Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization. Objective reduction methods in evolutionary manyobjective optimization.
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 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 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 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, As- soc Prof Dr Long Nguyen and researchers from the Evolutionary Compu- tation 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 Abstract Multi-objective optimization problems often have more than one ob- jective need to be optimized simultaneously One of the most suitable methods to solve these problems is using multi-objective evolutionary al- gorithms 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 prob- lems, multi-objective evolutionary algorithms meet several difficulties, es- pecially in determining the “good” individuals for the generation In or- der 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 al- gorithms Second, for problems containing redundant objectives, the al- gorithms 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 compo- nent 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 re- dundant objectives and keeping the essential ones Although many obv vi jective reductions have been proposed, most first components are multi- objective 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 reduc- tion into manyobjective evolutionary algorithms give a better result in removing redundant objectives than doing that into manyobjective evo- lutionary algorithms Second, it proposes (1) an objective reduction al- gorithm named COR The algorithm basing on a complete Pareto many- objective evolutionary algorithm, can self-determine the number of clus- ters 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 PCSLPCA and PCS-Cluster to remov- ing redundant objectives and keeping essential ones as solving redundant many-objectives problems While (2) PCS-LPCA using PCSEA to gener- ate a solution set composed a partial PF, then using linear PCA to analyze objective values of obtained solutions which are generated by PCSEA algo- rithm; (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 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 1.1 Problem statement 1.2 Motivation 1.3 Aim and objectives of the study 1.3.1 Aim of the study 1.3.2 Objectives of the study 1.3.3 Research questions .8 1.4 Contributions .8 1.5 Structure of the thesis vii Chapter Literature Review 11 1.1 Background 12 1.1.1 Op timization 12 1.1.2 M ulti-objective optimization 15 1.1.3 M achine learning algorithms used in this study .25 1.2 Related works .28 1.2.1 Ma ny-objective optimization 28 1.2.2 Ob jective reduction 34 1.3 Benchmarks and performance measures 46 1.3.1 Be nchmark methods .46 1.3.2 Be nchmark problems 47 1.3.3 Pe rformance measures 47 Chapter The complete PF-based objective reduction algorithms 49 2.1 Efficiency in many - algorithms in objective reduction .51 2.1.1 Th e proposed method .51 2.1.2 Ex perimental design .53 2.1.3 Re sults and discussions 54 2.2 COR objective reduction algorithm .63 2.2.1 Th e proposed algorithm 63 2.2.2 Ex perimental design .67 2.2.3 Re sults and discussions 68 ... comparing the individuals during evolving or improve the existing multi -objective evolutionary al- gorithms Second, for problems containing redundant objectives, the al- gorithms use objective reduction. .. 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. ..MINISTRY OF EDUCATION AND TRAINING MINISTRY OF NATIONAL DEFENCE MILITARY TECHNICAL ACADEMY NGUYEN XUAN HUNG Objective reduction methods in evolutionary many -objective optimization