Linear programming: foundations and extensions
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Kluwer Acad. Publ.
1998
|
| Ausgabe: | 3. print. |
| Schriftenreihe: | International series in operations research & management science
4 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 418 S. graph. Darst. |
| ISBN: | 0792398041 |
Internformat
MARC
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| 100 | 1 | |a Vanderbei, Robert J. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Linear programming |b foundations and extensions |c Robert J. Vanderbei |
| 250 | |a 3. print. | ||
| 264 | 1 | |a Boston [u.a.] |b Kluwer Acad. Publ. |c 1998 | |
| 300 | |a XVIII, 418 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a International series in operations research & management science |v 4 | |
| 650 | 4 | |a Linear programming | |
| 650 | 4 | |a Mathematical optimization | |
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Datensatz im Suchindex
| _version_ | 1804126535722991616 |
|---|---|
| adam_text | Contents
Preface xv
Part 1. Basic Theory—The Simplex Method and Duality 1
Chapter 1. Introduction 3
1. Managing a Production Facility 3
1.1. Production Manager as Optimist 3
1.2. Comptroller as Pessimist 4
2. The Linear Programming Problem 6
Exercises 8
Notes 10
Chapter 2. The Simplex Method 11
1. An Example 11
1.1. Dictionaries, Bases, Etc. 14
2. The Simplex Method 14
3. Initialization 17
4. Unboundedness 19
5. Geometry 20
Exercises 21
Notes 24
Chapter 3. Degeneracy 25
1. Definition of Degeneracy 25
2. Two Examples of Degenerate Problems 26
3. The Perturbation/Lexicographic Method 29
4. Bland sRule 32
5. Fundamental Theorem of Linear Programming 34
6. Geometry 35
Exercises 39
Notes 40
vii
viii Contents
Chapter 4. Efficiency of the Simplex Method 41
1. Performance Measures 41
2. Measuring the Size of a Problem 42
3. Measuring the Effort to Solve a Problem 42
4. Worst Case Analysis of the Simplex Method 43
Exercises 48
Notes 49
Chapter 5. Duality Theory 51
1. Motivation—Finding Upper Bounds 51
2. The Dual Problem 53
3. The Weak Duality Theorem 54
4. The Strong Duality Theorem 55
5. Complementary Slackness 62
6. The Dual Simplex Method 64
7. A Dual Based Phase I Algorithm 66
8. The Dual of a Problem in General Form 68
9. Resource Allocation Problems 70
10. Lagrangian Duality 74
Exercises 75 .
Notes 80
Chapter 6. The Simplex Method in Matrix Notation 81
1. Matrix Notation 81
2. The Primal Simplex Method 83
3. An Example 88
3.1. First Iteration 89
3.2. Second Iteration 90
3.3. Third Iteration 92
3.4. Fourth Iteration 93 :
4. The Dual Simplex Method 94
5. Two Phase Methods 95
Exercises 97
Notes 98
Chapter 7. Sensitivity and Parametric Analyses 101
1. Sensitivity Analysis 101
1.1. Ranging 102
2. Parametric Analysis and the Homotopy Method 105
3. The Primal Dual Simplex Method 109
Exercises 111
Notes 113
Contents ix
Chapter 8. Implementation Issues 115
1. Solving Systems of Equations: LU Factorization 116
2. Exploiting Sparsity 120
3. Reusing a Factorization 126
4. Performance Tradeoffs 130
5. Updating a Factorization 131
6. Shrinking the Bump 135
7. Partial Pricing 137
8. Steepest Edge 138
Exercises 139
Notes 140
Chapter 9. Problems in General Form 143
1. The Primal Simplex Method 143
2. The Dual Simplex Method 145
Exercises 151
Notes 152
Chapter 10. Convex Analysis 153
1. Convex Sets 153
2. Caratheodory s Theorem 155
3. The Separation Theorem 156
4. Farkas Lemma 158
5. Strict Complementarity 159
Exercises 162
Notes 162
Chapter 11. Game Theory 163
1. Matrix Games 163
2. Optimal Strategies 165
3. The Minimax Theorem 167
4. Poker 171
Exercises 174
Notes 176
Chapter 12. Regression 177
1. Measures of Mediocrity 177
2. Multidimensional Measures: Regression Analysis 179
3. L2 Regression 180
4. L1 Regression 183
5. Iteratively Reweighted Least Squares 184
6. An Example: How Fast is the Simplex Method? 185
7. Which Variant of the Simplex Method is Best? 189
x Contents
Exercises 190
Notes 195 j
Part 2. Network Type Problems 197
Chapter 13. Network Flow Problems 199
1. Networks 199
2. Spanning Trees and Bases 203
3. The Primal Dual Network Simplex Method 207
4. The Integrality Theorem 218
4.1. Konig s Theorem 218
Exercises 219
Notes 222
Chapter 14. Applications 223
1. The Transportation Problem 223
2. The Assignment Problem 225
3. The Shortest Path Problem 226
3.1. Network Flow Formulation 227
3.2. A Label Correcting Algorithm 227
3.2.1. Method of Successive Approximation 228 ;
3.2.2. Efficiency 228
3.3. A Label Setting Algorithm 228
4. Upper Bounded Network Flow Problems 230
5. The Maximum Flow Problem 232
Exercises 234
Notes 235
Chapter 15. Structural Optimization 237
1. An Example 237
2. Incidence Matrices 239
3. Stability 240
4. Conservation Laws 242
5. Minimum Weight Structural Design 245
6. Anchors Away 247
Exercises 250
Notes 250
Part 3. Interior Point Methods 253
Chapter 16. The Central Path 255
Warning: Nonstandard Notation Ahead 255
1. The Barrier Problem 256
Contents xi
2. Lagrange Multipliers 257
3. Lagrange Multipliers Applied to the Barrier Problem 261
4. Second Order Information 263
5. Existence 263
Exercises 266
Notes 267
Chapter 17. A Path Following Method 269
1. Computing Step Directions 269
2. Newton s Method 271
3. Estimating an Appropriate Value for the Barrier Parameter 272
4. Choosing the Step Length Parameter 273
5. Convergence Analysis 273
5.1. Measures of Progress 275
5.2. Progress in One Iteration 275
5.3. Stopping Rule 278
5.4. Progress Over Several Iterations 278
Exercises 280
Notes 283
Chapter 18. The KKT System 285
1. The Reduced KKT System 285
2. The Normal Equations 286
3. Step Direction Decomposition 288
Exercises 291
Notes 291
Chapter 19. Implementation Issues 293
1. Factoring Positive Definite Matrices 293
1.1. Stability 296
2. Quasidefinite Matrices 297
2.1. Instability 300
3. Problems in General Form 303
Exercises 308
Notes 310
Chapter 20. The Affine Scaling Method 311
1. The Steepest Ascent Direction 311
2. The Projected Gradient Direction 313
3. The Projected Gradient Direction with Scaling 315
4. Convergence 319
5. Feasibility Direction 321
6. Problems in Standard Form 322
xii Contents
Exercises 323
Notes 324
Chapter 21. The Homogeneous Self Dual Method 325
1. From Standard Form to Self Dual Form 325
2. Homogeneous Self Dual Problems 326
2.1. Step Directions 328
2.2. Predictor Corrector Algorithm 330
2.3. Convergence Analysis 333
2.4. Complexity of the Predictor Corrector Algorithm 335
2.5. The KKT System 336
3. Back to Standard Form 337
3.1. The Reduced KKT System 337
4. Simplex Method vs Interior Point Methods 339
Exercises 343
Notes 345
Part 4. Extensions 347
Chapter 22. Integer Programming 349
1. Scheduling Problems 349
2. The Traveling Salesman Problem 351
3. Fixed Costs 354
4. Nonlinear Objective Functions 354
5. Branch and Bound 356
Exercises 368
Notes 369
Chapter 23. Quadratic Programming 371
1. The Markowitz Model 371
2. The Dual 375
3. Convexity and Complexity 378
4. Solution Via Interior Point Methods 380
5. Practical Considerations 382
Exercises 385
Notes 387
Chapter 24. Convex Programming 389
1. Differentiable Functions and Taylor Approximations 389
2. Convex and Concave Functions 390
3. Problem Formulation 390
4. Solution Via Interior Point Methods 391
5. Successive Quadratic Approximations 393
Contents xiii
Exercises 393
Notes 395
Appendix A. Source Listings 397
1. The Primal Dual Simplex Method 398
2. The Homogeneous Self Dual Simplex Method 401
Answers to Selected Exercises 405
Bibliography 407
Index 412
|
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| genre | 1\p (DE-588)4151278-9 Einführung gnd-content |
| genre_facet | Einführung |
| id | DE-604.BV011948234 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:19:00Z |
| institution | BVB |
| isbn | 0792398041 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008077910 |
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| physical | XVIII, 418 S. graph. Darst. |
| publishDate | 1998 |
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| series | International series in operations research & management science |
| series2 | International series in operations research & management science |
| spelling | Vanderbei, Robert J. Verfasser aut Linear programming foundations and extensions Robert J. Vanderbei 3. print. Boston [u.a.] Kluwer Acad. Publ. 1998 XVIII, 418 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series in operations research & management science 4 Linear programming Mathematical optimization Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Innere-Punkte-Methode (DE-588)4352322-5 gnd rswk-swf Netzwerkfluss (DE-588)4126130-6 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Lineare Optimierung (DE-588)4035816-1 s Innere-Punkte-Methode (DE-588)4352322-5 s 2\p DE-604 Netzwerkfluss (DE-588)4126130-6 s 3\p DE-604 International series in operations research & management science 4 (DE-604)BV011630976 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008077910&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Vanderbei, Robert J. Linear programming foundations and extensions International series in operations research & management science Linear programming Mathematical optimization Lineare Optimierung (DE-588)4035816-1 gnd Innere-Punkte-Methode (DE-588)4352322-5 gnd Netzwerkfluss (DE-588)4126130-6 gnd |
| subject_GND | (DE-588)4035816-1 (DE-588)4352322-5 (DE-588)4126130-6 (DE-588)4151278-9 |
| title | Linear programming foundations and extensions |
| title_auth | Linear programming foundations and extensions |
| title_exact_search | Linear programming foundations and extensions |
| title_full | Linear programming foundations and extensions Robert J. Vanderbei |
| title_fullStr | Linear programming foundations and extensions Robert J. Vanderbei |
| title_full_unstemmed | Linear programming foundations and extensions Robert J. Vanderbei |
| title_short | Linear programming |
| title_sort | linear programming foundations and extensions |
| title_sub | foundations and extensions |
| topic | Linear programming Mathematical optimization Lineare Optimierung (DE-588)4035816-1 gnd Innere-Punkte-Methode (DE-588)4352322-5 gnd Netzwerkfluss (DE-588)4126130-6 gnd |
| topic_facet | Linear programming Mathematical optimization Lineare Optimierung Innere-Punkte-Methode Netzwerkfluss Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008077910&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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