## 1 Introduction
However, DMS-PSO is not conducive to the evolution of some subswarms because it adopts the same learning template for different subswarms. A heterogeneous comprehensive learning and dynamic multiswarm PSO (HCLDMS-PSO) [12] is presented, where a comprehensive learning (CL) strategy was used as the exploitation subswarm learning model and an improved DMS strategy was designed to structure the exploration subswarm learning model. The two subswarms rely on the different learning exemplars to update independently to ensure that the population does not lose its diversity. Exploration and exploitation are two important stages for the evolution process of population-based algorithms [13; 14]. In the exploration stage, to seek the global optimal solution, individuals need to reach different regional positions as much as possible, and better diversity is obtained at the population level. In the exploitation stage, particles search in a small area for the sake of obtaining a high-precision optimal solution faster, which can effectively improve the local search ability of the algorithm but sacrifices the diversity of the population and increases the risk of premature convergence. Therefore, the probability of the premature convergence can be significantly reduced by improving the diversity of the population. The multiswarm strategy divides the entire population into several subswarms. Individuals within a subswarm cooperate and share information with other subswarms to produce new offspring individuals, which makes each subswarm evolve along different paths and effectively maintains the diversity of the population during the evolution. To date, the multiswarm strategy has been widely used in population-based evolutionary algorithms because it has natural advantages in maintaining population diversity. Various population division strategies have been proposed such as stochastic division, clustering, fitness sharing, etc., in which the algorithm extends the learning beyond the dominant individuals by improving the probability of poor individuals entering the next generation. Their common starting point is to maintain the diversity of the population. In addition, considering that the update mode of all individuals is the same for the algorithm of a single learning model, the population information cannot be fully used to guide individual evolution, which cannot intelligently deal with different complex situations. In other words, when the algorithm is in different evolutionary stages, or is used to solve different types of optimization problems, the guiding effects of the different learning models are quite different. Therefore, some scholars have begun to explore PS based on a variety of different learning models. For instance, in the triple archives PSO (TAPSO) [15], three different learning models, the Diffident Model, the Mild Model, and the Confident Model, are provided to update particles so that the particles in the population show different search behaviors, meeting the needs of the algorithm in its different evolutionary stages. In the multiswarm PSO with dynamic learning strategy (PSO-DLS) [16], particles take the local optimal position in their subswarms (focusing on exploitation) and a united local best position among subswarms (focusing on exploration), as examples. During the search process, the population can maintain diversity without being affected by the global best position of all the dominant particles. In summary, multiple population structures can offer significant diversity at the population level, and excellent learning strategies can more extensively mine population historical knowledge, which are effective in helping find the global optimum, especially for problems with multiple local optimal solutions. However, most studies have a fixed number of populations. If the population size is large, it helps with the initial exploration of the algorithm, but limits the local development ability in the later stages of evolution, and vice versa.