Modified Brain Storm Optimization Algorithm for Solving Multimodal Multiobjective Optimization Problems.

Finding more solutions from different Pareto-optimal sets (PSs) in the decision space and maintaining the diversity of solutions at the Pareto front (PF) in the objective space are the two objectives of multimodal multiobjective optimization (MMO). This paper proposes a brain storm optimization algorithm with a non-dominated special crowding distance sorting strategy (BSO-SCD) to ensure a sufficient number of good enough solutions and diversity in both the decision and objective spaces. Based on the brain storm optimization (BSO) algorithm, the BSO-SCD algorithm enhances the ability of diversity maintenance and has three steps. Firstly, k-means clustering divides the population into multiple subpopulations in the decision space to help locate various optimal solutions. Secondly, the non-dominated special crowding distance sorting (SCD) strategy, considering the diversity of decision space and objective space simultaneously, is used to find each subpopulation cluster center and stored in an archive. Finally, the solutions in the archive are selected after the iteration. The performance of the BSO-SCD algorithm and the other five algorithms is verified on 12 MMO benchmark functions. Experimental results show that the BSO-SCD algorithm could find as many equivalent PSs as possible in the decision space and guarantee a good PF distribution in the objective space.