Nova Southeastern University NSUWorks CEC Theses and Dissertations College of Engineering and Computing 2010 A Species-Conserving Genetic Algorithm for Multimodal Optimization Michael Scott rB own Nova Southeastern University, [email protected] This document is a product of extensive research conducted at the Nova Southeastern University College of Engineering and Computing. For more information on research and degree programs at the NSU College of Engineering and Computing, please click here. Follow this and additional works at: https://nsuworks.nova.edu/gscis_etd Part of the Computer Sciences Commons Share Feedback About This Item NSUWorks Citation Michael Scott rB own. 2010. A Species-Conserving Genetic Algorithm for Multimodal Optimization. Doctoral dissertation. Nova Southeastern University. Retrieved from NSUWorks, Graduate School of Computer and Information Sciences. (104) https://nsuworks.nova.edu/gscis_etd/104. 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A Species-Conserving Genetic Algorithm for Multimodal Optimization by Michael Scott Brown A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science Graduate School of Computer and Information Sciences Nova Southeastern University 2010 We hereby certify that this dissertation, submitted by Michael Scott Brown, conforms to acceptable standards and is fully adequate in scope and quality to fulfill the dissertation requirements for the degree of Doctor of Philosophy. _____________________________________________ ________________ Michael Laszlo, Ph.D. Date Chairperson of Dissertation Committee _____________________________________________ ________________ James Cannady, Ph.D. Date Dissertation Committee Member _____________________________________________ ________________ Sumitra Mukherjee, Ph.D. Date Dissertation Committee Member Approved: _____________________________________________ ________________ Leonidas Irakliotis, Ph.D. Date Dean Graduate School of Computer and Information Sciences Nova Southeastern University 2010 An Abstract of a Dissertation Submitted to Nova Southeastern University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A Species-Conserving Genetic Algorithm for Multimodal Optimization by Michael Scott Brown September 2010 The problem of multimodal functional optimization has been addressed by much research producing many different search techniques. Niche Genetic Algorithms is one area that has attempted to solve this problem. Many Niche Genetic Algorithms use some type of radius. When multiple optima occur within the radius, these algorithms have a difficult time locating them. Problems that have arbitrarily close optima create a greater problem. This paper presents a new Niche Genetic Algorithm framework called Dynamic-radius Species-conserving Genetic Algorithm. This new framework extends existing Genetic Algorithm research. This new framework enhances an existing Niche Genetic Algorithm in two ways. As the name implies the radius of the algorithm varies during execution. A uniform radius can cause issues if it is not set correctly during initialization. A dynamic radius compensates for these issues. The framework does not attempt to locate all of the optima in a single pass. It attempts to find some optima and then uses a tabu list to exclude those areas of the domain for future iterations. To exclude these previously located optima, the framework uses a fitness sharing approach and a seed exclusion approach. This new framework addresses many areas of difficulty in current multimodal functional optimization research. This research used the experimental research methodology. A series of classic benchmark functional optimization problems were used to compare this framework to other algorithms. These other algorithms represented classic and current Niche Genetic Algorithms. Results from this research show that this new framework does very well in locating optima in a variety of benchmark functions. In functions that have arbitrarily close optima, the framework outperforms other algorithms. Compared to other Niche Genetic Algorithms the framework does equally well in locating optima that are not arbitrarily close. Results indicate that varying the radius during execution and the use of a tabu list assists in solving functional optimization problems for continuous functions that have arbitrarily close optima. Acknowledgments I would like to thank Dr. Laszlo for being my dissertation chair. His help cannot be overstated. I would also like to thank Dr. Cannady and Dr. Mukherjee for serving on my committee. I would also like to thank my parents, Thomas and Sandra Brown. You always told me that I could do anything if I only worked hard and never gave up. You were right. This dissertation is dedicated to the three ladies in my life, Truocmai Dinh, Jessica Maikhanh Brown and Tiffany Maitam Brown, my wife and two daughters. All that I do, I do for you. Table of Contents Abstract iii List of Tables vii List of Figures ix Chapters 1. Introduction 1 Problem Statement 1 Research Goal 3 Approach 5 Relevance and Significance 8 Barriers and Issues 9 Definition of Terms 12 Summary 14 2. Review of the Literature 16 Relevant Research Other than NGAs 16 Fitness Sharing Methods 18 Crowding Methods 24 Other Niche Genetic Algorithm Methods 36 Summary 48 3. Methodology 49 Research Method Employed 49 Specific Procedures Employed 50 Format for Presenting Results 66 Resources Required 67 Summary 67 4. Results 69 Parameter Settings and Implementation Methods 69 Results of Algorithms on F1 77 Results of Algorithms on F2 79 Results of Algorithms on F3 80 Results of Algorithms on F4 81 Results of Algorithms on F5 82 Results of Algorithms on F6 84 Results of Algorithms on F7 85 Results of Algorithms on F8 87 Summary of Results 88 5. Conclusions, Implications, Recommendations and Summary 90 Conclusions 90 Implications 93 v Recommendations 94 Summary 95 Appendixes A. Ranking of Algorithms 102 Reference List 110 vi List of Tables Tables 1. Tabu Search Decision 17 2. Sequential Niche Technique Algorithm 22 3. Genetic Algorithm with Species Algorithm 28 4. Species Conserving Genetic Algorithm Seed Selection 30 5. Species Conserving Genetic Algorithm Species Conservation 31 6. Crowding Clustering Genetic Algorithm 34 7. Enhanced Evolutionary Tabu Search Algorithm 39 8. Genetic Algorithm and Particle Swarm Optimization Algorithm 41 9. Cellular Genetic Algorithm 42 10. DSGA Parameters 51 11. DSGA Algorithm 52 12. DSGA Algorithm Initialization 53 13. Seed Selection 54 14. Seed Conservation for each Generation 55 15. Test Functions 60 16. Benchmark Algorithm Comparison 64 17. Fitness Functions 73 18. Results for Equation F1 78 19. Results for Equation F2 79 20. Results for Equation F3 80 21. Results for Equation F4 82 vii 22. Results for Equation F5 83 23. Results for Equation F6 84 24. Results for Equation F7 86 25. Results for Equation F8 88 26. Ranking of Algorithms 102 viii List of Figures Figures 1. Graph of y = x sin (x 2) 2 2. Graph of f(x) = 4 (x – 0.5) 2 26 3. Graph of f(x) = 2.8 (x - 0.6) 2 26 4. Chart of Recall and Precision for F1 78 5. Chart of Recall and Precision for F2 80 6. Chart of Recall and Precision for F3 81 7. Chart of Recall and Precision for F4 82 8. Chart of Recall and Precision for F5 83 9. Chart of Recall and Precision for F6 85 10. Chart of Recall for F7 85 11. Chart of Recall and Precision for F8 88 ix 1 Chapter 1 Introduction This introductory chapter is organized into seven sections. The first section defines the problem that this research addressed. The second section describes the research goal. The third section provides an overview of the approach. The fourth section explains the relevance and significance. The fifth section describes the barriers and issues that need to be overcome by this research. The sixth section provides definitions of terms used in the dissertation. The final section is a summary of this chapter. Problem Statement Genetic Algorithms (GA) have a difficult time solving problems with multiple correct answers. When traditional GAs attempt to solve multimodal problems they often converge to only one of the possible correct or good solutions. A current area of research in GAs is called Niche Genetic Algorithms (NGA), which hopes to address this problem. NGAs can be used to solve problems that seek local optima where multiple exist. Currently there are many NGAs. Two prominent approaches to developing NGAs are crowding and sharing (Deb & Goldberg, 1989). In crowding algorithms, members of one population coexist with members of the next population. Older individuals of the population are selected for removal based on how similar they are to newer members. A variety of NGAs use some type of crowding scheme (Cavicchio, 1970; De Jong, 1975; Jelasity & Dombi 1998; Li, Balazs, Parks & Clarkson, 2002; Ling, Wa, Yang & Wang, 2 2008; Raghuwanshi & Kakde, 2007). The second approach is through fitness sharing. In sharing schemes, the fitness of an individual is dependent on its distance to other individuals in the population. This increases the chance that species will form around niches by rewarding genetic isolation. There are many methods based on sharing schemes (Beasley, Bull & Martin, 1993b; Bernier, 1996; Goldberg & Richardson,