Research On Multimodal Multi-objective Optimization Algorithm Based On Two-stage Search

In multi-objective optimization problems,it is often necessary to optimize multiple objectives that are in conflict and eventually find a set of optimal solution sets.However,in practical production life,not only do we need to get effective solutions,but we also want to get solutions that are not single.Therefore,it is necessary to consider how to provide decision makers with a sufficient variety of solutions so that they can choose the most appropriate solution according to the specific situation and their own experience.This turns into a multimodal multi-objective optimization problem.Multi-modal multi-objective optimization problems are widely used in practical engineering and scientific research,and the main characteristic of this type of problem is that multiple different solutions in the decision space have the same performance in the objective space.Traditional multimodal multi-objective optimization algorithms do not obtain multiple equivalent Pareto optimal solution sets with uniform distribution in the decision space when solving such problems.To this end,this paper applies a new environment selection strategy based on a two-stage search framework to more realistically reflect the crowding degree of individuals in the population,so that individuals with better crowding in the population decision space can be selected,thus improving the performance of the algorithm on the decision space.The specific research is as follows.(1)Design a multimodal multi-objective optimization algorithm based on two-stage search(TSSMMO).the algorithm divides the search process into two stages.In the first stage,the population uses a differential evolution algorithm to generate children,and the environment is selected using the crowding distance to measure the crowding of individuals in the decision space and the target space,and the special crowding distance metric to balance the crowding of individuals in the two spaces.After the population convergence,the set of non-dominated solutions obtained by the algorithm is saved to assist the second stage of population search for regions that have not been fully explored.In the second stage of the algorithm search,a new environment selection strategy is proposed that will simultaneously consider the non-dominated solution set obtained in the first stage of the search when calculating the special crowding distance of individuals.This will expand the search area so that the population in the second stage will select individuals far from the non-dominated solution set in the first stage and search for suitable individuals in regions that have not been fully explored in the first stage.Finally,the populations obtained in the two phases are combined and selected to obtain the final solution set.To evaluate the performance of the algorithm,experiments are conducted on 11 test problems in this paper,and by comparing with related algorithms,it is shown that the algorithm is effective in solving multimodal multi-objective optimization problems.(2)Based on the TSSMMO algorithm,a multimodal multi-objective optimization algorithm based on neighborhood-based variational operations(NMOMMO)is proposed to further improve the performance of the algorithm on the objective space and decision space.The algorithm improves the variation strategy in the process of two-stage search of the TSSMMO algorithm: the population,in the process of selecting parents to generate offspring,will form a small habitat from the neighborhood of the current individual,and adaptively select random individuals or select individuals with larger crowding as the base vector during the variation operation according to the optimization process of two-stage search.To improve the convergence performance of the algorithm,in the final environment selection stage,the algorithm will preferentially select the final solution set among the non-dominated solution sets obtained during the two-stage search process.Experiments show that the NMOMMO proposed in this paper performs better in solving multimodal multi-objective optimization problems and can find as many equivalent Pareto optimal solution sets with uniform distribution as possible while guaranteeing the performance of the objective space.