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Multimodal optimization (MMO), means the optimization problem where there are multiple global or local optimal solutions (Goldberg & Richardson, 1987). In actual production, many problems require not only a single global optimal solution, but also multiple potential local optimal solutions in parallel. Multimodal optimization has been widely used in scientifically researches and produces such as image segmentation, neural network structure, optimization of weights and maximizing the benefits of factory production products (Zhang, Wang & Ji., 2015; W., 2013). These problems have the characteristics of non-linear, large-scale, dynamic, and multimodal. Therefore, this kind of problem is called a multimodal optimization problem, and the extracted function from it is called a multimodal function. The traditional mathematical method of finding extreme points by derivation can only find a single global optimal solution, which is no longer applicable to such problems. For this type of problem where the derivative of the objective function cannot be obtained, a direct search can only be performed in the search space.
Particle swarm optimization (PSO) is a population-based optimization method based on collective artificial intelligence (Kennedy & Ederhar, 1995). The PSO algorithm mimics the behavior of bird flocks foraging. Particles update their positions and velocities based on their own experience and the experience of the entire population. After experiments of changing the parameter setting, it is found that the PSO performs optimally among many optimization algorithms. The algorithm was already applied in many fields like the multi-objective optimizations, the neural network training and the image segmentation (Zhang, W., Li, G., Zhang, W., Liang, J., & Yen, G. G., 2019; H. Melo, & J. Watada, 2016; Farshi, T. R., Drake, J. H., & Zcan, E, 2020).
The particle in the PSO algorithm is memorable, that is, all particles have the memory of the previous optimal solutions of their own and the global optimal solution of the entire particle swarm, which is very advantageous for solving optimization problems. However, the PSO algorithm has the disadvantages of easily losing diversity of the solution and premature convergence to local optimal solutions (Kennedy & Ederhar, 2002). It can be known from the formula of updating the velocities and positions of particles in the PSO algorithm that when a particle is located at a certain global or local optimal solution, the velocity depends only on the previous velocity, so the velocity may be small and the position hardly changes. After that, other particles will converge towards this particle and will not move after convergence. Therefore, the PSO algorithm is prone to the premature phenomenon, and after locating an optimal solution, it will not search for other possible optimal solutions. According to these disadvantages, in order for the PSO algorithm to find multiple optimal solutions, it is necessary to solve the problem of premature convergence.