Integer programming:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Wiley
[2021]
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| Edition: | Second edition |
| Series: | Wiley Interscience series in discrete mathematics and optimization
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xix, 316 Seiten Diagramme |
| ISBN: | 9781119606536 |
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vii Contents Preface to the Second Edition xii Preface to the First Edition xiii Abbreviations and Notation xvii About the Companion Website xix 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1 Introduction 1 What Is an Integer Program? 3 Formulating IPs and BIPs 5 The Combinatorial Explosion 8 Mixed Integer Formulations 9 Alternative Formulations 12 Good and Ideal Formulations 15 Notes 18 Exercises 19 2 Optimality, Relaxation, and Bounds 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Optimality and Relaxation 25 Linear Programming Relaxations 27 Combinatorial Relaxations 28 Lagrangian Relaxation 29 Duality 30 Linear Programming and Polyhedra 32 Primal Bounds:Greedy and Local Search 34 Notes 38 Exercises 38 1 Formulations 25
viii I Contents 3 3.1 3.2 3.3 3.4 3.4.1 3.4.2 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 5 5.1 5.2 5.3 5.4 5.4.1 5.4.2 5.5 5.6 5.7 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 43 Properties of Easy Problems 43 IPs with Totally Unimodular Matrices 44 Minimum Cost Network Flows 46 Special Minimum Cost Flows 48 Shortest Path 48 Maximum s -1 Flow 49 Optimal Trees 50 Submodularity and Matroids 54 Two Harder Network Flow Problems 57 Notes 59 Exercises 60 Well-Solved Problems Matchings and Assignments 63 Augmenting Paths and Optimality 63 Bipartite Maximum Cardinality Matching 65 The Assignment Problem 67 Matchings in Nonbipartite Graphs 73 Notes 74 Exercises 75 79 Some Motivation: Shortest Paths 79 Uncapacitated Lot-Sizing 80 An Optimal Subtree of a Tree 83 Knapsack Problems 84 0-1 Knapsack Problems 85 Integer Knapsack Problems 86 The Cutting Stock Problem 89 Notes 91 Exercises 92 Dynamic Programming Complexity and Problem Reductions 95 Complexity 95 Decision Problems, and Classes MP and P 96 Polynomial Reduction and the Class J^PC 98 Consequences of P = ^P or P ƒ MP 103 Optimization and Separation 104 The Complexity of Extended Formulations 105 Worst-Case Analysis of Heuristics 106 Notes 109 Exercises 110
Contents I ix 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 113 Divide and Conquer 113 Implicit Enumeration 114 Branch and Bound: an Example 116 LP-Based Branch and Bound 120 Using a Branch-and֊Bound/Cut System Preprocessing or Presolve 129 Notes 134 Exercises 135 8 Cutting Plane Algorithms 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.7.1 8.7.2 8.7.3 8.7.4 8.8 8.9 8.10 Introduction 139 Some Simple Valid Inequalities 140 Valid Inequalities 143 A Priori Addition of Constraints 147 Automatic Reformulation or Cutting Plane Algorithms Gomory’s Fractional Cutting Plane Algorithm 150 Mixed Integer Cuts 153 The Basic Mixed Integer Inequality 153 The Mixed Integer Rounding (MIR) Inequality 155 The Gomory Mixed Integer Cut 155 Split Cuts 156 Disjunctive Inequalities and Lift-and֊Project 158 Notes 161 Exercises 162 9 Strong Valid Inequalities 9.1 9.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.4.1 9.4.2 9.5 9.5.1 9.6 9.7 9.8 Introduction 167 Strong Inequalities 168 0-1 Knapsack Inequalities 175 Cover Inequalities 175 Strengthening Cover Inequalities 176 Separation for Cover Inequalities 178 Mixed 0-1 Inequalities 179 Flow Cover Inequalities 179 Separation for Flow Cover Inequalities 181 The Optimal Subtour Problem 183 Separation for Generalized Subtour Constraints Branch-and-Cut 186 Notes 189 Exercises 190 7 Branch and Bound 123 139 167 183 149
Contents 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 11 195 Lagrangian Relaxation 195 The Strength of the Lagrangian Dual 200 Solving the Lagrangian Dual 202 Lagrangian Heuristics 205 Choosing a Lagrangian Dual 207 Notes 209 Exercises 210 Lagrangian Duality 11.1 11.2 11.3 11.4 11.5 11.5.1 11.5.2 11.5.3 11.6 11.7 11.7.1 11.7.2 11.7.3 11.8 11.9 213 Introduction 213 The Dantzig-Wolfe Reformulation of an IP 215 Solving the LP Master Problem: Column Generation 216 Solving the Master Problem: Branch-and-Price 219 Problem Variants 222 Handling Multiple Subproblems 222 Partitioning/Packing Problems with Additional Variables 223 Partitioning/Packing Problems with Identical Subsets 224 Computational Issues 225 Branch-Cut-and-Price: An Example 226 A Capacitated Vehicle Routing Problem 226 Solving the Subproblems 229 The Load Formulation 230 Notes 231 Exercises 232 12 Benders’ Algorithm 12.1 12.2 12.3 12.4 12.5 12.5.1 12.5.2 12.6 12.7 Introduction 235 Benders’ Reformulation 236 Benders’ with Multiple Subproblems 240 Solving the Linear Programming Subproblems 242 Integer Subproblems: Basic Algorithms 244 Branching in the (x, η, y)-Space 244 Branching in (x, /?)֊Space and “No-Good” Cuts 246 Notes 247 Exercises 248 13 Primal Heuristics 251 Introduction 251 Greedy and Local Search Revisited 252 Improved Local Search Heuristics 255 13.1 13.2 13.3 Column (and Row) Generation Algorithms 235
Contents I xi 13.3.1 13.3.2 13.3.3 13.4 13.4.1 13.4.2 13.5 13.6 13.7 Tabu Search 255 Simulated Annealing 256 Genetic Algorithms 257 Heuristics Inside МІР Solvers 259 Construction Heuristics 259 Improvement Heuristics 261 User-Defined МІР heuristics 262 Notes 265 Exercises 266 14 From Theory to Solutions 269 Introduction 269 Software for Solving Integer Programs 269 How Do We Find an Improved Formulation? 272 Multi-item Single Machine Lot-Sizing 277 A Multiplexer Assignment Problem 282 Integer Programming and Machine Learning 285 Notes 287 Exercises 287 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 References Index 311 291 |
| adam_txt |
vii Contents Preface to the Second Edition xii Preface to the First Edition xiii Abbreviations and Notation xvii About the Companion Website xix 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1 Introduction 1 What Is an Integer Program? 3 Formulating IPs and BIPs 5 The Combinatorial Explosion 8 Mixed Integer Formulations 9 Alternative Formulations 12 Good and Ideal Formulations 15 Notes 18 Exercises 19 2 Optimality, Relaxation, and Bounds 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Optimality and Relaxation 25 Linear Programming Relaxations 27 Combinatorial Relaxations 28 Lagrangian Relaxation 29 Duality 30 Linear Programming and Polyhedra 32 Primal Bounds:Greedy and Local Search 34 Notes 38 Exercises 38 1 Formulations 25
viii I Contents 3 3.1 3.2 3.3 3.4 3.4.1 3.4.2 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 5 5.1 5.2 5.3 5.4 5.4.1 5.4.2 5.5 5.6 5.7 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 43 Properties of Easy Problems 43 IPs with Totally Unimodular Matrices 44 Minimum Cost Network Flows 46 Special Minimum Cost Flows 48 Shortest Path 48 Maximum s -1 Flow 49 Optimal Trees 50 Submodularity and Matroids 54 Two Harder Network Flow Problems 57 Notes 59 Exercises 60 Well-Solved Problems Matchings and Assignments 63 Augmenting Paths and Optimality 63 Bipartite Maximum Cardinality Matching 65 The Assignment Problem 67 Matchings in Nonbipartite Graphs 73 Notes 74 Exercises 75 79 Some Motivation: Shortest Paths 79 Uncapacitated Lot-Sizing 80 An Optimal Subtree of a Tree 83 Knapsack Problems 84 0-1 Knapsack Problems 85 Integer Knapsack Problems 86 The Cutting Stock Problem 89 Notes 91 Exercises 92 Dynamic Programming Complexity and Problem Reductions 95 Complexity 95 Decision Problems, and Classes MP and P 96 Polynomial Reduction and the Class J^PC 98 Consequences of P = ^P or P ƒ MP 103 Optimization and Separation 104 The Complexity of Extended Formulations 105 Worst-Case Analysis of Heuristics 106 Notes 109 Exercises 110
Contents I ix 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 113 Divide and Conquer 113 Implicit Enumeration 114 Branch and Bound: an Example 116 LP-Based Branch and Bound 120 Using a Branch-and֊Bound/Cut System Preprocessing or Presolve 129 Notes 134 Exercises 135 8 Cutting Plane Algorithms 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.7.1 8.7.2 8.7.3 8.7.4 8.8 8.9 8.10 Introduction 139 Some Simple Valid Inequalities 140 Valid Inequalities 143 A Priori Addition of Constraints 147 Automatic Reformulation or Cutting Plane Algorithms Gomory’s Fractional Cutting Plane Algorithm 150 Mixed Integer Cuts 153 The Basic Mixed Integer Inequality 153 The Mixed Integer Rounding (MIR) Inequality 155 The Gomory Mixed Integer Cut 155 Split Cuts 156 Disjunctive Inequalities and Lift-and֊Project 158 Notes 161 Exercises 162 9 Strong Valid Inequalities 9.1 9.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.4.1 9.4.2 9.5 9.5.1 9.6 9.7 9.8 Introduction 167 Strong Inequalities 168 0-1 Knapsack Inequalities 175 Cover Inequalities 175 Strengthening Cover Inequalities 176 Separation for Cover Inequalities 178 Mixed 0-1 Inequalities 179 Flow Cover Inequalities 179 Separation for Flow Cover Inequalities 181 The Optimal Subtour Problem 183 Separation for Generalized Subtour Constraints Branch-and-Cut 186 Notes 189 Exercises 190 7 Branch and Bound 123 139 167 183 149
Contents 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 11 195 Lagrangian Relaxation 195 The Strength of the Lagrangian Dual 200 Solving the Lagrangian Dual 202 Lagrangian Heuristics 205 Choosing a Lagrangian Dual 207 Notes 209 Exercises 210 Lagrangian Duality 11.1 11.2 11.3 11.4 11.5 11.5.1 11.5.2 11.5.3 11.6 11.7 11.7.1 11.7.2 11.7.3 11.8 11.9 213 Introduction 213 The Dantzig-Wolfe Reformulation of an IP 215 Solving the LP Master Problem: Column Generation 216 Solving the Master Problem: Branch-and-Price 219 Problem Variants 222 Handling Multiple Subproblems 222 Partitioning/Packing Problems with Additional Variables 223 Partitioning/Packing Problems with Identical Subsets 224 Computational Issues 225 Branch-Cut-and-Price: An Example 226 A Capacitated Vehicle Routing Problem 226 Solving the Subproblems 229 The Load Formulation 230 Notes 231 Exercises 232 12 Benders’ Algorithm 12.1 12.2 12.3 12.4 12.5 12.5.1 12.5.2 12.6 12.7 Introduction 235 Benders’ Reformulation 236 Benders’ with Multiple Subproblems 240 Solving the Linear Programming Subproblems 242 Integer Subproblems: Basic Algorithms 244 Branching in the (x, η, y)-Space 244 Branching in (x, /?)֊Space and “No-Good” Cuts 246 Notes 247 Exercises 248 13 Primal Heuristics 251 Introduction 251 Greedy and Local Search Revisited 252 Improved Local Search Heuristics 255 13.1 13.2 13.3 Column (and Row) Generation Algorithms 235
Contents I xi 13.3.1 13.3.2 13.3.3 13.4 13.4.1 13.4.2 13.5 13.6 13.7 Tabu Search 255 Simulated Annealing 256 Genetic Algorithms 257 Heuristics Inside МІР Solvers 259 Construction Heuristics 259 Improvement Heuristics 261 User-Defined МІР heuristics 262 Notes 265 Exercises 266 14 From Theory to Solutions 269 Introduction 269 Software for Solving Integer Programs 269 How Do We Find an Improved Formulation? 272 Multi-item Single Machine Lot-Sizing 277 A Multiplexer Assignment Problem 282 Integer Programming and Machine Learning 285 Notes 287 Exercises 287 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 References Index 311 291 |
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| spelling | Wolsey, Laurence A. 1945- (DE-588)129756326 aut Integer programming Laurence A. Wolsey ; UCLouvain Second edition New York Wiley [2021] xix, 316 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Wiley Interscience series in discrete mathematics and optimization Ganzzahlige Optimierung - Lehrbuch Ganzzahlige Optimierung (DE-588)4155950-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Ganzzahlige Optimierung (DE-588)4155950-2 s DE-604 Erscheint auch als Online-Ausgabe 978-1-119-60647-5 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032330580&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wolsey, Laurence A. 1945- Integer programming Ganzzahlige Optimierung - Lehrbuch Ganzzahlige Optimierung (DE-588)4155950-2 gnd |
| subject_GND | (DE-588)4155950-2 (DE-588)4123623-3 |
| title | Integer programming |
| title_auth | Integer programming |
| title_exact_search | Integer programming |
| title_exact_search_txtP | Integer programming |
| title_full | Integer programming Laurence A. Wolsey ; UCLouvain |
| title_fullStr | Integer programming Laurence A. Wolsey ; UCLouvain |
| title_full_unstemmed | Integer programming Laurence A. Wolsey ; UCLouvain |
| title_short | Integer programming |
| title_sort | integer programming |
| topic | Ganzzahlige Optimierung - Lehrbuch Ganzzahlige Optimierung (DE-588)4155950-2 gnd |
| topic_facet | Ganzzahlige Optimierung - Lehrbuch Ganzzahlige Optimierung Lehrbuch |
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