Metaheuristics for Finding Multiple Solutions

Foreword
Preface
Contents
Multimodal Optimization: Formulation, Heuristics, and a Decade of Advances
	1 Introduction
	2 Definitions
		2.1 The General Optimisation Problem
		2.2 The Multimodal Optimization Problem
	3 Performance Measures
	4 Benchmark Suites and Problem Generators
	5 Popular Algorithmic Approaches and History of the Field
	6 Niching Competition Result Analysis
	7 Conclusion
	References
Representation, Resolution, and Visualization in Multimodal Optimization
	1 Multimodal Optimization: The What and the Why
		1.1 What Is a Mode?
		1.2 Why Optimize Multiple Modes?
	2 Representation, Resolution, and Basic Visualizations Plots
	3 Visualizing the Multimodal Landscape: Local Optima Networks
	4 Conclusion
	References
Finding Representative Solutions in Multimodal Optimization for Enhanced Decision-Making
	1 Introduction
	2 Related Work
		2.1 Classic Niching Methods
		2.2 Recent Development
		2.3 Differential Evolution
		2.4 Hopkins-Statistic
		2.5 Adaptable Non-maximal Suppression
	3 Suppression-Radius-Based Niching (SRN)
		3.1 Phase I—Identifying Representative Areas
		3.2 Phase II—Guided Search Toward Representative Areas
	4 Experiments
		4.1 Experimental Design
		4.2 Incorporating a User Specified Number of Optima
		4.3 Automatic Estimation of the Number of Optima
		4.4 No Specification of Number of Representatives
	5 Conclusions
	References
Lifting the Multimodality-Fog in Continuous Multi-objective Optimization
	1 Introduction
	2 Related Work
	3 Multimodality in MO Optimization
		3.1 Theoretical Foundations
		3.2 Visualizing Landscapes of Multi-objective Gradients
	4 On the Properties of State-of-the-Art Benchmarks
		4.1 A Visual Overview
		4.2 Interpretation and Categorization
	5 How Multi-objective Optimization Algorithms Can Capitalize from Basins of Attraction
	6 Conclusion
	References
Towards Basin Identification Methods with Robustness Against Outliers
	1 Introduction
	2 Nearest-Better Clustering
	3 Related Research
	4 Ideas for New Basin Identification Methods
	5 Experiments
		5.1 Determining Regression Models
		5.2 Validation
	6 Conclusions
	References
Deflection and Stretching Techniques for Detection of Multiple Minimizers in Multimodal Optimization Problems
	1 Introduction
	2 Deflection Technique
		2.1 Basic Scheme
		2.2 Variants and Applications
	3 Stretching Technique
		3.1 Basic Scheme
		3.2 Variants and Applications
	4 Experimental Evaluation
	5 Conclusions
	References
Multimodal Optimization by Evolution Strategies with Repelling Subpopulations
	1 Introduction
	2 Niching with Repelling Subpopulations
		2.1 Core Algorithm
		2.2 Main Niching Ideas
		2.3 Evolution of Subpopulations
		2.4 Restart Strategy with Increasing Population
		2.5 Adaptation of the Normalized Taboo Distance
		2.6 Boosting Time Efficiency
		2.7 Initialization of Subpopulations
		2.8 Parameter Setting
	3 Numerical Evaluation
	4 Summary and Conclusions
	References
Two-Phase Real-Valued Multimodal Optimization with the Hill-Valley Evolutionary Algorithm
	1 Introduction
	2 Framework for Two-Phase MMO EAs
		2.1 Initial Population Sampling
	3 Fitness-Informed Clustering
		3.1 Nearest-Better Clustering
		3.2 Hierarchical Gaussian Mixture Learning
		3.3 Hill-Valley Clustering
	4 Core Search Algorithms
		4.1 Termination Criteria for Core Search Algorithms
	5 Experiments
		5.1 Experiment 1: Clustering Comparison
		5.2 Experiment 2: Core Search Algorithms and Clustering Methods
		5.3 Experiment 3: MMO EA Comparison
		5.4 Experiment 4: Larger Budget
	6 Conclusion
	References
Probabilistic Multimodal Optimization
	1 Introduction
	2 Probability Distribution-Based Niching
		2.1 Existing Niching Methods
		2.2 Locality Sensitive Hashing (LSH)
		2.3 Fast Niching
		2.4 Extensive Experiments
	3 Probability Distribution-Based Optimization
		3.1 Estimation of Distribution Algorithms (EDAs)
		3.2 Ant Colony Optimization (ACO)
		3.3 Multimodal Estimation of Distribution Algorithms (MEDAs)
		3.4 Adaptive Multimodal Ant Colony Optimization (AM-ACO)
		3.5 Extensive Comparison
	4 Applications
	5 Discussion and Future Work
	6 Conclusion
	References
Reduced Models of Gene Regulatory Networks: Visualising Multi-modal Landscapes
	1 Introduction
	2 Data-Driven Application: Gene Regulatory Network Models
		2.1 Introduction to Gene Regulatory Networks and Circadian Rhythms
		2.2 Boolean Delay Equations
		2.3 An Exemplar Computational Model of Circadian Rhythms Based on BDEs
		2.4 Parameter Optimisation of the BDE Model
	3 Landscape Analysis
	4 Local Optima Networks
	5 Discussion
	References
Grammar-Based Multi-objective Genetic Programming with Token Competition and Its Applications in Financial Fraud Detection
	1 Introduction
	2 Background
		2.1 Multi-objective Optimization Problems
		2.2  Genetic Programming (GP)
		2.3 Financial Fraud Detection
	3 Approach
		3.1 Grammar-Based Multi-objective Genetic Programming (GBMGP) with Token Competition
		3.2 Statistical Selection Learning
	4 Experiments and Results
		4.1 Introduction to Experiment Preparation
		4.2 Parameter Settings
		4.3 Results and Analysis
	5 Conclusion
		5.1 Contributions
		5.2 Directions for Future Research
	References
Phenotypic Niching Using Quality Diversity Algorithms
	1 Introduction
	2 The Search for Diversity
		2.1 Genetic Diversity
		2.2 Phenotypic Diversity
	3 Quality Diversity
		3.1 First Algorithms
		3.2 General Description
		3.3 A Practical Example
		3.4 Success Stories
	4 Insights
		4.1 Alignment of Quality and Diversity
		4.2 Stepping Stones
		4.3 Alignment of Genome and Phenotype
		4.4 Exploitation and Exploration
	5 Comparing Performance
		5.1 Performance Metrics
		5.2 Benchmarks
	6 Conclusions and Open Challenges
	References