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.
The @math PSO niching algorithms using ring topology, such as @math , @math , @math - @math and @math - @math , were also proposed by Li @cite_31 for multimodal function optimization. These ring topology based PSO niching algorithms also remove the need of specifying any niching parameter. Taking @math for example, a particle's neighboring best point @math , shown in Eq. , is set as the best one among the best previous positions of its two immediate neighbors (i.e., left and right neighbors identified by population indices). With the ring topology methods, these @math PSO algorithms are able to form multiple subpopulations over successive iterations. Li showed that the @math PSO algorithms using ring topology can provide comparable or better performance than SPSO and FERPSO on some test functions.
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Niching is an important technique for multimodal optimization. Most existing niching methods require specification of certain niching parameters in order to perform well. These niching parameters, often used to inform a niching algorithm how far apart between two closest optima or the number of optima in the search space, are typically difficult to set as they are problem dependent. This paper describes a simple yet effective niching algorithm, a particle swarm optimization (PSO) algorithm using a ring neighborhood topology, which does not require any niching parameters. A PSO algorithm using the ring topology can operate as a niching algorithm by using individual particles' local memories to form a stable network retaining the best positions found so far, while these particles explore the search space more broadly. Given a reasonably large population uniformly distributed in the search space, PSO algorithms using the ring topology are able to form stable niches across different local neighborhoods, eventually locating multiple global local optima. The complexity of these niching algorithms is only O(N), where N is the population size. Experimental results suggest that PSO algorithms using the ring topology are able to provide superior and more consistent performance over some existing PSO niching algorithms that require niching parameters.