OPTIMIZATION EVOLUTIONARY ALGORITHMS. Biologically-Inspired and. Computer Intelligence. Wiley. Population-Based Approaches to.

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1 EVOLUTIONARY OPTIMIZATION ALGORITHMS Biologically-Inspired and Population-Based Approaches to Computer Intelligence Dan Simon Cleveland State University Wiley

2 DETAILED TABLE OF CONTENTS Acknowledgments Acronyms List of Algorithms xxi xxiii xxvii PART I INTRODUCTION TO EVOLUTIONARY OPTIMIZATION 1 Introduction Terminology Why Another Book on Evolutionary Algorithms? Prerequisites Homework Problems Notation Outline of the Book A Course Based on This Book 8 2 Optimization Unconstrained Optimization Constrained Optimization Multi-Objective Optimization Multimodal Optimization Combinatorial Optimization 20 vii

3 viii DETAILED TABLE OF CONTENTS 2.6 Hill Climbing Biased Optimization Algorithms The Importance of Monte Carlo Simulations Intelligence Adaptation Randomness Communication Feedback Exploration and Exploitation Conclusion 29 Problems 30 PART II CLASSIC EVOLUTIONARY ALGORITHMS 3 Genetic Algorithms The History of Genetics Charles Darwin Gregor Mendel The Science of Genetics The History of Genetic Algorithms A Simple Binary Genetic Algorithm A Genetic Algorithm for Robot Design Selection and Crossover Mutation GA Summary GA Tuning Parameters and Examples A Simple Continuous Genetic Algorithm Conclusion 59 Problems 60 4 Mathematical Models of Genetic Algorithms Schema Theory Markov Chains Markov Model Notation for Evolutionary Algorithms Markov Models of Genetic Algorithms Selection Mutation Crossover Dynamic System Models of Genetic Algorithms Selection Mutation Crossover 87

4 DETAILED TABLE OF CONTENTS ix 4.6 Conclusion 92 Problems 93 5 Evolutionary Programming Continuous Evolutionary Programming Finite State Machine Optimization Discrete Evolutionary Programming The Prisoner's Dilemma The Artificial Ant Problem Conclusion 113 Problems Evolution Strategies The (1+1) Evolution Strategy The 1/5 Rule: A Derivation The (p+1) Evolution Strategy (// + A) and (n, A) Evolution Strategies Self-Adaptive Evolution Strategies Conclusion 136 Problems Genetic Programming Lisp: The Language of Genetic Programming The Fundamentals of Genetic Programming Fitness Measure Termination Criteria Terminal Set Function Set Initialization Genetic Programming Parameters Genetic Programming for Minimum Time Control Genetic Programming Bloat Evolving Entities other than Computer Programs Mathematical Analysis of Genetic Programming Definitions and Notation Selection and Crossover Mutation and Final Results Conclusion 173 Problems 175

5 X DETAILED TABLE OF CONTENTS 8 Evolutionary Algorithm Variations Initialization Convergence Criteria Problem Representation Using Gray Coding Elitism Steady-State and Generational Algorithms Population Diversity Duplicate Individuals Niche-Based and Species-Based Recombination Niching Selection Options Stochastic Universal Sampling Over-Selection Sigma Scaling Rank-Based Selection Linear Ranking Tournament Selection Stud Evolutionary Algorithms Recombination Single-Point Crossover (Binary or Continuous EAs) Multiple-Point Crossover (Binary or Continuous EAs) Segmented Crossover (Binary or Continuous EAs) Uniform Crossover (Binary or Continuous EAs) Multi-Parent Crossover (Binary or Continuous EAs) Global Uniform Crossover (Binary or Continuous EAs) Shuffle Crossover (Binary or Continuous EAs) Flat Crossover and Arithmetic Crossover (Continuous EAs) Blended Crossover (Continuous EAs) Linear Crossover (Continuous EAs) Simulated Binary Crossover (Continuous EAs) Summary Mutation Uniform Mutation Centered at Xi(fc) Uniform Mutation Centered at the Middle of the Search Domain Gaussian Mutation Centered at Xj(fc) Gaussian Mutation Centered at the Middle of the Search Domain Conclusion 215 Problems 217

6 DETAILED TABLE OF CONTENTS xi PART III MORE RECENT EVOLUTIONARY ALGORITHMS 9 Simulated Annealing Annealing in Nature A Simple Simulated Annealing Algorithm Cooling Schedules Linear Cooling Exponential Cooling Inverse Cooling Logarithmic Cooling Inverse Linear Cooling Dimension-Dependent Cooling Implementation Issues Candidate Solution Generation Reinitialization Keeping Track of the Best Candidate Solution Conclusion 238 Problems Ant Colony Optimization Pheromone Models Ant System Continuous Optimization Other Ant Systems Max-Min Ant System Ant Colony System Even More Ant Systems Theoretical Results Conclusion 262 Problems Particle Swarm Optimization A Basic Particle Swarm Optimization Algorithm Velocity Limiting Inertia Weighting and Constriction Coefficients Inertia Weighting The Constriction Coefficient PSO Stability Global Velocity Updates The Fully Informed Particle Swarm Learning from Mistakes 285

7 Xli DETAILED TABLE OF CONTENTS 11.7 Conclusion 288 Problems Differential Evolution A Basic Differential Evolution Algorithm Differential Evolution Variations Trial Vectors Mutant Vectors Scale Factor Adjustment Discrete Optimization Mixed-Integer Differential Evolution Discrete Differential Evolution Differential Evolution and Genetic Algorithms Conclusion 309 Problems Estimation of Distribution Algorithms Estimation of Distribution Algorithms: Basic Concepts A Simple Estimation of Distribution Algorithm Computations of Statistics First-Order Estimation of Distribution Algorithms The Univariate Marginal Distribution Algorithm (UMDA) The Compact Genetic Algorithm (cga) Population Based Incremental Learning (PBIL) Second-Order Estimation of Distribution Algorithms Mutual Information Maximization for Input Clustering (MIMIC) Combining Optimizers with Mutual Information Trees (COMIT) The Bivariate Marginal Distribution Algorithm (BMDA) Multivariate Estimation of Distribution Algorithms The Extended Compact Genetic Algorithm (ECGA) Other Multivariate Estimation of Distribution Algorithms Continuous Estimation of Distribution Algorithms The Continuous Univariate Marginal Distribution Algorithm Continuous Population Based Incremental Learning Conclusion 347 Problems 348

8 DETAILED TABLE OF CONTENTS xiii 14 Biogeography-Based Optimization Biogeography Biogeography is an Optimization Process Biogeography-Based Optimization BBO Extensions Migration Curves Blended Migration Other Approaches to BBO BBO and Genetic Algorithms Conclusion 370 Problems Cultural Algorithms Cooperation and Competition Belief Spaces in Cultural Algorithms Cultural Evolutionary Programming The Adaptive Culture Model Conclusion 393 Problems Opposition-Based Learning Opposition Definitions and Concepts Reflected Opposites and Modulo Opposites Partial Opposites Type 1 Opposites and Type 2 Opposites Quasi Opposites and Super Opposites Opposition-Based Evolutionary Algorithms Opposition Probabilities Jumping Ratio Oppositional Combinatorial Optimization Dual Learning Conclusion 417 Problems Other Evolutionary Algorithms Tabu Search Artificial Fish Swarm Algorithm Random Behavior Chasing Behavior Swarming Behavior Searching Behavior 425

9 xiv DETAILED TABLE OF CONTENTS Leaping Behavior A Summary of the Artificial Fish Swarm Algorithm Group Search Optimizer Shuffled Frog Leaping Algorithm The Firefly Algorithm Bacterial Foraging Optimization Artificial Bee Colony Algorithm Gravitational Search Algorithm Harmony Search Teaching-Learning-Based Optimization Conclusion 444 Problems 446 PART IV SPECIAL TYPES OF OPTIMIZATION PROBLEMS 18 Combinatorial Optimization The Traveling Salesman Problem TSP Initialization Nearest-Neighbor Initialization Shortest-Edge Initialization Insertion Initialization Stochastic Initialization TSP Representations and Crossover Path Representation Adjacency Representation Ordinal Representation Matrix Representation TSP Mutation Inversion Insertion Displacement Reciprocal Exchange An Evolutionary Algorithm for the Traveling Salesman Problem The Graph Coloring Problem Conclusion 477 Problems Constrained Optimization Penalty Function Approaches Interior Point Methods Exterior Methods 485

10 DETAILED TABLE OF CONTENTS XV 19.2 Popular Constraint-Handling Methods Static Penalty Methods Superiority of Feasible Points The Eclectic Evolutionary Algorithm Co-evolutionary Penalties Dynamic Penalty Methods Adaptive Penalty Methods Segregated Genetic Algorithm Self-Adaptive Fitness Formulation Self-Adaptive Penalty Function Adaptive Segregational Constraint Handling Behavioral Memory Stochastic Ranking The Niched-Penalty Approach Special Representations and Special Operators Special Representations Special Operators Genocop Genocop II Genocop III Other Approaches to Constrained Optimization Cultural Algorithms Multi-Objective Optimization Ranking Candidate Solutions Maximum Constraint Violation Ranking Constraint Order Ranking e-level Comparisons A Comparison Between Constraint-Handling Methods Conclusion 511 Problems Multi-Objective Optimization Pareto Optimality The Goals of Multi-Objective Optimization Hypervolume Relative Coverage Non-Pareto-Based Evolutionary Algorithms Aggregation Methods The Vector Evaluated Genetic Algorithm (VEGA) Lexicographic Ordering The e-constraint Method Gender-Based Approaches 534

11 xvi DETAILED TABLE OF CONTENTS 20.4 Pareto-Based Evolutionary Algorithms Evolutionary Multi-Objective Optimizers The e-based Multi-Objective Evolutionary Algorithm (e-moea) The Nondominated Sorting Genetic Algorithm (NSGA) The Multi-Objective Genetic Algorithm (MOGA) The Niched Pareto Genetic Algorithm (NPGA) The Strength Pareto Evolutionary Algorithm (SPEA) The Pareto Archived Evolution Strategy (PAES) Multi-Objective Biogeography-Based Optimization Vector Evaluated BBO Nondominated Sorting BBO Niched Pareto BBO Strength Pareto BBO Multi-Objective BBO Simulations Conclusion 556 Problems Expensive, Noisy, and Dynamic Fitness Functions Expensive Fitness Functions Fitness Function Approximation Approximating Transformed Functions How to Use Fitness Approximations in Evolutionary Algorithms Multiple Models Overfitting Evaluating Approximation Methods Dynamic Fitness Functions The Predictive Evolutionary Algorithm Immigrant Schemes Memory-Based Approaches Evaluating Dynamic Optimization Performance Noisy Fitness Functions Resampling Fitness Estimation The Kalman Evolutionary Algorithm Conclusion 600 Problems 603

12 DETAILED TABLE OF CONTENTS xvii PART V APPENDICES Appendix A: Some Practical Advice 607 A.l Check for Bugs 607 A.2 Evolutionary Algorithms are Stochastic 608 A.3 Small Changes can have Big Effects 608 A.4 Big changes can have Small Effects 609 A.5 Populations Have Lots of Information 609 A.6 Encourage Diversity 609 A.7 Use Problem-Specific Information 609 A.8 Save your Results Often 610 A.9 Understand Statistical Significance 610 A.10 Write Well 610 A.ll Emphasize Theory 610 A. 12 Emphasize Practice 611 Appendix B: The No Free Lunch Theorem and Performance Testing 613 B. l The No Free Lunch Theorem 614 B.2 Performance Testing 621 B.2.1 Overstatements Based on Simulation Results 621 B.2.2 How to Report (and How Not to Report) Simulation Results 623 B.2.3 Random Numbers 628 B.2.4 T-Tests 631 B. 2.5 F-Tests 636 B. 3 Conclusion 640 Appendix C: Benchmark Optimization Functions 641 C. l Unconstrained Benchmarks 642 C. l.l The Sphere Function 642 C.1.2 The Ackley Function 643 C.l.3 The Ackley Test Function 644 C.l.4 The Rosenbrock Function 644 C.1.5 The Fletcher Function 645 C.l.6 The Griewank Function 646 C.1.7 The Penalty #1 C.l.8 The Penalty #2 Function 647 Function 647 C.l.9 The Quartic Function 648 C.l. 10 The Tenth Power Function 649 C.l. 11 The Rastrigin Function 650 C.l. 12 The Schwefel Double Sum Function 650 C.1.13 The Schwefel Max Function 651

13 XVlii DETAILED TABLE OF CONTENTS C.1.14 The Schwefel Absolute Function 652 C.1.15 The Schwefel Sine Function 652 C.1.16 The Step Function 653 C.1.17 The Absolute Function 654 C.1.18 Shekel's Foxhole Function 654 C.1.19 The Michalewicz Function 655 C.1.20 The Sine Envelope Function 655 C.1.21 The Eggholder Function 656 C.1.22 The Weierstrass Function 657 Constrained Benchmarks 657 C.2.1 The C01 Function 658 C.2.2 The C02 Function 658 C.2.3 The C03 Function 659 C.2.4 The C04 Function 659 C.2.5 The C05 Function 659 C.2.6 The C06 Function 660 C.2.7 The C07 Function 660 C.2.8 The C08 Function 660 C.2.9 The C09 Function 661 C.2.10 The CIO Function 661 C.2.11 The Cll Function 661 C.2.12 The C12 Function 662 C.2.13 The C13 Function 662 C.2.14 The C14 Function 662 C.2.15 The C15 Function 663 C.2.16 The C16 Function 663 C.2.17 The C17 Function 664 C.2.18 The C18 Function 664 C.2.19 Summary of Constrained Benchmarks 664 Multi-Objective Benchmarks 665 C.3.1 Unconstrained Multi-Objective Optimization Problem C.3.2 Unconstrained Multi-Objective Optimization Problem C.3.3 Unconstrained Multi-Objective Optimization Problem C.3.4 Unconstrained Multi-Objective Optimization Problem C.3.5 Unconstrained Multi-Objective Optimization Problem C.3.6 Unconstrained Multi-Objective Optimization Problem C.3.7 Unconstrained Multi-Objective Optimization Problem C.3.8 Unconstrained Multi-Objective Optimization Problem C.3.9 Unconstrained Multi-Objective Optimization Problem C.3.10 Unconstrained Multi-Objective Optimization Problem

14 DETAILED TABLE OF CONTENTS xix C.4 Dynamic Benchmarks 672 C.4.1 The Complete Dynamic Benchmark Description 672 C.4.2 A Simplified Dynamic Benchmark Description 677 C.5 Noisy Benchmarks 677 C.6 Traveling Salesman Problems 678 C.7 Unbiasing the Search Space 680 C.7.1 Offsets 681 C.7.2 Rotation Matrices 682 References 685 Topic Index 727

ARTIFICIAL INTELLIGENCE (CSCU9YE ) LECTURE 5: EVOLUTIONARY ALGORITHMS

ARTIFICIAL INTELLIGENCE (CSCU9YE ) LECTURE 5: EVOLUTIONARY ALGORITHMS Gabriela Ochoa http://www.cs.stir.ac.uk/~goc/ OUTLINE Optimisation problems Optimisation & search Two Examples The knapsack problem