Multi-Objective Optimization using Evolutionary Algorithms

Transcription

1 Multi-Objective Optimization using Evolutionary Algorithms Kalyanmoy Deb Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India JOHN WILEY & SONS, LTD Chichester New York Weinheim Brisbane Singapore Toronto

2 Contents Foreword Preface xv xvii 1 Prologue Single and Multi-Objective Optimization Fundamental Differences Two Approaches to Multi-Objective Optimization Why Evolutionary? Rise of Multi-Objective Evolutionary Algorithms Organization of the Book 9 Exercise Problems 11 2 Multi-Objective Optimization Multi-Objective Optimization Problem Linear and Nonlinear MOOP Convex and Nonconvex MOOP Principles of Multi-Objective Optimization Illustrating Pareto-Optimal Solutions Objectives in Multi-Objective Optimization Non-Conflicting Objectives Difference with Single-Objective Optimization Two Goals Instead of One Dealing with Two Search Spaces No Artificial Fix-Ups Dominance and Pareto-Optimality Special Solutions Concept of Domination Properties of Dominance Relation Pareto-Optimality Strong Dominance and Weak Pareto-Optimality Procedures for Finding a Non-Dominated Set Non-Dominated Sorting of a Population 40

3 viii CONTENTS 2.5 Optimality Conditions Summary 46 Exercise Problems 46 3 Classical Methods Weighted Sum Method Hand Calculations Advantages Disadvantages Difficulties with Nonconvex Problems e-constraint Method Hand Calculations Advantages Disadvantages Weighted Metric Methods Hand Calculations Advantages Disadvantages Rotated Weighted Metric Method Dynamically Changing the Ideal Solution Benson's Method Advantages Disadvantages Value Function Method Advantages Disadvantages Goal Programming Methods Weighted Goal Programming Lexicographic Goal Programming Min-Max Goal Programming Interactive Methods Review of Classical Methods Summary 77 Exercise Problems 78 4 Evolutionary Algorithms Difficulties with Classical Optimization Algorithms Genetic Algorithms Binary Genetic Algorithms Real-Parameter Genetic Algorithms Constraint-Handling in Genetic Algorithms Evolution Strategies Non-Recombinative Evolution Strategies Recombinative Evolution Strategies 136

4 CONTENTS ix Self-Adaptive Evolution Strategies Connection Between Real-Parameter GAs and Self-Adaptive ESs Evolutionary Programming (EP) Genetic Programming (GP) Multi-Modal Function Optimization Diversity Through Mutation Preselection Crowding Model ; Sharing Function Model Ecological GA Other Models Need for Mating Restriction Summary 163 Exercise Problems Non-Elitist Multi-Objective Evolutionary Algorithms Motivation for Finding Multiple Pareto-Optimal Solutions Early Suggestions Example Problems Minimization Example Problem: Min-Ex Maximization Example Problem: Max-Ex Vector Evaluated Genetic Algorithm Hand Calculations Computational Complexity Advantages Disadvantages Simulation Results Non-Dominated Selection Heuristic Mate Selection Heuristic Vector-Optimized Evolution Strategy Advantages and Disadvantages Weight-Based Genetic Algorithm Sharing Function Approach Vector Evaluated Approach Random Weighted GA Multiple Objective Genetic Algorithm Hand Calculations Computational Complexity Advantages Disadvantages Simulation Results Dynamic Update of the Sharing Parameter Non-Dominated Sorting Genetic Algorithm 209

5 x CONTENTS Hand Calculations Computational Complexity Advantages Disadvantages Simulation Results Niched-Pareto Genetic Algorithm Hand Calculations Computational Complexity Advantages Disadvantages Simulation Results Predator-Prey Evolution Strategy Hand Calculations Advantages Disadvantages Simulation Results A Modified Predator-Prey Evolution Strategy Other Methods Distributed Sharing GA Distributed Reinforcement Learning Approach Neighborhood Constrained GA Modified NESSY Algorithm Nash GA Summary 234 Exercise Problems Elitist Multi-Objective Evolutionary Algorithms Rudolph's Elitist Multi-Objective Evolutionary Algorithm Hand Calculations Computational Complexity Advantages Disadvantages Elitist Non-Dominated Sorting Genetic Algorithm Crowded Tournament Selection Operator Hand Calculations Computational Complexity Advantages Disadvantages Simulation Results Distance-Based Pareto Genetic Algorithm Hand Calculations Computational Complexity Advantages 258

6 6.3.4 Disadvantages Simulation Results Strength Pareto Evolutionary Algorithm Clustering Algorithm Hand Calculations Computational Complexity Advantages Disadvantages Simulation Results Thermodynamical Genetic Algorithm Computational Complexity Advantages and Disadvantages Pareto-Archived Evolution Strategy Hand Calculations Computational Complexity Advantages Disadvantages Simulation Results Multi-Membered PAES Multi-Objective Messy Genetic Algorithm Original Single-Objective Messy GAs Modification for Multi-Objective Optimization Other Elitist Multi-Objective Evolutionary Algorithms Non-Dominated Sorting in Annealing GA Pareto Converging GA Multi-Objective Micro-GA Elitist MOEA with Coevolutionary Sharing Summary 285 Exercise Problems Constrained Multi-Objective Evolutionary Algorithms An Example Problem Ignoring Infeasible Solutions Penalty Function Approach Simulation Results Jimenez-Verdegay-Gomez-Skarmeta's Method Hand Calculations Advantages Disadvantages Simulation Results Constrained Tournament Method Constrained Tournament Selection Operator Hand Calculations 305 xi

7 xii CONTENTS Advantages and Disadvantages Simulation Results Ray-Tai-Seow's Method Hand Calculations Computational Complexity Advantages Disadvantages Simulation Results Summary 312 Exercise Problems Salient Issues of Multi-Objective Evolutionary Algorithms Illustrative Representation of Non-Dominated Solutions Scatter-Plot Matrix Method Value Path Method Bar Chart Method Star Coordinate Method Visual Method Performance Metrics Metrics Evaluating Closeness to the Pareto-Optimal Front Metrics Evaluating Diversity Among Non-Dominated Solutions Metrics Evaluating Closeness and Diversity Test Problem Design Difficulties in Converging to the Pareto-Optimal Front Difficulties in Maintaining Diverse Pareto-Optimal Solutions Tunable Two-Objective Optimization Problems Test Problems with More Than Two Objectives Test Problems for Constrained Optimization Comparison of Multi-Objective Evolutionary Algorithms Zitzler, Deb and Thiele's Study Veldhuizen's Study Knowles and Corne's Study Deb, Agrawal, Pratap and Meyarivan's Study Constrained Optimization Studies Objective Versus Decision-Space Niching Searching for Preferred Solutions Post-Optimal Techniques Optimization-Level Techniques Exploiting Multi-Objective Evolutionary Optimization Constrained Single-Objective Optimization Goal Programming Using Multi-Objective Optimization Scaling Issues Non-Dominated Solutions in a Population 416

8 CONTENTS xiii Population Sizing Convergence Issues Convergent MOEAs An MOEA with Spread Controlling Elitism Controlled Elitism in NSGA-II Multi-Objective Scheduling Algorithms Random-Weight Based Genetic Local Search Multi-Objective Genetic Local Search NSGA and Elitist NSGA (ENGA) Summary 438 Exercise Problems Applications of Multi-Objective Evolutionary Algorithms An Overview of Different Applications Mechanical Component Design Two-Bar Truss Design Gear Train Design Spring Design Truss-Structure Design A Combined Optimization Approach Microwave Absorber Design Low-Thrust Spacecraft Trajectory Optimization A Hybrid MOEA for Engineering Shape Design Better Convergence Reducing the Size of the Non-Dominated Set Optimal Shape Design Hybrid MOEAs Summary Epilogue 481 References 489 Index 509