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MINISTRY OF EDUCATION AND TRAINING MINISTRY OF NATIONAL DEFENCE MILITARY TECHNICAL ACADEMY NGUYEN XUAN HUNG Objective reduction methods in evolutionary many objective optimization DOCTORAL THESIS IN M[.]
MINISTRY OF EDUCATION AND TRAINING MINISTRY OF NATIONAL DEFENCE MILITARY TECHNICAL ACADEMY NGUYEN XUAN HUNG Objective reduction methods in evolutionary many-objective optimization DOCTORAL THESIS IN MATHEMATICS Hanoi - 2022 TIEU LUAN MOI download : skknchat123@gmail.com moi nhat MINISTRY OF EDUCATION AND TRAINING MINISTRY OF NATIONAL DEFENCE MILITARY TECHNICAL ACADEMY NGUYEN XUAN HUNG Objective reduction methods in evolutionary many-objective optimization Major: Mathematical Foundation for Informatics Code: 46 01 10 DOCTORAL THESIS IN MATHEMATICS SUPERVISOR: Assoc.Prof., Dr Bui Thu Lam Hanoi - 2022 TIEU LUAN MOI download : skknchat123@gmail.com moi nhat Originality Statement I guarantee that this is a work which is researched by me, under the guidance of Assoc Prof Dr Bui Thu Lam Research results published in the thesis are truthful The documents used in the thesis have clear origins Hanoi, November 2022 Author Nguyen Xuan Hung iii TIEU LUAN MOI download : skknchat123@gmail.com moi nhat Acknowledgments The research included in this thesis could not have been performed successfully but for many individuals’ assistance First of all, I would like to express my sincere thanks to my supervisor, Assoc Prof Dr Bui Thu Lam whose whole-hearted, enthusiastic, and academic efforts in guiding my PhD progress I would like to express my deep gratitude to Dr Cao Truong Tran, who has helped and guided me in constructing, analyzing and writing papers as well as the thesis in a scientific, objective and convincing manner Without his help and assistance, I would not have been able to complete this thesis I would also like to extend my hearty thanks the scientists who have devoted to reviewing, giving feed-backs to my thesis seminar, faculty-level thesis defence and double-anonymous peer review; and giving invaluable remarks on my works so that I could fulfill my thesis I would like to pay my deep tributes to Dr Nguyen Manh Hung, Assoc Prof Dr Long Nguyen and researchers from the Evolutionary Computation Research Group for their encouragement and assistance during my research process; and Dr Tran Le Duyen from Military Science Academy for proofreading the thesis thoroughly Last but not least, I also would like to acknowledge the encouragement and support of my family members, especially my wife, who have stood by me side-by-side and served as both material and spiritual shelters for me to accomplish this thesis iv TIEU LUAN MOI download : skknchat123@gmail.com moi nhat Abstract Multi-objective optimization problems often have more than one objective need to be optimized simultaneously One of the most suitable methods to solve these problems is using multi-objective evolutionary algorithms The algorithms work by simulating evolution of a population of individuals in a number of generations, by selecting a number of “good” solutions to the next in each generation As the number of objectives is greater than three, the problems are considered as many-objective optimization ones Dealing with these problems, multi-objective evolutionary algorithms meet several difficulties, especially in determining the “good” individuals for the generation In order to alleviate the difficulties, many-objective evolutionary algorithms are proposed These algorithms can be roughly categorized in two approaches First, the algorithms modify “relation” when comparing the individuals during evolving or improve the existing multi-objective evolutionary algorithms Second, for problems containing redundant objectives, the algorithms use objective reduction techniques to remove these redundant objectives before solving them The algorithms belonging to the second approach are called objective reduction ones The objective reduction contains two components The first component is multi-objective evolutionary algorithm for generating non-dominated solutions The second one, dimensionality objective reduction, analyzes the objective values of obtained non-dominated solutions to removing redundant objectives and keeping the essential ones Although many obv TIEU LUAN MOI download : skknchat123@gmail.com moi nhat vi jective reductions have been proposed, most first components are multiobjective evolutionary algorithms while existing many the state-of-the-art many-objective evolutionary algorithms Moreover, many of them have not considered reducing objectives or validated by testing redundant problems Last but not least, the existing objective reductions are often validated by testing with redundant problems on a small number of objectives The thesis first investigates the efficiency of combining existing manyobjective evolutionary algorithms and dimensionality objective reductions More specifically, it shows that integrating dimensionality objective reduction into many-objective evolutionary algorithms give a better result in removing redundant objectives than doing that into many-objective evolutionary algorithms Second, it proposes (1) an objective reduction algorithm named COR The algorithm basing on a complete Pareto manyobjective evolutionary algorithm, can self-determine the number of clusters to partition a set of objects (presenting objectives in problems) to remove the redundant objectives Third, the thesis proposes two objective reduction algorithms (ORAs), viz PCS-LPCA and PCS-Cluster to removing redundant objectives and keeping essential ones as solving redundant many-objectives problems While (2) PCS-LPCA using PCSEA to generate a solution set composed a partial PF, then using linear PCA to analyze objective values of obtained solutions which are generated by PCSEA algorithm; (3) PCS-Cluster using PCSEA to generate a solution set composed a partial PF, then using clustering machine learning algorithms to analyze the set in order to keep the essential objectives TIEU LUAN MOI download : skknchat123@gmail.com moi nhat Contents Page Originality Statement iii Acknowledgments iv Abstract v Contents vii Acronyms x List of Tables xii List of Figures xiv List of Algorithms xv Introduction 0.1 Problem statement 0.2 Motivation 0.3 Aim and objectives of the study 0.3.1 Aim of the study 0.3.2 Objectives of the study 0.3.3 Research questions 0.4 Contributions 0.5 Structure of the thesis vii TIEU LUAN MOI download : skknchat123@gmail.com moi nhat CONTENTS viii Chapter Literature Review 11 1.1 Background 12 1.1.1 Optimization 12 1.1.2 Multi-objective optimization 15 1.1.3 Machine learning algorithms used in this study 25 1.2 Related works 28 1.2.1 Many-objective optimization 28 1.2.2 Objective reduction 34 1.3 Benchmarks and performance measures 46 1.3.1 Benchmark methods 46 1.3.2 Benchmark problems 47 1.3.3 Performance measures 47 Chapter The complete PF-based objective reduction algorithms 2.1 Efficiency in many- algorithms in objective reduction 49 51 2.1.1 The proposed method 51 2.1.2 Experimental design 53 2.1.3 Results and discussions 54 2.2 COR objective reduction algorithm 63 2.2.1 The proposed algorithm 63 2.2.2 Experimental design 67 2.2.3 Results and discussions 68 Chapter The partial PF-based objective reduction algorithms 73 3.1 PCS-LPCA objective reduction algorithm 74 3.1.1 The proposed algorithm 74 3.1.2 Experimental design 78 3.1.3 Results and discussions 79 TIEU LUAN MOI download : skknchat123@gmail.com moi nhat CONTENTS ix 3.2 PCS-Cluster objective reduction algorithm 90 3.2.1 The proposed algorithm 90 3.2.2 Experiment 94 3.2.3 Results and discussions 95 Conclusion and future works 110 Publications 113 Bibliography 114 Appendix A Representations of non-dominated solutions 123 Appendix B Several machine learning algorithms 127 TIEU LUAN MOI download : skknchat123@gmail.com moi nhat ACRONYMS Acronym Meaning COR a clustering objective reduction algorithm for many- problems DBSCAN Density-Based Spatial Clustering of Applications with Noise DRA Dimensionality reduction algorithm DTLZ DTLZ1 problem set [28] EA Evolutionary Algorithm EC Evolution Computing GD Generational Distance GrEA Grid based Evolutionary Algorithm HV Hypervolume indicator IGD Inverted Generational Distance k -means a method for partitioning n objects into k clusters KnEA Knee point driven Evolutionary Algorithm L-PCA Linear Principal Component Analysis many- algorithm Many-objective Evolutionary Algorithm many- problem Many-objective Optimization Problem MaOO Many-objective Optimization MOEA/D Multi-objective Evolutionary Algorithm Based on Decomposition MOO Multi-objective Optimization MOSS Minimum Objective Subset problem multi- algorithm Multi-objective Evolutionary Algorithm multi-/many- algorithm Multi/Many-objective Evolutionary Algorithm multi- problem Multi-objective Optimization Problem NSGA-II Non-dominated Sorting Genetic Algorithm II NSGA-III Reference-point based many-objective NSGA-II ODR Objective dimensionality reduction ORA Objective reduction algorithm (continued on next page) proposed by Deb, Thiele, Laumanns, and Zitzler x TIEU LUAN MOI download : skknchat123@gmail.com moi nhat