El-Ghazali Talbi (Ed.) Metaheuristics for Bi-level Optimization ^ Springer
Contents 1 A Taxonomy of Metaheuristics for Bi-level Optimization 1 El-Ghazali Talbi 1.1 Introduction 1 1.2 Bi-level Optimization Concepts 3 1.2.1 Bi-level Optimization Problems 6 1.2.2 Complexity and Optimality Conditions 9 1.3 Relationships with Other Problems 10 1.3.1 Bi-level versus Stackelberg Games 10 1.3.2 Bi-level versus Multi-objective Problems 11 1.3.3 Bi-level versus Set-Valued Optimization Problems 12 1.4 Applications of Bi-level Optimization 13 1.5 Metaheuristics 15 1.5.1 S-metaheuristics 17 1.5.2 P-metaheuristics 18 1.6 Metaheuristics for Bi-level Optimization 21 1.6.1 Metaheuristics Based on Nested Approach 22 1.6.2 Metaheuristics Based on Reformulation to Single-Level Optimization Problems 25 1.6.3 Metaheuristics Based on Transformation to Multi-objective Optimization Problems 27 1.6.4 Co-evolutionary Metaheuristics 28 1.7 Performance Assessment of Bi-level Metaheuristics 30 1.7.1 Performance Indicators 31 1.8 Conclusions and Perspectives 33 References 34 2 A Genetic Algorithm for Power System VulnerabUity Analysis under Multiple Contingencies 41 Jose M. Arroyo, Francisco J. Fernandez 2.1 Introduction 41 2.2 Power System Vulnerability Analysis 43 2.3 Attacker-Defender Models 44
XII Contents 2.4 Genetic Algorithm: An Evolutionary Metaheuristic 46 2.4.1 Structure of a Genetic Algorithm 47 2.4.2 Proposed Genetic Algorithm for Attacker-Defender Models 49 2.5 Application: Maximum Vulnerability Model with Line Switching 53 2.5.1 Problem Formulation 54 2.5.2 Particular Features of the Genetic Algorithm Approach 55 2.6 Numerical Results 57 2.7 Conclusions 63 References 65 3 A Bilevel Particle Swarm Optimization Algorithm for Supply Chain Management Problems 69 Yannis Marinakis, Magdalene Marinaki 3.1 Introduction 69 3.2 Bilevel Programming 71 3.3 Supply Chain Management Problems 72 3.3.1 Vehicle Routing Problem 72 3.3.2 Location Routing Problem 75 3.4 Bilevel Particle Swarm Optimization Algorithm 78 3.4.1 General Description of the Bilevel Particle Swarm Optimization Algorithm (PSOBilevel) 78 3.4.2 Particle Swarm Optimization 79 3.4.3 Expanding Neighborhood Search 82 3.5 Results 83 3.6 Conclusions and Future Research 90 References 90 4 CoBRA: A Coevolutionary Metaheuristic for Bi-level Optimization 95 Frangois Legillon, Arnaud Liefooghe, El-Ghazali Talbi 4.1 Introduction 95 4.2 Bi-level Optimization 96 4.2.1 General Principles of Bi-level Optimization 97 4.2.2 Meta-heuristic Approaches for Bi-level Optimization 97 4.3 CoBRA, a Coevolutionary Meta-heuristic for Bi-level Optimization 99 4.3.1 General Principles 99 4.3.2 CoBRA Components 99 4.3.3 General Algorithm 100 4.4 Performance Assessment and Bi-level Optimization 4.4.1 Motivations 101 4.4.2 Rationality 103 4.4.3 Discussion 103 ' 101
Contents XIII 4.5 Application to Bi-level Transportation 104 4.5.1 A Bi-level Multi-depot Vehicle Routing Problem 105 4.5.2 A Multi-objective Bi-level Multi-depot Vehicle Routing Problem : 107 4.6 Experimental Analysis 108 4.6.1 Experimental Design 108 4.6.2 CoBRA instantiation for BiMDVRP an M-BiMDVRP... 108 4.6.3 Experimental Results 110 4.7 Conclusions and Future Works 112 References 113 5 A Matheuristic for Leader-Follower Games Involving Facility Location-Protection-Interdiction Decisions 115 Deniz Aksen, Necati Aras 5.1 Introduction and Background 116 5.1.1 Man-Made Attacks as a Source of Disruption 116 5.1.2 Preliminary Interdiction Models 116 5.1.3 Recent Network and Power Grid Interdiction Models 117 5.1.4 Protection-Interdiction Models 118 5.1.5 Location-Interdiction and Network Design-Interdiction Models 120 5.1.6 The Triple Problem of Facility Location-Protection-Interdiction 121 5.2 Two Service Network Design Models 122 5.2.1 A Location-Protection-Interdiction Model for Coverage-Type Networks 122 5.2.2 A Location-Protection-Interdiction Model for Median-Type Networks 125 5.3 A Tabu Search Based Matheuristic for the BFCLP 130 5.3.1 Background of Tabu Search 132 5.3.2 Key Features of the Tabu Search Algorithm TSH 132 5.3.3 Partial Validation of TSH Solutions with Exhaustive Search 137 5.4 Computational Results 139 5.4.1 Random Generation of Test Instances 139 5.4.2 Preliminary Analysis of the Neighborhood Structure 140 5.4.3 Test results of TSH 140 5.4.4 Comparison of the Best TSH Solutions with the ESV-p Results 142 5.4.5 Sensitivity of the TSH Solutions to Problem Parameters 145 5.4.6 Effect of the Protection Costs on the Best Facility Configuration 146 5.5 Conclusions 147 References 148
XIV Contents 6 A Metaheuristic Framework for Bi-level Programming Problems with Multi-disciplinary Applications 153 Andrew Koh 6.1 Introduction 153 6.2 The Bi-level Programming Problem 155 6.2.1 A General BLPP 155 6.2.2 Mathematical Programs with Equilibrium Constraints... 155 6.2.3 Solution Algorithms for the BLPP 156 6.3 Differential Evolution for Bi-Level Programming (DEBLP) 157 6.3.1 Differential Evolution 158 6.3.2 Control Parameters of DE 161 6.3.3 Implicit Assumptions of DEBLP 161 6.4 Applications to Transportation Systems Management 162 6.4.1 The Lower Level Program in Transportation 162 6.4.2 Continuous Optimal Toll Pricing Problem (COTP) 164 6.4.3 Continuous Network Design Problem 165 6.5 Applications to Parameter Estimation Problems 168 6.5.1 Formulation of EIV Model 168 6.5.2 Examples 169 6.6 Handling Upper Level Constraints 171 6.6.1 Overview of Constraint Handling Techniques with Meta-Heuristics 171 6.6.2 Stochastic Ranking 172 6.6.3 Revised DEBLP with Stochastic Ranking 173 6.7 Applications to Generalised Nash Equilibrium Problems 173 6.7.1 The Generalised Nash Equilibrium Problem 174 6.7.2 Nikaido Isoda Function 175 6.7.3 Solution of the GNEP 175 6.7.4 Examples 176 6.7.5 Discussion 177 6.8 Summary and Conclusions 178 6.8.1 Summary 178 6.8.2 Further Research 179 References 181 7 Matheuristics and Exact Methods for the Discrete (r p)-centroid Problem 189 Ekaterina Alekseeva, Yury Kochetov 7.1 Introduction 189 7.2 The Problem Statement 191 7.2.1 The Bi-level Mixed Integer Linear Formulation 191 7.2.2 The Min-Max Formulation 194 7.2.3 The Single-Level Mixed Integer Linear Formulation 195 7.2.4 The Brief Overview of Related Works 196 7.3 Complexity Status 198
Contents XV 7.4 Heuristics for the Discrete (r p)-centroid Problem 199 7.4.1 Median Heuristics 200 7.4.2 Alternative Heuristics 203 7.4.3 Hybrid Heuristics : 205 7.5 Exact Methods 207 7.5.1 The Branch-and-Cut Method 209 7.5.2 An Iterative Exact Method 211 7.6 Computational Experiments 213 7.7 Conclusions 216 References 217 8 Exact Solution Methodologies for Linear and (Mixed) Integer Bilevel Programming 221 Georgios K.D. Saharidis, Antonio J. Conejo, George Kozanidis 8.1 Introduction 222 8.2 Description of the Bilevel Problem 223 8.3 Solution Approaches for the Linear Bilevel Problem 226 8.4 Solution Approaches for the Mixed Integer Bilevel Problem 234 8.5 Bilevel Programming Applications 236 8.6 Summary 241 References 241 9 Bilevel Multi-Objective Optimization and Decision Making 247 Ankur Sinha, Kalyanmoy Deb 9.1 Introduction 248 9.2 Multi-objective Bilevel Optimization Problems 249 9.2.1 Real World Problems 251 9.3 Existing Classical and Evolutionary Methodologies 253 9.3.1 Theoretical Developments 253 9.3.2 Algorithmic Developments 254 9.3.3 Evolutionary Methods 255 9.3.4 Development of Tunable Test Problems 256 9.4 Hybrid Bilevel Evolutionary Multi-Objective Optimization (H-BLEMO) Algorithm 259 9.4.1 Update of Population Sizes 261 9.4.2 Termination Criteria 262 9.4.3 Step-by-Step Procedure 263 9.4.4 Algorithmic Complexity 265 9.5 Results on Test Problems 266 9.6 Scalability Study 268 9.7 Comparison with a Nested Algorithm 269 9.8 Incorporating Decision Maker Preferences in H-BLEMO 271 9.9 Progressively Interactive Hybrid Bilevel Evolutionary Multi-Objective Optimization Algorithm (PI-HBLEMO) 272 9.9.1 Step 3: Preference Elicitation and Construction of a Value Function 273
XVI Contents 9.9.2 Termination Criterion 275 9.9.3 Modified Domination Principle 275 9.10 Results 276 9.11 Accuracy and DM Calls 278 9.12 Conclusions 280 References 281 Index 285