Whale swarm algorithm with the mechanism of identifying and escaping from extreme points for multimodal function optimization

  • Bing Zeng
  • Xinyu Li
  • Liang GaoEmail author
  • Yuyan Zhang
  • Haozhen Dong
Original Article


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 \(T_{\mathrm{s}}\) and fitness threshold \(T_{\mathrm{f}}\)), 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.


Whale swarm algorithm Multimodal optimization Metaheuristic algorithm Niching Extreme point 



This study was funded by the National Natural Science Foundation of China (NSFC) (51825502, 51775216, and 51721092), Natural Science Foundation of Hubei Province (2018CFA078), and the Program for HUST Academic Frontier Youth Team.

Compliance with ethical standards

Conflict of interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “Whale swarm algorithm with the mechanism of identifying and escaping from extreme points for multimodal function optimization.”


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and EngineeringHuazhong University of Science and TechnologyWuhanChina

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