Metaheuristics for hard optimization simulated annealing tabu search evolutionary and genetic algorithms ant colonies, methods and case studi

Metaheuristics for Hard Optimization J Dr´eo A P´etrowski P Siarry E Taillard Metaheuristics for Hard Optimization Simulated Annealing, Tabu Search, Evolutionary and Genetic Algorithms, Ant Colonies, … Methods and Case Studies With 140 Figures 123 Johann Dr´eo Professor Patrick Siarry Universit´e Paris XII, Facult´e des Sciences, LiSSi 61 avenue du Ge´n´eral de Gaulle, 94010 Cr´eteil, France Alain P´etrowski Institut National des Télécommunications, rue Charles Fourier, 91011 Evry, France Professor Eric Taillard EIVD, Ecole d’Ingénieurs du Canton de Vaud route de Cheseaux 1, 1400 Yverdon-les-Bains, Switzerland Translator: Amitava Chatterjee Originally published in French by Eyrolles, Paris (2003) under the title: “M´etaheuristiques pour l’optimisation difficile" Book coordinated by Patrick Siarry Library of Congress Control Number: 2005930496 ISBN-10 3-540-23022-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-23022-9 Springer Berlin Heidelberg New York This work is subject to 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Services Cover design: de’blik, Berlin Printed on acid-free paper SPIN 11009153 62/3141/SPI 543210 Preface Metaheuristics for Hard Optimization comprises of three parts The first part is devoted to the detailed presentation of the four most widely known metaheuristics: • • • • the the the the simulated annealing method; tabu search; genetic and evolutionary algorithms; ant colony algorithms Each one of these metaheuristics is actually a family of methods, of which we try to discuss the essential elements Some common features clearly appear in most metaheuristics, such as the use of diversification, to force the exploration of regions of the search space, rarely visited until now, and the use of intensification, to go thoroughly into some promising regions Another common feature is the use of memory to archive the best encountered solutions One common drawback for most metaheuristics still is the delicate tuning of numerous parameters; theoretical results available by now are not sufficient to really help in practice the user facing a new hard optimization problem In the second part, we present some other metaheuristics, less widespread or emergent: some variants of simulated annealing; noising method; distributed search; Alienor method; particle swarm optimization; estimation of distribution methods; GRASP method; cross-entropy method; artificial immune systems; differential evolution Then we describe some extensions of metaheuristics for continuous optimization, multimodal optimization, multiobjective optimization and contrained evolutionary optimization We present some of the existing techniques and some ways of research The last chapter is devoted to the problem of the choice of a metaheuristic; we describe an unifying method called “Adaptive Memory Programming”, which tends to attenuate the difficulty of this choice The delicate subject of a rigorous statistical comparison between stochastic iterative methods is also discussed VI Preface The last part of the book concentrates on three case studies: • the optimization of the 3G mobile networks (UMTS) using the genetic algorithms After a brief presentation of the operation of UMTS networks and of the quantities involved in the analysis of their performances, the chapter discusses the optimization problem for planning the UMTS network; an efficient method using a genetic algorithm is presented and illustrated through one example of a realistic network; • the application of genetic algorithms to the problems of management of the air traffic One details two problems of air traffic management for which a genetic algorithm based solution has been proposed: the first application deals with the en route conflict resolution problem; the second one discusses the traffic management in an airport platform; • constrained programming and ant colony algorithms applied to vehicle routing problems It is shown that constraint programming provides a modelling procedure, making it possible to represent the problems in an expressive and concise way; the use of ant colony algorithms allows to obtain heuristics which can be simultaneously robust and generic in nature One appendix of the book is devoted to the modeling of simulated annealing through the Markov chain formalism Another appendix gives a complete implementation in C++ language for robust tabu search method Cr´eteil, Evry, Yerdon-les-Bains September 2005 Johann Dr´eo Patrick Siarry Alain P´etrowski Eric Taillard Contents Introduction Part I Presentation of the Main Metaheuristics Simulated Annealing 1.1 Introduction 1.2 Presentation of the method 1.2.1 Analogy between an optimization problem and some physical phenomena 1.2.2 Real annealing and simulated annealing 1.2.3 Simulated annealing algorithm 1.3 Theoretical approaches 1.3.1 Theoretical convergence of simulated annealing 1.3.2 Configuration space 1.3.3 Rules of acceptance 1.3.4 Annealing scheme 1.4 Parallelization of the simulated annealing algorithm 1.5 Some applications 1.5.1 Benchmark problems of combinatorial optimization 1.5.2 Layout of electronic circuits 1.5.3 Search for an equivalent schema in electronics 1.5.4 Practical applications in various fields 1.6 Advantages and disadvantages of the method 1.7 Simple practical suggestions for the beginners 1.8 Annotated bibliography 23 23 24 24 25 25 27 27 28 30 30 32 35 35 36 40 42 44 44 45 Tabu Search 2.1 Introduction 2.2 The quadratic assignment problem 2.3 Basic tabu search 47 47 49 51 VIII Contents 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.3.1 Neighborhood 2.3.2 Moves, neighborhood 2.3.3 Evaluation of the neighborhood Candidate list Short-term memory 2.5.1 Hashing tables 2.5.2 Tabu list 2.5.3 Duration of tabu conditions 2.5.4 Aspiration conditions Convergence of tabu search Long-term memory 2.7.1 Frequency-based memory 2.7.2 Obligation to carry out move Strategic oscillations Conclusion Annotated bibliography 51 52 54 56 57 57 59 60 66 66 69 69 71 72 72 72 Evolutionary Algorithms 75 3.1 From genetics to engineering 75 3.2 The generic evolutionary algorithm 77 3.2.1 Selection operators 77 3.2.2 Variation operators 78 3.2.3 The generational loop 78 3.2.4 Solving a simple problem 79 3.3 Selection operators 81 3.3.1 Selection pressure 81 3.3.2 Genetic drift 82 3.3.3 Proportional selection 83 3.3.4 Tournament selection 88 3.3.5 Truncation selection 90 3.3.6 Replacement selections 90 3.3.7 Fitness function 92 3.4 Variation operators and representation 93 3.4.1 Generalities about the variation operators 93 3.4.2 Binary representation 97 3.4.3 Real representation 101 3.4.4 Some discrete representations for the permutation problems 108 3.4.5 Representation of parse trees for the genetic programming 113 3.5 Particular case of the genetic algorithms 118 3.6 Some considerations on the convergence of the evolutionary algorithms 119 3.7 Conclusion 120 3.8 Glossary 121 Contents IX 3.9 Annotated bibliography 122 Ant Colony Algorithms 123 4.1 Introduction 123 4.2 Collective behavior of social insects 124 4.2.1 Self-organization and behavior 124 4.2.2 Natural optimization: pheromonal trails 127 4.3 Optimization by ant colonies and the traveling salesman problem 129 4.3.1 Basic algorithm 130 4.3.2 Variants 131 4.3.3 Choice of the parameters 134 4.4 Other combinatorial problems 134 4.5 Formalization and properties of ant colony optimization 135 4.5.1 Formalization 135 4.5.2 Pheromones and memory 137 4.5.3 Intensification/diversification 137 4.5.4 Local search and heuristics 138 4.5.5 Parallelism 138 4.5.6 Convergence 139 4.6 Prospect 139 4.6.1 Continuous optimization 139 4.6.2 Dynamic problems 147 4.6.3 Metaheuristics and ethology 147 4.6.4 Links with other metaheuristics 148 4.7 Conclusion 149 4.8 Annotated bibliography 150 Part II Variants, Extensions and Methodological Advices Some Other Metaheuristics 153 5.1 Introduction 153 5.2 Some variants of simulated annealing 154 5.2.1 Simulated diffusion 154 5.2.2 Microcanonic annealing 155 5.2.3 The threshold method 157 5.2.4 “Great deluge” method 157 5.2.5 Method of the “record to record travel” 157 5.3 Noising method 159 5.4 Method of distributed search 159 5.5 “Alienor” method 160 5.6 Particle swarm optimization method 162 5.7 The estimation of distribution algorithm 166 5.8 GRASP method 169 X Contents 5.9 5.10 5.11 5.12 5.13 “Cross-Entropy” method 170 Artificial immune systems 172 Method of differential evolution 173 Algorithms inspired by the social insects 175 Annotated bibliography 176 Extensions 179 6.1 Introduction 179 6.2 Adaptation for the continuous variable problems 179 6.2.1 General framework of “difficult” continuous optimization179 6.2.2 Some continuous metaheuristics 185 6.3 Multimodal optimization 196 6.3.1 The problem 196 6.3.2 Niching with the sharing method 196 6.3.3 Niching with the deterministic crowding method 199 6.3.4 The clearing procedure 201 6.3.5 Speciation 203 6.4 Multiobjective optimization 206 6.4.1 Formalization of the problem 206 6.4.2 Simulated annealing based methods 208 6.4.3 Multiobjective evolutionary algorithms 211 6.5 Constrained evolutionary optimization 216 6.5.1 Penalization methods 217 6.5.2 Superiority of the feasible individuals 219 6.5.3 Repair methods 220 6.5.4 Variation operators satisfying the constraint structures 221 6.5.5 Other methods dealing with constraints 223 6.6 Conclusion 223 6.7 Annotated bibliography 224 Methodology 225 7.1 Introduction 225 7.2 Problem modeling 227 7.3 Neighborhood choice 228 7.3.1 “Simple” neighborhoods 228 7.3.2 Ejection chains 230 7.3.3 Decomposition into subproblems: POPMUSIC 231 7.3.4 Conclusions on modeling and neighborhood 233 7.4 Improving method, simulated annealing, taboo search ? 235 7.5 Adaptive Memory Programming 235 7.5.1 Ant colonies 236 7.5.2 Evolutionary or memetic algorithms 236 7.5.3 Scatter Search 236 7.5.4 Vocabulary building 238 7.5.5 Path relinking 239 Contents XI 7.6 Iterative heuristics comparison 240 7.6.1 Comparing proportion 241 7.6.2 Comparing iterative optimization methods 243 7.7 Conclusion 244 7.8 Annotated bibliography 247 Part III Case Studies Optimization of UMTS Radio Access Networks with Genetic Algorithms 251 8.1 Introduction 251 8.2 Introduction to mobile radio networks 252 8.2.1 Cellular network 252 8.2.2 Characteristic of the radio channel 253 8.2.3 Radio interface of the UMTS 255 8.3 Definition of the optimization problem 261 8.3.1 Radio planning of a UMTS network 261 8.3.2 Definition of the optimization problem 262 8.4 Application of the genetic algorithm to automatic planning 265 8.4.1 Coding 265 8.4.2 Genetic operators 266 8.4.3 Evaluation of the individuals 267 8.5 Results 267 8.5.1 Optimization of the capacity 269 8.5.2 Optimization of the loads 270 8.5.3 Optimization of the intercellular interferences 272 8.5.4 Optimization of the coverage 272 8.5.5 Optimization of the probability of access 273 8.6 Conclusion 274 Genetic Algorithms Applied to Air Traffic Management 277 9.1 En route conflict resolution 277 9.1.1 Complexity of the conflict resolution problem 280 9.1.2 Existing resolution methods 280 9.1.3 Modeling of the problem 281 9.1.4 Implementation of the genetic algorithm 285 9.1.5 Theoretical study of a simple example 288 9.1.6 Numerical application 292 9.1.7 Remarks 295 9.2 Ground Traffic optimization 296 9.2.1 Modeling 296 9.2.2 BB: the 1-against-n resolution method 300 9.2.3 GA and GA+BB : genetic algorithms 301 9.2.4 Experimental results 303 ... P´etrowski P Siarry E Taillard Metaheuristics for Hard Optimization Simulated Annealing, Tabu Search, Evolutionary and Genetic Algorithms, Ant Colonies, … Methods and Case Studies With 140 Figures... the the the simulated annealing method; tabu search; genetic and evolutionary algorithms; ant colony algorithms Each one of these metaheuristics is actually a family of methods, of which we try... widely known metaheuristics: • • • • the simulated annealing method; tabu search; the evolutionary algorithms; ant colony algorithms Each one of these metaheuristics is actually a family of methods,