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 speciation technique was also introduced into the conventional PSO by Li @cite_22 to solve multimodal optimization problems. In each iteration of Species-based PSO (SPSO), after the population is divided into multiple species and the species seeds are determined, each species seed is assigned to its member individuals as the @math . Then, each individual updates its position according to the iterative equations of velocity and position in the @math PSO. The experimental results showed that SPSO is comparable or better than SNGA @cite_2 , SCGA and NichePSO @cite_11 over a set of multimodal functions.
@cite_11 @cite_22 @cite_2
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This paper proposes an improved particle swarm optimizer using the notion of species to determine its neighbourhood best values, for solving multimodal optimization problems. In the proposed species- based PSO (SPSO), the swarm population is divided into species sub- populations based on their similarity. Each species is grouped around a dominating particle called the species seed. At each iteration step, species seeds are identified from the entire population and then adopted as neig- hbourhood bests for these individual species groups separately. Species are formed adaptively at each step based on the feedback obtained from the multimodal fitness landscape. Over successive iterations, species are able to simultaneously optimize towards multiple optima, regardless of if they are global or local optima. Our experiments demonstrated that SPSO is very effective in dealing with multimodal optimization functions with lower dimensions. A technique is described that allows unimodal function optimization methods to be extended to locate all optima of multimodal problems efficiently. We describe an algorithm based on a traditional genetic algorithm (GA). This technique involves iterating the GA but uses knowledge gained during one iteration to avoid re-searching, on subsequent iterations, regions of problem space where solutions have already been found. This gain is achieved by applying a fitness derating function to the raw fitness function, so that fitness values are depressed in the regions of the problem space where solutions have already been found. Consequently, the likelihood of discovering a new solution on each iteration is dramatically increased. The technique may be used with various styles of GAs or with other optimization methods, such as simulated annealing. The effectiveness of the algorithm is demonstrated on a number of multimodal test functions. The technique is at least as fast as fitness sharing methods. It provides an acceleration of between 1 and l0p on a problem with p optima, depending on the value of p and the convergence time complexity.