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.
Thomsen @cite_10 proposed Crowding-based DE (CDE) algorithm by introducing crowding method into the conventional DE for multimodal function optimization. In CDE algorithm, the similarity of two individuals is measured by the Euclidean distance between two individuals. The fitness value of an offspring is only compared with that of the most similar individual in the current population, and the offspring replaces the most similar individual if it has better fitness. This replacement scheme can make the population remain diversity in the search space, which has a great contribution to the location of multiple optima. Thomsen showed that CDE algorithm performs better than a fitness sharing DE variant over a group of multimodal functions.
@cite_10
2017258920
Multimodal optimization is an important area of active research within the evolutionary computation community. The ability of algorithms to discover and maintain multiple optima is of great importance - in particular when several global optima exist or when other high-quality solutions might be of interest. The differential evolution algorithm (DE) is extended with a crowding scheme making it capable of tracking and maintaining multiple optima. The introduced CrowdingDE algorithm is compared with a DE using the well-known sharing scheme that penalizes similar candidate solutions. In conclusion, the introduced CrowdingDE outperformed the sharing-based DE algorithm on fourteen commonly used benchmark problems.