A Two-archive Algorithm With Decomposition And Fitness Allocation For Multi-modal Multi-objective Optimization

Traditional Multi-objective Optimization Problems(MOPs)has a real Pareto Front(PF)in the objective space by the only one Pareto optimal Set(PS)in the decision space mapping.When the number of mapped Pareto optimal solution sets is greater than or equal to 2,MOPs are defined as multi-modal multi-objective optimization problems(MMOPs).Classical Multi-objective Evolutionary Algorithms(MOEAs),such as NSGA-II,SPEA2 and MOEA/D,can effectively solve MOPs.However,when dealing with MMOPs,the lack of a diversity maintenance mechanism in the decision space leads to the diversity of the algorithm in the decision space is lost.Therefore,balancing the diversity of the algorithm between the decision space and the objective space in a multi-modal and multi-objective environment is an urgent challenge to be solved.In addition,while ensuring that the solution set finally obtained by algorithm has good diversity in the two spaces,MOEAs also need to ensure that the final solution converges to each PS.How to balance the convergence and diversity in the two spaces must be considered when designing MOEAs..This paper proposes an Evolutionary Multi-modal Multi-Objective Algorithms(EMMOAs)based on two archive sets to solve MMOPs.We named the proposed EMMOA A Two-Archive Algorithm with Decomposition and Fitness Allocation for Multi-modal Multi-objective Optimization(TA&DF).The overall framework of the algorithm is based on two archive sets,the Convergence Archive(CA)promotes the convergence of the population,and the Diversity Archive(DA)maintains the diversity of the population.The update of the two archive sets is based on the decomposition framework.In CA,TA&DF proposes a new fitness selection scheme by analyzing the convergence of the objective space and the diversity of the decision space during the evolution of the population,and it prompts the population to search for different PSs.In DA,TA&DF uses the crowding distance based on the decision space to maintain the diversity of the decision space.In addition,we use different neighborhood criteria in the two archives to ensure the convergence and diversity of the population.The method proposed in this paper is compared with 5 advanced EMMOAs(MO?PSO?Ring?SCD,DNEA,Tri MOEA-TA&R,DN-NSGA-II and Omni?optimizer)on MMF,Omni,SYM?PART and IDMP series test functions.By comparing the performance metric results of IGDM and IGDX,it is found that the method proposed in this paper has excellent performance under different test functions.In addition,in order to verify the effectiveness of the new fitness scheme proposed in this paper,this scheme has been horizontally compared with the fitness scheme of SPEA2 and the PBI aggregation function of MOEA/D on the above series of test functions and evaluation indicators.The fitness selection scheme is effective in solving MMOPs.