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Table of Contents Foreword 13 Introduction 17 PART ?. Particle Swarm Optimization 21 Chapter 1. What Is a Difficult Problem? 23 1.1. An intrinsic definition 23 1.2. Estimation and practical measurement 25 1.3. For "amatheurs": some estimates of difficulty 26 1.3.1. Function 27 1.3.2. Function 27 1.3.3. Function 27 1.3.4. Traveling salesman on D cities 28 1.4. Summary 28 Chapter 2. On a Table Corner 29 2.1. Apiarian metaphor 29 2.2. An aside on the spreading of a rumor 30 2.3. Abstract formulation 30 2.4. What is really transmitted 34 2.5. Cooperation versus competition 35 2.6. For "amatheurs": a simple calculation of propagation of rumor 35 2.7. Summary 36 Chapter 3. First Formulations 37 3.1. Minimal version 37 3.1.1. Swarm size 37 3.1.2. Information links 38 3.1.3. Initialization 38 3.1.4. Equations of motion 39 3.1.5. Interval confinement 40 3.1.6. Proximity distributions 42 3.2. Two common errors 44 3.3. Principal drawbacks of this formulation 45 3.3.1. Distribution bias 45 3.3.2. Explosion and maximum velocity 48 3.4. Manual parameter setting 48 3.5. For "amatheurs": average number of informants 49 3.6. Summary 50 Chapter 4. Benchmark Set 51 4.1. What is the purpose of test functions? 51 4.2. Six reference functions 52 4.3. Representations and comments 52 4.4. For "amatheurs": estimates of levels of difficulty 56 4.4.1. Theoretical difficulty 56 4.4.1.1. Tripod 56 4.4.1.2. Alpine 10D 57 4.4.1.3. Rosenbrock 57 4.4.2. Difficulty according to the search effort 58 4.5. Summary 58 Chapter 5. Mistrusting Chance 59 5.1. Analysis of an anomaly 59 5.2. Computing randomness 61 5.3. Reproducibility 61 5.4. On numerical precision 62 5.5. The rare KISS 62 5.5.1. Brief description 63 5.5.2. Test of KISS 64 5.6. On the comparison of results 64 5.7. For "amatheurs": confidence in the estimate of a rate of failure 65 5.8. C programs 68 5.9. Summary 70 Chapter 6. First Results 71 6.1. A simple program 71 6.2. Overall results 71 6.3. Robustness and performance maps 73 6.4. Theoretical difficulty and noted difficulty 80 6.5. Source code of OEP 0 80 6.6. Summary 85 Chapter 7. Swarm: Memory and Graphs of Influence 87 7.1. Circular neighborhood of the historical PSO 87 7.2. Memory-swarm 88 7.3. Fixed topologies 90 7.4. Random variable topologies 92 7.4.1. Direct recruitment 92 7.4.2. Recruitment by common channel of communication 92 7.5. Influence of the number of informants 93 7.5.1. In fixed topology 93 7.5.2. In random variable topology 95 7.6. Influence of the number of memories 95 7.7. Reorganizations of the memory-swarm 97 7.7.1. Mixing of the memories 97 7.7.2. Queen and other centroids 98 7.7.3. Comparative results 98 7.8. For "amatheurs": temporal connectivity in random recruitment 99 7.9. Summary 101 Chapter 8. Distributions of Proximity 103 8.1. The random possibilities 103 8.2. Review of rectangular distribution 104 8.3. Alternatives distributions of possibles 105 8.3.1. Ellipsoidal positive sectors 105 8.3.2. Independent gaussians 106 8.3.3. Local by independent Gaussians 107 8.3.4. The class of one-dimensional distributions 107 8.3.5. Pivots 108 8.3.6. Adjusted ellipsoids 112 8.4. Some comparisons of results 113 8.5. For "amatheurs" 116 8.5.1. Squaring of a hypersphere 116 8.5.2. From sphere to ellipsoid 117 8.5.3. Random volume for an adjusted ellipsoid 117 8.5.4. Uniform distribution in a D-sphere 118 8.6. C program of isotropic distribution 118 8.7. Summary 119 Chapter 9. Optimal Parameter Settings 121 9.1. Defense of manual parameter setting 121 9.2. Better parameter settings for the benchmark set 122 9.2.1. Search space 122 9.2.2. To optimize the optimizer 123 9.2.3. Analysis of results 125 9.2.3.1. Rate of failure 125 9.2.3.2. Distribution 125 9.2.3.3. Topology and the number of informants 125 9.2.3.4. Informants K 125 9.2.3.5. Coefficient??? 126 9.2.3.6. Informants N and memories M 126 9.3. Towards adaptation 127 9.4. For "amatheurs": number of graphs of information 127 9.5. Summary 128 Chapter 10. Adaptations 129 10.1. Demanding criteria 129 10.1.1. Criterion 1 129 10.1.2. Criterion 2 129 10.2. Rough sketches 130 10.2.1. Weighting with temporal decrease 130 10.2.2. Selection and replacement 131 10.2.3. Parametric adaptations 132 10.2.4. Nonparametric adaptations 133 10.3. For "amatheurs" 135 10.3.1. Formulas of temporal decrease 135 10.3.2. Parametric adaptations 136 10.3.2.1. Case 1 ( ) 137 10.3.2.2. Case 2 ( ) 137 10.4. Summary 138 Chapter 11. TRIBES or co-operatin of Tribes 139 11.1. Towards an ultimate program 139 11.2. Description of TRIBES 141 11.2.1. Tribes 141 11.2.2. The tribal relationships 141 11.2.3. Quality of a particle 141 11.2.4. Quality of a tribe 142 11.2.5. Evolution of the tribes 142 11.2.5.1. Removal of a particle 142 11.2.5.2. Generation of a particle 144 11.2.6. Strategies of displacement 145 11.2.7. Best informant 146 11.2.7.1. Direct comparison, general case 147 11.2.7.2. Comparison by pseudo-gradients, metric spaces 147 11.3. Results of the benchmark set 147 11.4. Summary 149 Chapter 12. On the Constraints 151 12.1. Some preliminary reflections 151 12.2. Representation of the constraints 152 12.3. Imperative constraints and indicative constraints 153 12.4. Interval confinement 154 12.5. Discrete variable 154 12.5.1. Direct method 155 12.5.1.1. List not ordered (and not orderable) 155 12.5.1.2. Ordered list 155 12.5.2. Indirect method 155 12.6. Granularity confinement 156 12.7. "All different" confinement 156 12.8. Confinement by dichotomy 157 12.9. Multicriterion treatment 158 12.10. Treatment by penalties 161 12.11. C source code. Dichotomic search in a list 162 12.12. For "amatheurs" 162 12.13. Summary 165 Chapter 13. Problems and Application 167 13.1. Ecological niche 167 13.2. Typology and choice of problems 168 13.3. Canonical representation of a problem of optimization 169 13.4. Knapsack 169 13.5. Magic squares 170 13.6. Quadratic assignment 171 13.7. Traveling salesman 172 13.8. Hybrid JM 173 13.9. Training of a neural network 174 13.9.1. Exclusive OR 175 13.9.2. Diabetes among Pima Indians 176 13.9.3. Servomechanism 176 13.9.4. Comparisons 176 13.10. Pressure vessel 177 13.10.1. Continuous relaxed form 179 13.10.2. Complete discrete form 180 13.11. Compression spring 182 13.12. Moving peaks 185 13.13. For "amatheurs": the magic of squares 188 13.14. Summary 188 chapter 14. Conclusion 189 14.1. End of the beginning 189 14.2. Mono, poly, meta 189 14.3. The beginning of the end? 190 PART ??. Outlines 193 Chapter 15. On Parallelism 195 15.1. The short-sighted swarm 195 15.2. A parallel model 195 15.3. A counter-intuitive result 196 15.4. Qualitative explanation 197 15.5. For "amatheurs": probability of questioning an improved memory 198 15.6. Summary 199 Chapter 16. Combinatorial Problems 201 16.1. Difficulty of chaos 201 16.2. Like a crystal 202 16.3. Confinement method 203 16.4. Canonical PSO 204 16.5. Summary 210 Chapter 17. Dynamics of a Swarm 211 17.1. Motivations and tools 211 17.2. An example with the magnifying glass 212 17.2.1. One particle 212 17.2.2. Two particles 214 17.3. Energies 217 17.3.1. Definitions 217 17.3.2. Evolutions 218 17.4. For experienced "amatheurs": convergence and constriction 220 17.4.1. Criterion of convergence 220 17.4.2. Coefficients of constriction 221 17.4.3. Positive discriminant 222 17.5. Summary 223 Chapter 18. Techniques and Alternatives 225 18.1. Reprise 225 18.2. Stop-restart/reset 226 18.2.1. A criterion of abandonment 226 18.2.2. Guided re-initialization 227 18.3. Multi-swarm 227 18.4. Dynamic optimization 228 18.5. For "amatheurs" 229 18.5.1. Maximum flight and criterion of abandonment 229 18.5.2. Dilation 230 18.6. Summary 230 To Further Information 231 Bibliography 233 Index 239
Library of Congress Subject Headings for this publication:
Mathematical optimization.
Particles (Nuclear physics).
Swarm intelligence.