Linear programming: 2 Theory and extensions
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2003
|
| Series: | Springer series in operations research
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 379 - 438 |
| Physical Description: | XXV, 448 S. graph. Darst. |
| ISBN: | 0387986138 |
Staff View
MARC
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| 245 | 1 | 0 | |a Linear programming |n 2 |p Theory and extensions |c George B. Dantzig ; Mukund N. Thapa |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2003 | |
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| 650 | 7 | |a dégénérescence |2 inriac | |
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| 650 | 7 | |a problème transport |2 inriac | |
| 650 | 7 | |a programmation linéaire |2 inriac | |
| 650 | 7 | |a programmation stochastique |2 inriac | |
| 650 | 7 | |a théorie probabilité |2 inriac | |
| 650 | 7 | |a variation |2 inriac | |
| 650 | 7 | |a écoulement multidimensionnel |2 inriac | |
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Record in the Search Index
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| adam_text | Contents
LIST OF FIGURES xv
LIST OF TABLES xvii
PREFACE xix
DEFINITION OF SYMBOLS xxiii
1 GEOMETRY OF LINEAR INEQUALITY SYSTEMS AND THE
SIMPLEX METHOD 1
1.1 CONVEXITY AND LINEAR INEQUALITY SYSTEMS 1
1.1.1 Affine Convex Combinations 1
1.1.2 Two dimensional Convex Regions 3
1.1.3 Line Segments, Rays, and Half Lines 5
1.1.4 General Convex Regions 6
1.1.5 Hyperplanes and Half Spaces 7
1.1.6 Convexity of Half Spaces and Hyperplanes 8
1.1.7 Convexity of the Set of Feasible Solutions of an LP 9
1.1.8 Convex Polyhedrons, Polytopes, and Cones 9
1.1.9 Separating Hyperplane 11
1.2 SIMPLEX DEFINED 13
1.3 GLOBAL MINIMUM, EXTREME POINTS, AND EDGES 14
1.4 THE SIMPLEX METHOD VIEWED AS THE STEEPEST DE¬
SCENT ALONG EDGES 20
1.5 THE SIMPLEX INTERPRETATION OF THE SIMPLEX METHOD 24
1.6 NOTES k SELECTED BIBLIOGRAPHY 31
1.7 PROBLEMS 31
2 DUALITY AND THEOREMS OF THE ALTERNATIVES 43
2.1 THE DUALITY THEOREM 43
2.2 ADDITIONAL THEOREMS ON DUALITY 47
2.2.1 Unboundedness Theorem 47
2.2.2 Miscellaneous Theorems for the Standard Form 48
ix
x CONTENTS
2.3 COMPLEMENTARY SLACKNESS 49
2.4 THEOREMS OF THE ALTERNATIVES 50
2.4.1 Gordan s Theorem 51
2.4.2 Farkas s Lemma 52
2.4.3 Stiemke s Theorem 53
2.4.4 Motzkin s Transposition Theorem 54
2.4.5 Ville s Theorem 55
2.4.6 Tucker s Strict Complementary Slackness Theorem 56
2.5 NOTES SELECTED BIBLIOGRAPHY 58
2.6 PROBLEMS 59
3 EARLY INTERIOR POINT METHODS 67
3.1 VON NEUMANN S METHOD 70
3.1.1 The von Neumann Algorithm 73
3.1.2 Improving the Rate of Convergence 81
3.1.3 Von Neumann Algorithm as a Variant of the Simplex Algorithm 83
3.2 DIKIN S METHOD 84
3.2.1 Dikin s Algorithm 87
3.2.2 Convergence of Dikin s Algorithm 89
3.3 KARMARKAR S METHOD 100
3.3.1 Development of the Algorithm 100
3.3.2 Proof of Convergence 105
3.3.3 The Algorithm Summarized 114
3.3.4 Converting a Standard LP to a Starting Form for the Algorithmic
3.3.5 Computational Comments 116
3.3.6 Complexity of von Neumann versus Karmarkar Algorithms . 118
3.4 NOTES SELECTED BIBLIOGRAPHY 119
3.5 PROBLEMS 121
4 INTERIOR POINT METHODS 123
4.1 NEWTON S METHOD 123
4.2 THE LINEAR LEAST SQUARES PROBLEM 127
4.3 BARRIER FUNCTION METHODS 128
4.3.1 The Logarithmic Barrier Function 128
4.3.2 Properties of Barrier Function Methods 130
4.4 THE PRIMAL LOGARITHMIC BARRIER METHOD FOR SOLV¬
ING LINEAR PROGRAMS 131
4.4.1 Details of the Method 131
4.4.2 Initial Feasible Solution 134
4.5 PRIMAL DUAL LOGARITHMIC BARRIER METHODS 134
4.6 RECOVERING A BASIC FEASIBLE SOLUTION 137
4.7 COMPUTATIONAL COMMENTS 139
4.8 NOTES SELECTED BIBLIOGRAPHY 140
4.9 PROBLEMS 146
CONTENTS xi
5 DEGENERACY 149
5.1 EXAMPLES OF CYCLING 149
5.2 ON RESOLVING DEGENERACY 153
5.3 DANTZIG S INDUCTIVE METHOD 154
5.4 WOLFE S RULE 156
5.5 BLAND S RULE 158
5.6 KRISHNA S EXTRA COLUMN RULE 160
5.7 ON AVOIDING DEGENERATE PIVOTS 164
5.8 NOTES SELECTED BIBLIOGRAPHY 166
5.9 PROBLEMS 167
6 VARIANTS OF THE SIMPLEX METHOD 173
6.1 INTRODUCTION 173
6.2 MAX IMPROVEMENT PER ITERATION 176
6.3 DUAL SIMPLEX METHOD 179
6.4 PARAMETRIC LINEAR PROGRAMS 183
6.4.1 Parameterizing the Objective Function 183
6.4.2 Parameterizing the Right Hand Side 187
6.5 SELF DUAL PARAMETRIC ALGORITHM 188
6.6 THE PRIMAL DUAL ALGORITHM 191
6.7 THE PHASE I LEAST SQUARES ALGORITHM 197
6.8 NOTES SELECTED BIBLIOGRAPHY 200
6.9 PROBLEMS 202
7 TRANSPORTATION PROBLEM AND VARIATIONS 207
7.1 THE CLASSICAL TRANSPORTATION PROBLEM 207
7.1.1 Mathematical Statement 208
7.1.2 Properties of the System 208
7.2 FINDING AN INITIAL SOLUTION 213
7.3 FINDING AN IMPROVED BASIC SOLUTION 214
7.4 DEGENERACY IN THE TRANSPORTATION PROBLEM 216
7.5 TRANSSHIPMENT PROBLEM 219
7.5.1 Formulation 219
7.5.2 Reduction to the Classical Case by Computing Minimum Cost
Routes 222
7.5.3 Reduction to the Classical Case by the Transshipment Pro¬
cedure 222
7.6 TRANSPORTATION PROBLEMS WITH BOUNDED PARTIAL
SUMS 225
7.7 NOTES SELECTED BIBLIOGRAPHY 227
7.8 PROBLEMS 228
xjj CONTENTS
8 NETWORK FLOW THEORY 231
8.1 THE MAXIMAL FLOW PROBLEM 232
8.1.1 Decomposition of Flows 233
8.1.2 The Augmenting Path Algorithm for Maximal Flow 234
8.1.3 Cuts in a Network 239
8.2 SHORTEST ROUTE 241
8.3 MINIMUM COST FLOW PROBLEM 242
8.4 NOTES k SELECTED BIBLIOGRAPHY 243
8.5 PROBLEMS 245
9 GENERALIZED UPPER BOUNDS 251
9.1 PROBLEM STATEMENT 251
9.2 BASIC THEORY 253
9.3 SOLVING SYSTEMS WITH GUB EQUATIONS 253
9.4 UPDATING THE BASIS AND WORKING BASIS 257
9.5 NOTES SELECTED BIBLIOGRAPHY 264
9.6 PROBLEMS 264
10 DECOMPOSITION OF LARGE SCALE SYSTEMS 265
10.1 WOLFE S GENERALIZED LINEAR PROGRAM 267
10.2 DANTZIG WOLFE (D W) DECOMPOSITION PRINCIPLE .... 280
10.2.1 D W Principle 284
10.2.2 D W Decomposition Algorithm and Variants 289
10.2.2.1 The D W Algorithm 289
10.2.2.2 Variants of the D W Algorithm 290
10.2.3 Optimality and Dual Prices 290
10.2.4 D W Initial Solution 291
10.2.5 D W Algorithm Illustrated 292
10.3 BENDERS DECOMPOSITION 299
10.3.1 Dual of D W Decomposition 299
10.3.2 Derivation of Benders Decomposition 300
10.4 BLOCK ANGULAR SYSTEM 306
10.5 STAIRCASE STRUCTURED PROBLEMS 308
10.5.1 Using Benders Decomposition 309
10.5.2 Using D W Decomposition 310
10.5.3 Using D W Decomposition with Alternate Stages Forming the
Subproblems 312
10.6 DECOMPOSITION USED IN CENTRAL PLANNING 313
10.7 NOTES k SELECTED BIBLIOGRAPHY 315
10.8 PROBLEMS 317
CONTENTS xiii
11 STOCHASTIC PROGRAMMING: INTRODUCTION 323
11.1 OVERVIEW 324
11.2 UNCERTAIN COSTS 326
11.2.1 Minimum Expected Costs 326
11.2.2 Minimum Variance 327
11.3 UNCERTAIN DEMANDS 329
11.4 NOTES SELECTED BIBLIOGRAPHY 332
11.5 PROBLEMS 332
12 TWO STAGE STOCHASTIC PROGRAMS 335
12.1 THE DETERMINISTIC TWO STAGE LP PROBLEM 335
12.2 THE ANALOGOUS STOCHASTIC TWO STAGE LP PROBLEM . 336
12.3 LP EQUIVALENT OF THE STOCHASTIC PROBLEM (EQ LP) . 337
12.3.1 LP Equivalent Formulation 337
12.3.2 Geometric Description of Benders Decomposition Algorithm . 338
12.3.3 Decomposition Algorithm 341
12.3.4 Theory behind the Algorithm 348
12.4 SOLVING STOCHASTIC TWO STAGE PROBLEMS USING SAM¬
PLING 349
12.4.1 Overview 349
12.4.2 Naive Sampling 350
12.4.3 Sampling Methodology 351
12.4.4 Estimating Upper Bound zUB for Min z given x = xk 351
12.4.5 Estimating Lower Bound zLB for Min z 352
12.5 USE OF IMPORTANCE SAMPLING 354
12.5.1 Crude (Naive) Monte Carlo Methods 355
12.5.2 Monte Carlo Methods using Importance Sampling 356
12.6 NOTES SELECTED BIBLIOGRAPHY 360
12.7 PROBLEMS 362
A PROBABILITY THEORY: OVERVIEW 367
A.I BASIC CONCEPTS, EXPECTED VALUE. AND VARIANCE ... 367
A.2 NORMAL DISTRIBUTION AND THE CENTRAL LIMIT THEO¬
REM 370
A.3 CHI SQUARE DISTRIBUTION. STUDENT S ^ DISTRIBUTION.
AND CONFIDENCE INTERVALS 373
A.3.1 Chi Square Distribution 373
A.3.2 Student s ^ Distribution 375
A.3.3 Confidence Intervals 376
A.4 NOTES SELECTED BIBLIOGRAPHY 377
REFERENCES 379
INDEX 439
List of Figures
1 1 Vector (xi,X2)T 2
1 2 Examples of Two Dimensional Convex Regions 3
1 3 Additional Examples of Two Dimensional Convex Regions 3
1 4 Example of an Unbounded Two Dimensional Convex Set and an
Unbounded Three Dimensional Convex Set 4
1 5 Examples of Two Dimensional Nonconvex Regions 4
1 6 Points Common to Convex Sets 5
1 7 Example of a Line Segment 6
1 8 A Three Dimensional Simplex 13
1 9 Local and Global Minima 14
1 10 Extreme Points 15
1 11 Joining Extreme Points 18
1 12 Geometrically the Iterates of the Simplex Algorithm Move Along the
Edges of the Convex Set 21
1 13 Geometric Picture of the Distance of a Point to a Boundary 22
1 14 Movement of 9 24
1 15 Geometrically a Linear Program is a Center of Gravity Problem . . 26
1 16 Simplex Associated with an Iteration of the Simplex Algorithm (m = 2) 27
1 17 Geometry of the Simplex Algorithm for the Product Mix Problem . 28
1 18 Simplex Associated with an Iteration of the Simplex Algorithm (m = 3) 30
1 19 Hyperplane Hj and Simplex S 40
2 1 Illustration of the Duality Gap 45
2 2 Find Basic Feasible Solutions of Dual of Two Variable Primal .... 63
3 1 The Two Dimensional Center of Gravity Problem: Find a Simplex
that Contains the Origin 73
3 2 Finding an Improved Approximation 74
3 3 Convergence under Existence of a Ball B 77
3 4 Degenerate Two Dimensional Case 79
3 5 Decreasing x to Improve the Rate of Convergence 81
3 6 Ellipsoid Centered at (1,2) 86
3 7 Ellipsoid Subproblem Centered at yl 87
xv
Xvi FIGURES
3 8 Comparison of a Move from a Point i* Near the Center Versus a
Point xl Near the Boundary 101
3 9 Bound for cTyt+1 105
4 1 Barrier Function Method: Approach of x* (//) to x* 129
7 1 Network Representation of the Transportation Problem 209
7 2 Example of Standard Transportation Array 209
7 3 Cycling in the Transportation Problem 216
7 4 Perturbing the Transportation Problem 217
7 5 Example of a Standard Transshipment Array 219
7 6 The Transshipment Problem 221
8 1 A Simple Directed Network 232
8 2 Decomposition of Flows 234
8 3 Original Network and Associated Network with 9 Flow 236
8 4 Example to Show Matching 245
8 5 Data for a Max Flow Problem 246
10 1 Illustration of the Resolution Theorem 283
12 1 Benders Decomposition Applied to EQ LP 339
List of Tables
2 1 Tucker Diagram (Partitioned) 54
5 1 Hoffman s Example of Cycling (Continued on the Right) 150
5 2 Hoffman s Example of Cycling (Continued from the Left) 151
5 3 Beale s Example of Cycling 152
5 4 Inductive Method and Wolfe s Rule Applied to Beale s Example . . . 157
6 1 Primal Dual Correspondences 175
6 2 Primal Simplex and Dual Simplex Methods 180
9 1 An Example of GUB Constraints and Key Basic Variables 254
9 2 An Example of a Reordered Basis for GUB Constraints 254
12 1 Data for the Grape Grower s Dilemma 364
12 2 Data for Dinner Set Production Schedule 364
xvii
|
| any_adam_object | 1 |
| author | Dantzig, George Bernard 1914-2005 Thapa, Mukund Narain |
| author_GND | (DE-588)124788734 (DE-588)124788696 |
| author_facet | Dantzig, George Bernard 1914-2005 Thapa, Mukund Narain |
| author_role | aut aut |
| author_sort | Dantzig, George Bernard 1914-2005 |
| author_variant | g b d gb gbd m n t mn mnt |
| building | Verbundindex |
| bvnumber | BV012306867 |
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| ctrlnum | (OCoLC)492141034 (DE-599)BVBBV012306867 |
| dewey-full | 519.72 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.72 |
| dewey-search | 519.72 |
| dewey-sort | 3519.72 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| indexdate | 2024-07-09T18:25:16Z |
| institution | BVB |
| isbn | 0387986138 |
| language | English |
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| series2 | Springer series in operations research |
| spelling | Dantzig, George Bernard 1914-2005 Verfasser (DE-588)124788734 aut Linear programming 2 Theory and extensions George B. Dantzig ; Mukund N. Thapa New York [u.a.] Springer 2003 XXV, 448 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in operations research Literaturverz. S. 379 - 438 Programmation linéaire ram algorithme simplexe inriac dualité inriac dégénérescence inriac méthode point intérieur inriac problème transport inriac programmation linéaire inriac programmation stochastique inriac théorie probabilité inriac variation inriac écoulement multidimensionnel inriac Linear programming Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Lineare Optimierung (DE-588)4035816-1 s DE-604 Thapa, Mukund Narain Verfasser (DE-588)124788696 aut (DE-604)BV011292956 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008342452&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dantzig, George Bernard 1914-2005 Thapa, Mukund Narain Linear programming Programmation linéaire ram algorithme simplexe inriac dualité inriac dégénérescence inriac méthode point intérieur inriac problème transport inriac programmation linéaire inriac programmation stochastique inriac théorie probabilité inriac variation inriac écoulement multidimensionnel inriac Linear programming Lineare Optimierung (DE-588)4035816-1 gnd |
| subject_GND | (DE-588)4035816-1 |
| title | Linear programming |
| title_auth | Linear programming |
| title_exact_search | Linear programming |
| title_full | Linear programming 2 Theory and extensions George B. Dantzig ; Mukund N. Thapa |
| title_fullStr | Linear programming 2 Theory and extensions George B. Dantzig ; Mukund N. Thapa |
| title_full_unstemmed | Linear programming 2 Theory and extensions George B. Dantzig ; Mukund N. Thapa |
| title_short | Linear programming |
| title_sort | linear programming theory and extensions |
| topic | Programmation linéaire ram algorithme simplexe inriac dualité inriac dégénérescence inriac méthode point intérieur inriac problème transport inriac programmation linéaire inriac programmation stochastique inriac théorie probabilité inriac variation inriac écoulement multidimensionnel inriac Linear programming Lineare Optimierung (DE-588)4035816-1 gnd |
| topic_facet | Programmation linéaire algorithme simplexe dualité dégénérescence méthode point intérieur problème transport programmation linéaire programmation stochastique théorie probabilité variation écoulement multidimensionnel Linear programming Lineare Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008342452&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011292956 |
| work_keys_str_mv | AT dantziggeorgebernard linearprogramming2 AT thapamukundnarain linearprogramming2 |