Table Of ContentSPRINGER BRIEFS IN APPLIED SCIENCES AND
TECHNOLOGY COMPUTATIONAL INTELLIGENCE
Oliver Kramer
A Brief
Introduction
to Continuous
Evolutionary
Optimization
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Oliver Kramer
A Brief Introduction to
Continuous Evolutionary
Optimization
123
Oliver Kramer
Department fürInformatik
CarlvonOssietzkyUniversityofOldenburg
Oldenburg
Germany
ISSN 2191-530X ISSN 2191-5318 (electronic)
ISBN 978-3-319-03421-8 ISBN 978-3-319-03422-5 (eBook)
DOI 10.1007/978-3-319-03422-5
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Acknowledgments
I thank the German Research Foundation (Deutsche Forschungsgemeinschaft,
DFG) and the German Academic Exchange Service (Deutscher Akademischer
Austauschdienst, DAAD) for the support in the last three years. Further, I thank
Günter Rudolph for the support and the discussions about evolutionary compu-
tation, and the International Computer Science Institute (ICSI) in Berkeley for
their excellent support duringmystayinCaliforniain2010,inparticularRichard
Karp and Nelson Morgan. I thank the Bauhaus-Universität Weimar for their
support in 2011. Last, I want to thank all graduate students, postdocs, professors,
and other scientists I have been collaborating with since the beginning of my
research activities. I had numerous interesting discussions and research collabo-
rationswithJörgBremer,FabianGieseke,JustinHeinermann,ChristianHinrichs,
Sascha Hunold, Jörg Lässig, Jendrik Poloczek, Benjamin Satzger, Dirk Sudholt,
and Nils André Treiber.
Cancun, June 2013 Oliver Kramer
v
Contents
Part I Foundations
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Evolutionary Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Hybrid Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Overview of Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Evolution Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Mutation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Covariance Matrix Adaptation Evolution Strategies. . . . . . . . . . 22
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Parameter Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 The (1þ1)-EA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 A Study on Mutation Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Meta-Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Rechenberg’s 1/5th Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Self-Adaptation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
vii
viii Contents
Part II Advanced Optimization
4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Adaptive Penalty Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Experimental Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Meta-Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Iterated Local Search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Iterated Local Search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Powell’s Conjugate Gradient Method. . . . . . . . . . . . . . . . . . . . 46
5.4 Powell Evolution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.5 Experimental Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.6 Perturbation Mechanism and Population Sizes . . . . . . . . . . . . . 52
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6 Multiple Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.3 Non-dominated Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.4 Rake Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.5 Experimental Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.6 Properties and Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Part III Learning
7 Kernel Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.2 Kernel Density Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.3 Kernel Shape Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.4 Experimental Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.5 Local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Contents ix
8 Particle Swarm Embeddings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.3 Iterative Particle Swarm Embeddings. . . . . . . . . . . . . . . . . . . . 78
8.4 Experimental Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Appendix A: Test Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Abstract
Practicaloptimizationproblemsareoftenhardtosolve,inparticularwhentheyare
black boxes and no further information about the problem is available except via
function evaluations. This work introduces a collection of heuristics and algo-
rithms for black box optimization with evolutionary algorithms in continuous
solution spaces. The book gives an introduction to evolution strategies and
parameter control. Heuristic extensions are presented that allow optimization in
constrained, multimodal, and multiobjective solution spaces. An adaptive penalty
function is introduced for constrained optimization. Meta-models reduce the
numberoffitnessandconstraintfunctioncallsinexpensiveoptimizationproblems.
Thehybridizationofevolutionstrategieswithlocalsearchallowsfastoptimization
insolutionspaceswithmanylocaloptima.Aselectionoperatorbasedonreference
lines in objective space is introduced to optimize multiple conflictive objectives.
Evolutionary search is employed for learning kernel parameters of the Nadaraya-
Watson estimator, and a swarm-based iterative approach is presented for opti-
mizinglatentpointsindimensionalityreductionproblems.Experimentsontypical
benchmark problems as well as numerous figures and diagrams illustrate the
behavior of the introduced concepts and methods.
xi
