Research Of Multi-modal Multi-objective Differential Evolution Considering Both Global And Local Pareto Sets

Multi-modal multi-objective optimization problems(MMOPs)refer to multi-objective optimization problems(MOPs)with multiple global or local Pareto solution sets(PSs).According to the existence of global and local PSs,MMOPs can be divided into two categories: 1)problems with only global PSs,and 2)problems with both global and local PSs.Existing multi-modal multi-objective evolutionary algorithms(MMEAs)are mostly designed for the first category of problems.These MMEAs may only be able to locate the global PSs of the problem when facing the second category of problems,without being able to locate and preserve the local PSs of the problem.However,local PSs are of great importance in solving MMOPs in many scenarios.Therefore,this paper focuses on MMOPs that contain both global and local PSs and proposes the differential evolution algorithm that considers both global and local PSs.The specific research contents are as follows:1)For MMOPs with both global and local PSs,a special sorting mechanism-based multi-modal multi-objective differential evolution algorithm(SSDE-MM)is proposed.This mechanism combines the global and local Pareto ranks to achieve the preservation of global and local Pareto optimal solutions.Specifically,by calculating the local Pareto rank of each individual,the algorithm can determine whether an individual has the potential to become a global or local Pareto optimal solution and preserve individuals with such potential.By calculating the global Pareto rank of each individual,the algorithm can evaluate the real convergence of individuals.In addition,an elite based differential mutation strategy,DE/elite/2,is designed to guide individuals towards high-quality and sparse areas,maintain population diversity,and generate high-quality offspring.The experimental results on the CEC2019 test problem set verify the effectiveness of SSDE-MM in solving MMOPs with both global and local PSs.2)Although the above SSDE-MM performs well in maintaining population diversity and searching for and preserving global and local PSs,this mechanism may affect the convergence performance of the algorithm to some extent.To improve the convergence performance of the algorithm,a two-stage differential evolution algorithm with balanced exploration and exploitation(TSDE-EE)is proposed.The entire evolution process of TSDE-EE is divided into two stages: exploration and exploitation.In the exploration stage,the algorithm focuses on population diversity and uses a clustering-based DE/rand/2 mutation strategy to generate diverse offspring.The special sorting mechanism and clustering-based special crowding distance method are used to fully explore the entire search space and accurately locate potential global and local PSs.In the exploitation stage,the algorithm focuses on population convergence and distribution,uses a clustering-based DE/best/2 mutation strategy to generate offspring that enhance convergence,and designs sub-population partitioning and offspring selection strategies to separate the global and local optimal solutions found in the exploration stage into different sub-populations,allowing the global optimal and local optimal sub-populations to evolve independently without interference.In addition,a novel crowding distance based environmental selection is designed in the exploitation stage to further improve population distribution and algorithm performance.The performance of the proposed algorithm is verified by the CEC2019 test suite and the polygon test suite.The experimental results show that TSDE＿EE not only has the ability to locate and preserve global and local PSs but also exhibits good convergence and distribution performance.