A Grid-based Niche Multi-modal Multi-objective Algorithm

Multi-Objective Optimization Problems(MOPs)widely exist in real life.One kind of optimization problems with two or more PS corresponding to the same PF in the decision space is called Multi-Modal Optimization Problems(MMOPS).The key to solve MMOPs is to search multiple equivalent Pareto optimal sets(PSs)and keep them to the end.Niche method is one of the most commonly used methods to solve this kind of problems.In the process of niche formation,each individual in the decision space is the core of each niche,different selection mechanisms are used to determine the remaining individuals of the niche,and the nearest individuals are collected into a niche.However,in the process of experimental exploration,we found that in the Multi-Modal Evolutionary Elgorithm(MMOEA)with European distance as the radius of the niche,When there are enough individuals in the population,it is easy to delete by mistake,that is,when two individuals are very close,the two individuals may exist in two niches at the same time,just like the overlapping part of two intersections in mathematics.Due to the diversity protection mechanism of the algorithm,one individual may be deleted,so we can not get a complete PS.In order to avoid this phenomenon as much as possible,we propose a GridBased Adaptive Crowding Distance Multi-Objective Multi-Modal Evolutionary Optimization Algorithm(GACDMMOEA).GACDMMOEA deals with the whole group and the gird was divided according to the max and min values of each dimension of the decision space.Each niche is a rectangular grid with clear niche boundary,and the adjacent niches will not overlap.In this way,it can avoid the situation that individuals in different PSs exist in the same niche,so as to reduce the accidental deletion of corresponding individuals in the same area in the objective space,At the same time,in order to make the population have good distribution in the objective space and decision space,we designed an adaptive environment selection mechanism-Grid Adaptive Crowded Distance(GACD),In the process of environmental selection,the distribution of individuals in objective decision spaces can be considered through the designed diversity selection mechanism,the individuals with good performance in the above two indicators are selected into the next generation of population,and the population is optimized from generation to generation through the algorithm.Finally,we will get a population that searches all equivalent PSs in the decision space and has a good PF surface at the same time.