This book presents the first comprehensive account of fast sequential Monte Carlo (SMC) methods for counting and optimization at an exceptionally accessible level. Written by authorities in the field, it places great emphasis on cross-entropy, minimum cross-entropy, splitting, and stochastic enumeration. The overall aim is to make SMC methods accessible to readers who want to apply and to accentuate the unifying and novel mathematical ideas behind SMC in their future studies or work.
Preface xi 1. Introduction to Monte Carlo Methods 1 2. Cross-Entropy Method 6 2.1. Introduction 6 2.2. Estimation of Rare-Event Probabilities 7 2.3. Cross-Entrophy Method for Optimization 18 2.4. Continuous Optimization 31 2.5. Noisy Optimization 33 3. Minimum Cross-Entropy Method 37 3.1. Introduction 37 3.2. Classic MinxEnt Method 39 3.3. Rare Events and MinxEnt 43 3.4. Indicator MinxEnt Method 47 3.5. IME Method for Combinatorial Optimization 52 4. Splitting Method for Counting and Optimization 56 4.1. Background 56 4.2. Quick Glance at the Splitting Method 58 4.3. Splitting Algorithm with Fixed Levels 64 4.4. Adaptive Splitting Algorithm 68 4.5. Sampling Uniformly on Discrete Regions 74 4.6. Splitting Algorithm for Combinatorial Optimization 75 4.7. Enhanced Splitting Method for Counting 76 4.8. Application of Splitting to Reliability Models 79 4.9. Numerical Results with the Splitting Algorithms 86 4.10. Appendix: Gibbs Sampler 104 5. Stochastic Enumeration Method 106 5.1. Introduction 106 5.2. OSLA Method and Its Extensions 110 5.3. SE Method 120 5.4. Applications of SE 127 5.5. Numerical Results 136 A. Additional Topics 148 A.1. Combinatorial Problems 148 A.1.1. Counting 149 A.1.2. Combinatorial Optimization 154 A.2. Information 162 A.2.1. Shannon Entropy 162 A.2.2. Kullback--Leibler Cross-Entropy 163 A.3. Efficiency of Estimators 164 A.3.1. Complexity 165 A.3.2. Complexity of Randomized Algorithms 166 Bibliography 169 Abbreviations and Acronyms 177 List of Symbols 178 Index 181
