Stable Mutations for Evolutionary Algorithms

Contents......Page 3
Abbreviations......Page 6
Symbols......Page 7
1 Introduction......Page 9
References......Page 15
2 Foundations of Evolutionary Algorithms......Page 17
2.1 Basic Concepts......Page 18
2.2 Standard Evolutionary Algorithms......Page 20
2.2.1 Genetic Algorithms......Page 21
2.2.2 Genetic Programming......Page 22
2.2.4 Evolutionary Strategies......Page 24
2.2.5 Evolutionary Search with Soft Selection......Page 27
References......Page 29
3.1.1 Definition of Stable Distributions......Page 31
3.1.2 Chosen Properties of Stable Distributions......Page 34
3.1.3 Simulation of α-Stable Random Variables......Page 36
3.1.4 Stable Random Variable and the Evolutionary Algorithm......Page 37
3.2.1 Definition of the Stable Random Vector......Page 41
3.2.2 Selected Properties of the Stable Random Vector......Page 42
3.2.3 Non-isotropic Stable Random Vector......Page 44
3.2.4 Isotropic Distributions......Page 48
3.2.5 Isotropic Distribution Based on an α-Stable Generator......Page 49
3.2.6 Isotropic Stable Random Vector......Page 51
3.3 Summary......Page 55
References......Page 56
4 Non-Isotropic Stable Mutation......Page 57
4.1 Dead Surrounding Effect......Page 58
4.1.1 Dead Surrounding Versus Extremum Localization......Page 59
4.1.2 Saddle Crossing......Page 63
4.1.3 Optimization of Multimodal Functions......Page 68
4.2.1 Local Convergence of (1+1)ESN;α......Page 74
4.2.2 Saddle Crossing......Page 77
4.2.3 Symmetry Effect Versus Global Extremum Searching......Page 79
References......Page 83
5.1.1 Probability Density Function of the Norm  of the Isotropic Stable Vector......Page 85
5.1.2 Soft Selection Versus the Dead Surrounding......Page 88
5.2.1 Progress Rate......Page 89
5.2.2 Convergence Analysis for n=3......Page 91
5.3.1 Local Convergence of the (1+1)ESI;α Strategy......Page 94
5.3.2 Saddle Crossing: The (1,2)ESI;α Strategy......Page 97
5.4 Robustness Analysis of Stable Mutation......Page 99
5.4.1 General Searching Space......Page 100
5.4.2 Isotropic Stable Mutation Effectiveness in mathcalF......Page 101
5.4.3 Simulation Experiments......Page 103
5.5.1 Adaptation Strategy for (1+1)ESI;α......Page 105
References......Page 113
6.1 Stable Distributions with the DSM......Page 115
6.2 Selection of the Optimal Stable Distribution......Page 118
6.3 DSM Adaptation......Page 120
6.3.1 DSM Estimation......Page 123
6.3.2 Simulation Example......Page 124
6.4 Summary......Page 127
References......Page 128
7.1 Dead Surrounding Effect......Page 130
7.2.1 Local Convergence of the (1+1)ESS,α Strategies......Page 132
7.2.2 Local Convergence of the (1+λ)ESS,α Strategies......Page 133
7.3.1 Precision of Local Extremum Localization......Page 135
7.3.2 Saddle Crossing......Page 137
7.4 Global Optimization......Page 138
References......Page 141
8.1 Motivation......Page 142
8.2.1 Von Mises–Fisher Distributions......Page 146
8.3 Simulation Experiments......Page 150
8.3.1 Conducting the Experiment......Page 153
8.3.2 Results......Page 154
References......Page 160
9 Conclusions......Page 162
References......Page 165
Saddle Crossing Problem......Page 166
Benchmark Functions......Page 168