ 100
3.4 The minimal counterexample method............. 109
3.5 The method of lift and project................ 111
3.6 Summary............................ 116
3.7 Exercises ............................ 117
3.8 Notes and sources ....................... 120
4 Duality in integer optimization 123
4.1 Duality of binary optimization ................ 124
4.2 Superadditive duality...................... 134
4.3 Lagrangean duality....................... 141
4.4 Geometry and solution of Lagrangean duals......... 149
viii Contents
4.5 Summary............................ 159
4.6 Exercises............................ 160
4.7 Notes and sources ....................... 164
5 Algorithms for solving relaxations 165
5.1 The key geometric result behind the ellipsoid method ... 166
5.2 The ellipsoid method for the feasibility problem....... 173
5.3 The ellipsoid method for optimization............ 181
5.4 From separation to optimization ............... 183
5.5 Summary............................ 191
5.6 Exercises ............................ 191
5.7 Notes and sources ....................... 196
6 Lattices and their applications 199
6.1 Integer points in lattices.................... 200
6.2 Reduced bases for lattices................... 211
6.3 Simultaneous diophantine approximation........... 228
6.4 The approximate nearest vector problem........... 230
6.5 The maximum volume inscribed ellipsoid .......... 233
6.6 Integer optimization in fixed dimension ........... 235
6.7 Summary............................ 242
6.8 Exercises ............................ 242
6.9 Notes and sources ....................... 245
7 Algebraic geometry and integer optimization 247
7.1 Background from algebraic geometry............. 248
7.2 Applications to binary optimization ............. 259
7.3 Gröbner bases for integer optimization............ 264
7.4 Applications of real algebraic geometry............ 269
7.5 Generating functions for integer points in polyhedra .... 274
7.6 Summary............................ 281
7.7 Exercises ............................ 282
7.8 Notes and sources ....................... 284
8 Geometry of integer optimization 285
8.1 Definitions and examples.................... 286
8.2 Integral generating sets in cones................ 289
8.3 Optimality conditions..................... 295
8.4 Prom cones to polyhedra.................... 298
8.5 Algorithms to compute integral generating sets....... 305
8.6 Total dual integrality...................... 312
8.7 Summary............................ 320
8.8 Exercises ............................ 321
8.9 Notes and sources ....................... 323
Contents ix
9 Cutting plane methods 325
9.1 Cutting planes from integral generating sets......... 326
9.2 The Gomory cutting plane algorithm............. 333
9.3 Cutting planes based on lattices................ 335
9.4 The convex hull of solutions.................. 341
9.5 Summary............................ 349
9.6 Exercises ............................ 349
9.7 Notes and sources ....................... 352
10 The integral basis method 355
10.1 Dynamic reformulation methods ............... 356
10.2 An integer simplex algorithm................. 363
10.3 The integral basis method................... 368
10.4 Algebraic reformulation strategies............... 378
10.5 Combinatorial reformulation strategies............ 383
10.6 Summary............................ 391
10.7 Exercises ............................ 391
10.8 Notes and sources ....................... 393
11 Enumerative and heuristic methods 395
11.1 Branch and bound....................... 396
11.2 Optimization based heuristic methods............ 406
11.3 Dynamic programming..................... 408
11.4 Approximate dynamic programming............. 414
11.5 Local search........................... 416
11.6 Simulated annealing...................... 418
11.7 Summary............................ 420
11.8 Exercises ............................ 421
11.9 Notes and sources ....................... 425
12 Approximation algorithms 427
12.1 Primal-dual methods...................... 429
12.2 Cut covering problems..................... 440
12.3 Randomized rounding of linear relaxations.......... 447
12.4 Randomized rounding of convex relaxations......... 461
12.5 Approximation schemes.................... 469
12.6 Limitations in approximability................ 475
12.7 Summary............................ 476
12.8 Exercises ............................ 478
12.9 Notes and sources ....................... 484
13 Mixed integer optimization 487
13.1 The mixed integer Farkas lemma............... 488
13.2 Mixed integer lattices ..................... 494
13.3 Mixed integer optimality conditions ............. 496
13.4 Reformulations for mixed integer sets............. 500
x Contents
13.5 Cutting planes for mixed integer optimization........ 505
13.6 Summary............................ 520
13.7 Exercises............................ 520
13.8 Notes and Sources....................... 522
14 Robust discrete optimization 525
14.1 Robust mixed integer optimization.............. 526
14.2 Robust binary optimization.................. 532
14.3 Robust network flows...................... 536
14.4 Robust inventory theory.................... 540
14.5 Summary............................. 546
14.6 Exercises ............................ 546
14.7 Notes and sources....................... 549
A Elements of polyhedral theory 551
A.1 Cones.............................. 552
A.2 Dimension of polyhedra.................... 553
A.3 Valid inequalities........................ 555
A.4 Exercises ............................ 558
B Efficient algorithms and complexity theory 559
B.1 Efficient algorithms....................... 560
B.2 Complexity theory....................... 563
References 571
Index 595
|
| adam_txt |
Titel: Optimization over integers
Autor: Bertsimas, Dimitris
Jahr: 2005
Contents
1 Formulations 1
1.1 Modeling techniques. 2
1.2 Guidelines for strong formulations . 8
1.3 Modeling with exponentially many constraints . 13
1.4 Modeling with exponentially many variables. 23
1.5 Summary. 25
1.6 Exercises . 25
1.7 Notes and sources . 33
2 Methods to enhance formulations 35
2.1 Methods to generate valid inequalities. 36
2.2 Methods to generate facet defining inequalities. 46
2.3 Valid inequalities in independence systems. 52
2.4 On the strength of valid inequalities. 56
2.5 Nonlinear formulations. 61
2.6 Summary. 69
2.7 Exercises . 69
2.8 Notes and sources . 76
3 Ideal formulations 79
3.1 Total unimodularity. 80
3.2 Dual methods. 87
3.3 Randomized rounding methods. 100
3.4 The minimal counterexample method. 109
3.5 The method of lift and project. 111
3.6 Summary. 116
3.7 Exercises . 117
3.8 Notes and sources . 120
4 Duality in integer optimization 123
4.1 Duality of binary optimization . 124
4.2 Superadditive duality. 134
4.3 Lagrangean duality. 141
4.4 Geometry and solution of Lagrangean duals. 149
viii Contents
4.5 Summary. 159
4.6 Exercises. 160
4.7 Notes and sources . 164
5 Algorithms for solving relaxations 165
5.1 The key geometric result behind the ellipsoid method . 166
5.2 The ellipsoid method for the feasibility problem. 173
5.3 The ellipsoid method for optimization. 181
5.4 From separation to optimization . 183
5.5 Summary. 191
5.6 Exercises . 191
5.7 Notes and sources . 196
6 Lattices and their applications 199
6.1 Integer points in lattices. 200
6.2 Reduced bases for lattices. 211
6.3 Simultaneous diophantine approximation. 228
6.4 The approximate nearest vector problem. 230
6.5 The maximum volume inscribed ellipsoid . 233
6.6 Integer optimization in fixed dimension . 235
6.7 Summary. 242
6.8 Exercises . 242
6.9 Notes and sources . 245
7 Algebraic geometry and integer optimization 247
7.1 Background from algebraic geometry. 248
7.2 Applications to binary optimization . 259
7.3 Gröbner bases for integer optimization. 264
7.4 Applications of real algebraic geometry. 269
7.5 Generating functions for integer points in polyhedra . 274
7.6 Summary. 281
7.7 Exercises . 282
7.8 Notes and sources . 284
8 Geometry of integer optimization 285
8.1 Definitions and examples. 286
8.2 Integral generating sets in cones. 289
8.3 Optimality conditions. 295
8.4 Prom cones to polyhedra. 298
8.5 Algorithms to compute integral generating sets. 305
8.6 Total dual integrality. 312
8.7 Summary. 320
8.8 Exercises . 321
8.9 Notes and sources . 323
Contents ix
9 Cutting plane methods 325
9.1 Cutting planes from integral generating sets. 326
9.2 The Gomory cutting plane algorithm. 333
9.3 Cutting planes based on lattices. 335
9.4 The convex hull of solutions. 341
9.5 Summary. 349
9.6 Exercises . 349
9.7 Notes and sources . 352
10 The integral basis method 355
10.1 Dynamic reformulation methods . 356
10.2 An integer simplex algorithm. 363
10.3 The integral basis method. 368
10.4 Algebraic reformulation strategies. 378
10.5 Combinatorial reformulation strategies. 383
10.6 Summary. 391
10.7 Exercises . 391
10.8 Notes and sources . 393
11 Enumerative and heuristic methods 395
11.1 Branch and bound. 396
11.2 Optimization based heuristic methods. 406
11.3 Dynamic programming. 408
11.4 Approximate dynamic programming. 414
11.5 Local search. 416
11.6 Simulated annealing. 418
11.7 Summary. 420
11.8 Exercises . 421
11.9 Notes and sources . 425
12 Approximation algorithms 427
12.1 Primal-dual methods. 429
12.2 Cut covering problems. 440
12.3 Randomized rounding of linear relaxations. 447
12.4 Randomized rounding of convex relaxations. 461
12.5 Approximation schemes. 469
12.6 Limitations in approximability. 475
12.7 Summary. 476
12.8 Exercises . 478
12.9 Notes and sources . 484
13 Mixed integer optimization 487
13.1 The mixed integer Farkas lemma. 488
13.2 Mixed integer lattices . 494
13.3 Mixed integer optimality conditions . 496
13.4 Reformulations for mixed integer sets. 500
x Contents
13.5 Cutting planes for mixed integer optimization. 505
13.6 Summary. 520
13.7 Exercises. 520
13.8 Notes and Sources. 522
14 Robust discrete optimization 525
14.1 Robust mixed integer optimization. 526
14.2 Robust binary optimization. 532
14.3 Robust network flows. 536
14.4 Robust inventory theory. 540
14.5 Summary. 546
14.6 Exercises . 546
14.7 Notes and sources. 549
A Elements of polyhedral theory 551
A.1 Cones. 552
A.2 Dimension of polyhedra. 553
A.3 Valid inequalities. 555
A.4 Exercises . 558
B Efficient algorithms and complexity theory 559
B.1 Efficient algorithms. 560
B.2 Complexity theory. 563
References 571
Index 595 |
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| index_date | 2024-07-02T16:24:06Z |
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| spelling | Bertsimas, Dimitris Verfasser aut Optimization over integers Dimitris Bertsimas ; Robert Weismantel Belmont, Mass. Dynamic Ideas 2005 X, 602 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Discrete programmering gtt Optimaliseren gtt Algorithms Integer programming Mathematical optimization Ganzzahlige Optimierung (DE-588)4155950-2 gnd rswk-swf Programmierung (DE-588)4076370-5 gnd rswk-swf Ganzzahlige Optimierung (DE-588)4155950-2 s Programmierung (DE-588)4076370-5 s b DE-604 Weismantel, Robert Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015411371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bertsimas, Dimitris Weismantel, Robert Optimization over integers Discrete programmering gtt Optimaliseren gtt Algorithms Integer programming Mathematical optimization Ganzzahlige Optimierung (DE-588)4155950-2 gnd Programmierung (DE-588)4076370-5 gnd |
| subject_GND | (DE-588)4155950-2 (DE-588)4076370-5 |
| title | Optimization over integers |
| title_auth | Optimization over integers |
| title_exact_search | Optimization over integers |
| title_exact_search_txtP | Optimization over integers |
| title_full | Optimization over integers Dimitris Bertsimas ; Robert Weismantel |
| title_fullStr | Optimization over integers Dimitris Bertsimas ; Robert Weismantel |
| title_full_unstemmed | Optimization over integers Dimitris Bertsimas ; Robert Weismantel |
| title_short | Optimization over integers |
| title_sort | optimization over integers |
| topic | Discrete programmering gtt Optimaliseren gtt Algorithms Integer programming Mathematical optimization Ganzzahlige Optimierung (DE-588)4155950-2 gnd Programmierung (DE-588)4076370-5 gnd |
| topic_facet | Discrete programmering Optimaliseren Algorithms Integer programming Mathematical optimization Ganzzahlige Optimierung Programmierung |
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