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_28 proposed Species-based DE (SDE) algorithm to solve multimodal optimization problems via introducing speciation technique. In SDE algorithm, when the number of member individuals of a species is less than a predefined value, the algorithm will randomly generate new individuals within the radius of species seed until the species size reaches the predefined value. Then, each identified species runs the conventional DE algorithm by itself. In addition, if the fitness of an offspring is the same as that of its species seed, this offspring will be replaced by a randomly generated new individual. These two mechanisms have improved the efficiency of SDE algorithm significantly.
@cite_28
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In this paper differential evolution is extended by using the notion of speciation for solving multimodal optimization problems. The proposed species-based DE (SDE) is able to locate multiple global optima simultaneously through adaptive formation of multiple species (or subpopulations) in an DE population at each iteration step. Each species functions as an DE by itself. Successive local improvements through species formation can eventually transform into global improvements in identifying multiple global optima. In this study the performance of SDE is compared with another recently proposed DE variant CrowdingDE. The computational complexity of SDE, the effect of population size and species radius on SDE are investigated. SDE is found to be more computationally efficient than CrowdingDE over a number of benchmark multimodal test functions.