A tri-objective differential evolution approach for multimodal optimization

The multimodal optimization problems (MMOPs) need to find multiple optima simultaneously, so the population diversity is a critical issue that should be considered in designing an evolutionary optimization algorithm for MMOPs. Taking advantage of evolutionary multiobjective optimization in maintaining good population diversity, this paper proposes a tri-objective differential evolution (DE) approach to solve MMOPs. Given an MMOP, we first transform it into a tri-objective optimization problem (TOP). The three optimization objectives are constructed based on 1) the objective function of an MMOP, 2) the individual distance information measured by a set of reference points, and 3) the shared fitness based on niching technique. The first two objectives are mutually conflicting so that the advantage of evolutionary multiobjective optimization can be fully used. The population diversity is greatly improved by the third objective constructed by the niching technique which is insensitive to niching parameters. Mathematical proofs are given to demonstrate that the Pareto-optimal front of the TOP contains all global optima of the MMOP. Subsequently, DE-based multiobjective optimization techniques are applied to solve the converted TOP. Moreover, a modified solution comparison criterion and an adaptive ranking strategy for DE are introduced to improve the accuracy of solutions. Experiments have been conducted on 44 benchmark functions to evaluate the performance of the proposed approach. The results show that the proposed approach achieves competitive performance compared with several state-of-the-art multimodal optimization algorithms.