ISBN: 3790814962
TITLE: Representationsl for Genetic and Evolutionary Algorithms
AUTHOR: Rothlauf
TOC:

1. Introduction 1
1.1 Purpose 2
1.2 Organization 4
2. Representations for Genetic and Evolutionary Algorithms 9
2.1 Genetic Representations 10
2.1.1 Genotypes and Phenotypes 10
2.1.2 Decomposition of the Fitness Function 11
2.1.3 Types of Representations 13
2.2 Genetic and Evolutionary Algorithms 15
2.2.1 Principles 15
2.2.2 Functionality 16
2.2.3 Schema Theorem and Building Block Hypothesis 19
2.3 Problem Difficulty 22
2.3.1 Reasons for Problem Difficulty 22
2.3.2 Measurements of Problem Difficulty 25
2.4 Existing Recommendations for the Design of Efficient Representations for Genetic and Evolutionary Algorithms 28
2.4.1 Goldberg's Meaningful Building Blocks and Minimal Alphabets 29
2.4.2 Palmer's Tree Encoding Issues 29
2.4.3 Ronald's Representational Redundancy 30
3 Three Elements of a Theory of Genetic and Evolutionary Representations 31
3.1 Redundancy 33
3.1.1 Definitions and Background 33
3.1.2 Decomposing Redundancy 36
3.1.3 Population Sizing 37
3.1.4 Run Duration and Overall Problem Complexity 39
3.1.5 Empirical Results 40
3.1.6 Conclusions, Restrictions and Further Research 44
3.2 Building Block-Scaling 45
3.2.1 Background 46
3.2.2 Domino Model without Genetic Drift 47
3.2.3 Population Sizing for Domino Model and Genetic Drift 50
3.2.4 Empirical Results 53
3.2.5 Conclusions 56
3.3 Distance Distortion 57
3.3.1 Influence of Representations on Problem Difficulty 59
3.3.2 Locality and Distance Distortion 61
3.3.3 Modifying BB-Complexity for the One-Max Problem 63
3.3.4 Empirical Results 67
3.3.5 Conclusions 71
3.4 Summary and Conclusions 73
4. Time-Quality Framework for a Theory-Based Analysis and Design of Representations 77
4.1 Solution Quality and Time to Convergence 78
4.2 Elements of the Framework 79
4.2.1 Redundancy 79
4.2.2 Scaling 80
4.2.3 Distance Distortion 81
4.3 The Framework 84
4.3.1 Uniformly Scaled Representations 85
4.3.2 Exponentially Scaled Representations 86
4.4 Implications for the Design of Representations 89
4.4.1 Uniformly Redundant Representations Are Robust 90
4.4.2 Exponentially Scaled Representations Are Fast, but Inaccurate 92
4.4.3 BB-Modifying Representations Are Difficult to Predict 94
4.5 Summary and Conclusions 96
5. Analysis of Binary Representations of Integers 99
5.1 Two Integer Optimization Problems 100
5.2 Binary String Representations 101
5.3 A Theoretical Comparison 105
5.3.1 Redundancy and the Unary Encoding 105
5.3.2 Scaling, Modification of Problem Difficulty, and the Binary Encoding 107
5.3.3 Modification of Problem Difficulty and the Gray Encoding 108
5.4 Empirical Results 111
5.5 Conclusions 116
6. Analysis of Tree Representations 119
6.1 The Tree Design Problem 120
6.1.1 Definition 120
6.1.2 Metrics and Distances 122
6.1.3 Tree Structures 123
6.1.4 Schema Analysis for Graphs 124
6.1.5 Scalable Test Problems for Graphs 125
6.1.6 Tree Encoding Issues 128
6.2 Prfer Numbers 130
6.2.1 Historical Review 130
6.2.2 Construction 132
6.2.3 Properties 134
6.2.4 The Low Locality of the Prfer Number Encoding 136
6.2.5 Summary and Conclusions 148
6.3 The Link and Node Biased Encoding 149
6.3.1 Introduction 150
6.3.2 Motivation and Functionality 151
6.3.3 Biased Initial Populations and Non-Uniformly Redundant Encodings 153
6.3.4 The Node-Biased Encoding 155
6.3.5 The Link-and-Node-Biased Encoding 159
6.3.6 Empirical Results 162
6.3.7 Conclusions 165
6.4 The Characteristic Vector Encoding 166
6.4.1 Encoding Trees with the Characteristic Vector 167
6.4.2 Repairing Invalid Solutions 168
6.4.3 Bias and Stealth Mutation 169
6.4.4 Summary 173
6.5 Conclusions 174
7. Design of Tree Representations 177
7.1 Network Random Keys (NetKeys) 178
7.1.1 Motivation 178
7.1.2 Functionality 179
7.1.3 Advantages 183
7.1.4 Bias 185
7.1.5 Population Sizing and Run Duration for the One-Max Tree Problem 187
7.1.6 Conclusions 189
7.2 A Direct Tree Representation (NetDir) 190
7.2.1 Historical Review 191
7.2.2 Properties of Direct Representations 191
7.2.3 Operators for NetDir 193
7.2.4 Summary 196
8. Performance of Genetic and Evolutionary Algorithms on Tree Problems 199
8.1 GEA Performance on Scalable Test Tree Problems 200
8.1.1 Analysis of Representations 200
8.1.2 One-Max Tree Problem 202
8.1.3 Deceptive Tree Problem 210
8.2 GEA Performance on the Optimal Communication Spanning Tree Problem 215
8.2.1 Problem Definition 216
8.2.2 Theoretical Predictions 216
8.2.3 Palmer's Test Instances 217
8.2.4 Raidl's Test Instances 221
8.2.5 Test Instances from Berry, Murtagh, and McMahon (1995) 225
8.2.6 Selected Real-World Test Instances 229
8.3 Summary 235
9. Summary, Conclusions and Future Work 237
9.1 Summary 237
9.2 Conclusions 239
9.3 Future Work 242
A. Optimal Communication Spanning Tree Test Instances 245
A.1 Palmer's Test Instances 245
A.2 Raidl's Test Instances 250
A.3 Berry's Test Instances 254
A.4 Real World Problems 256
References 263
List of Symbols 281
List of Acronyms 285
Index 287
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