Bi-stage learning differential evolution for multimodal optimization problems

Multimodal optimization problems (MMOPs) require the identification of multiple optimal solutions for decision makers. To address MMOPs, algorithms must enhance the population diversity to find more global optimal regions while simultaneously refine the solution accuracy on each optimum. Therefore, in this paper, we introduces a bi-stage learning differential evolution (BLDE) with two learning stages: the pre-learning Find stage and the post-learning Refine stage. First of all, a bi-stage learning niching technique (BLNT) is proposed, which forms wide niches for full exploration in the pre-learning Find stage, while adaptively adjusts the niche radius for each individual to refine its corresponding solution accuracy in the post-learning Refine stage. Subsequently, a bi-stage learning mutation strategy (BLMS) is developed, enabling each individual to adaptively choose the suitable mutation strategy, achieving effective guidance for evolution. Moreover, different from other DE-based multimodal algorithms with only one selection operator, a bi-stage learning selection strategy (BLSS) is proposed to determine the suitable selection operator in different learning stages and preserve the promising individuals. The widely-used multimodal benchmark functions from CEC2015 competition are employed to evaluate the performance of BLDE. The results demonstrate that BLDE generally outperforms or at least comparable with other state-of-the-art multimodal algorithms, including the champion of CEC2015 competition. Moreover, BLDE is further applied to the real-world multimodal nonlinear equation system (NES) problems to demonstrate its applicability.