Metaheuristics for Hard Optimization: Methods and Case Studies

372 Pages·2005·4.46 MB·English

by  Johann Dréo, Alain Pétrowski, Patrick Siarry, Eric Taillard, A. Chatterjee

MetaheuristicsforHardOptimization
J. Dre´o A. Pe´trowski
P. Siarry E. Taillard
Metaheuristics
for Hard Optimization

Simulated Annealing, Tabu Search, Evolutionary
and Genetic Algorithms, Ant Colonies,…
Methods and Case Studies
With140 Figures
123
JohannDre´o

ProfessorPatrickSiarry

Universite´ParisXII,Faculte´des Sciences, LiSSi
61avenueduGe´ne´raldeGau lle,94010Cre´teil,France
AlainPe´trowski
Institut National des Télécommunications,
9 rue Charles Fourier, 91011 Evry, France
ProfessorEricTaillard

EIVD, Ecole d’Ingénieurs du Canton de Vaud
routed eCheseaux 1,1400Yverdon-les-Bains,Switzerland
Translator:AmitavaChatterjee
OriginallypublishedinFrenchbyEyrolles,Paris(2003)underthetitle:
“Me´taheuristiquespourl’optimisationdifficile"
Book coordinated by Patrick Siarry
LibraryofCongressControlNumber:2005930496
ISBN-103-540-23022-XSpringerBerlinHeidelbergNewYork
ISBN-13978-3-540-23022-9SpringerBerlinHeidelbergNewYork
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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 simulated annealing method;
• the tabu search;
• the genetic and evolutionary algorithms;
• the 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 explo-
ration of regions of the search space, rarely visited until now, and the use of
intensification, to go thoroughly into some promising regions. Another com-
mon 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; distrib-
uted search; Alienor method; particle swarm optimization; estimation of dis-
tribution methods; GRASP method; cross-entropy method; artificial immune
systems; differential evolution.
Then we describe some extensions of metaheuristics for continuous op-
timization, multimodal optimization, multiobjective optimization and con-
trainedevolutionaryoptimization.Wepresentsomeoftheexistingtechniques
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
MemoryProgramming”,whichtendstoattenuatethedifficultyofthischoice.
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 net-
work; an efficient method using a genetic algorithm is presented and illus-
trated through one example of a realistic network;
• theapplicationofgeneticalgorithmstotheproblemsofmanagementofthe
air traffic. One details two problems of air traffic management for which
a genetic algorithm based solution has been proposed: the first applica-
tion 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
obtainheuristicswhichcanbesimultaneouslyrobustandgenericinnature.
Oneappendixofthebookisdevotedtothemodelingofsimulatedanneal-
ing 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 Johann Dr´eo
September 2005 Patrick Siarry
Alain P´etrowski
Eric Taillard
Contents
Introduction................................................... 1
Part I Presentation of the Main Metaheuristics
1 Simulated Annealing....................................... 23
1.1 Introduction ............................................ 23
1.2 Presentation of the method ............................... 24
1.2.1 Analogy between an optimization problem and some
physical phenomena ............................... 24
1.2.2 Real annealing and simulated annealing .............. 25
1.2.3 Simulated annealing algorithm ...................... 25
1.3 Theoretical approaches................................... 27
1.3.1 Theoretical convergence of simulated annealing........ 27
1.3.2 Configuration space................................ 28
1.3.3 Rules of acceptance ............................... 30
1.3.4 Annealing scheme ................................. 30
1.4 Parallelization of the simulated annealing algorithm.......... 32
1.5 Some applications ....................................... 35
1.5.1 Benchmark problems of combinatorial optimization .... 35
1.5.2 Layout of electronic circuits ........................ 36
1.5.3 Search for an equivalent schema in electronics......... 40
1.5.4 Practical applications in various fields................ 42
1.6 Advantages and disadvantages of the method ............... 44
1.7 Simple practical suggestions for the beginners ............... 44
1.8 Annotated bibliography .................................. 45
2 Tabu Search ............................................... 47
2.1 Introduction ............................................ 47
2.2 The quadratic assignment problem......................... 49
2.3 Basic tabu search........................................ 51
VIII Contents
2.3.1 Neighborhood..................................... 51
2.3.2 Moves, neighborhood .............................. 52
2.3.3 Evaluation of the neighborhood ..................... 54
2.4 Candidate list........................................... 56
2.5 Short-term memory...................................... 57
2.5.1 Hashing tables .................................... 57
2.5.2 Tabu list ......................................... 59
2.5.3 Duration of tabu conditions......................... 60
2.5.4 Aspiration conditions .............................. 66
2.6 Convergence of tabu search ............................... 66
2.7 Long-term memory ...................................... 69
2.7.1 Frequency-based memory........................... 69
2.7.2 Obligation to carry out move ....................... 71
2.8 Strategic oscillations ..................................... 72
2.9 Conclusion ............................................. 72
2.10 Annotated bibliography .................................. 72
3 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
4 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
5 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 “Cross-Entropy” method ................................170
5.10 Artificial immune systems ................................172
5.11 Method of differential evolution ...........................173
5.12 Algorithms inspired by the social insects ...................175
5.13 Annotated bibliography ..................................176
6 Extensions.................................................179
6.1 Introduction ............................................179
6.2 Adaptation for the continuous variable problems ............179
6.2.1 Generalframeworkof“difficult”continuousoptimization179
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
7 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
8 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
9 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