Research And Application Of Multimodal Multi-objective Optimization Algorithm Considering The Diversity Of Decision Space

In real-life and engineering applications,numerous multi-modal multiobjective optimization problems exist.These problems involve multiple Pareto optimal solutions that correspond to identical or similar locations within the objective space.Identifying all Pareto optimal solutions can aid decision-makers in selecting the most appropriate solutions.Traditional multi-objective evolutionary algorithms strive to maintain diversity and convergence in the objective space but overlook the development of decision space,resulting in an inability to obtain multiple identical or similar solutions.In contrast,multi-modal multi-objective evolutionary algorithms place greater emphasis on enhancing the population’s diversity within the decision space without disregarding performance in the objective space.This increased complexity in multi-modal multi-objective optimization problems affects the decision-maker’s solution selection compared to traditional multi-objective optimization problems.To address this issue,three distinct multi-modal multi-objective evolutionary algorithms were designed and implemented,and their effectiveness was validated through simulation comparisons with the latest multi-modal multi-objective evolutionary algorithms.The main research work includes:(1)A multi-modal multi-objective evolutionary algorithm,D-DNEAL,based on diversity index ranking is proposed.Initially,non-dominated sorting within niches is performed to identify global and local optimal solutions,and a diversity index is used to eliminate individuals with poor diversity,thus enhancing the algorithm’s diversity in decision space.Next,an improved double niche fitness sharing method is employed during the selection stage to increase search accuracy.A multi-frontier archiving mechanism is then utilized to preserve global and local Pareto optimal solutions.Finally,experimental results from comparisons with six cutting-edge multi-modal multi-objective evolutionary algorithms on test problems demonstrate that D-DNEAL effectively improves search capability within decision space.(2)A multi-modal multi-objective evolutionary algorithm,MMEA-SND,based on a special niche distance mechanism is proposed.In MMEA-SND,a diversity archive and a convergence archive are initially established.Diversity fitness is applied in the diversity archive to inhibit individual convergence to specific areas,while convergence fitness is introduced in the convergence archive to prevent nonconverging individuals.Next,a special niche distance mechanism is employed in environmental selection,using distance information from solutions in both spaces to balance the diversity of objective and decision spaces.The number of dominated solutions in the diversity archive is then used to determine whether it contains local optimal solutions;if so,the diversity archive is output,otherwise,the convergence archive is output.Finally,experimental results from comparisons with six cuttingedge multi-modal multi-objective evolutionary algorithms on test problems indicate that MMEA-SND exhibits excellent performance in most multi-modal multiobjective optimization problems.(3)A multi-modal multi-objective evolutionary algorithm,MOEA/D-j DE-A,based on archive updates is proposed.First,an external archive is created to store individuals exhibiting exceptional diversity in decision space during the search process.An improved crowding distance is applied to ensure individual diversity in the archive,enhancing the algorithm’s search capability in decision space.The population is then updated through the archive,ensuring diversity within the decision space and using a decomposition strategy to distribute the population evenly across the objective space.As the number of evaluations increases,so does the frequency of archive updates.Additionally,MOEA/D-j DE-A employs distance relationships between individuals to identify parental individuals and generates individuals with superior distribution in decision space through adaptive differential evolution algorithms.Ultimately,experimental results demonstrate that MOEA/D-j DE-A is competitive in balancing diversity in both spaces.Consequently,MOEA/D-j DE-A will be further applied to address location optimization problems,with results indicating that it can identify all optimal solutions.