Most real-world optimization problems often come with multiple global optima or local optima. Therefore, increasing niching metaheuristic algorithms, which devote to finding multiple optima in a single run, are developed to solve these multimodal optimization problems. However, there are two difficulties urgently to be solved for most existing niching metaheuristic algorithms: how to set the niching parameter values for different optimization problems and how to jump out of the local optima efficiently. These two difficulties limit their practicality largely. Based on Whale Swarm Algorithm (WSA) we proposed previously, this paper presents a new multimodal optimizer named WSA with Iterative Counter (WSA-IC) to address these two difficulties. On the one hand, WSA-IC improves the iteration rule of the original WSA for multimodal optimization, which removes the need of specifying different values of attenuation coefficient for different problems to form multiple subpopulations, without introducing any niching parameter. On the other hand, WSA-IC enables the identification of extreme points during the iterations relying on two new parameters (i.e., stability threshold (T_ s ) and fitness threshold (T_ f )), to jump out of the located extreme points. Moreover, the convergence of WSA-IC is proved. Finally, the proposed WSA-IC is compared with several niching metaheuristic algorithms on CEC2015 niching benchmark test functions and on five additional high-dimensional multimodal functions. The experimental results demonstrate that WSA-IC statistically outperforms other niching metaheuristic algorithms on most test functions.
Li @cite_35 proposed Fitness-Distance-Ratio based PSO (FERPSO) algorithm, which utilities FER to avoid specifying any niching parameter, for multimodal function optimization. The FER value between particle @math and particle @math is shown as follows.
@cite_35
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One of the most critical issues that remains to be fully addressed in existing multimodal evolutionary algorithms is the difficulty in pre-specifying parameters used for estimating how far apart optima are. These parameters are typically represented as some sorts of niching parameters in existing EAs. Without prior knowledge of a problem, it is almost impossible to determine appropriate values for such niching parameters. This paper proposes a PSO for multimodal optimization that removes the need of these niching parameters. Our results show that the proposed algorithm, Fitness Euclidean-distance Ratio based PSO (FER-PSO) is able to reliably locate multiple global optima on the search landscape over some widely used multimodal optimization test functions, given that the population size is sufficiently large.