## 1 Introduction  
Many real-world optimization problems such as classification problems, nonlinear processes, path selection, scheduling problems, structural designs, and text mining may have multimode, non-convex, nonlinear, discontinuous, and non-differentiable characteristics, which makes these problems extremely complex and difficult to solve by traditional methods [1]. To solve these problems better and more effectively, many researchers have developed population-based nature-inspired search optimization algorithms such as genetic algorithms (GAs) [2], particle swarm optimizations (PSOs) [3], ant colony optimizations (ACOs) [4], gravitation search algorithms (GSAs) [5], differential evolutions (DEs) [6], and distribution estimation algorithms (EDAs) [7], proposing various population organization patterns and learning mechanisms, cleverly utilizing individual historical knowledge to help population evolution. These population-based algorithms are capable of solving most optimization problems and are easy to implement. Due to the random initialization of the population and the introduction of random numbers into the search mechanism, their dependence on the initial scheme and gradient information is lessened to a great extent. Among the population-based algorithms, PSO [3] was proposed by [PERSON] and [PERSON] in 1995 based on research on the predation behavior of birds. It finds the best solution through information sharing and cooperation among the individuals in the population. Because of its simplicity, a few parameters adjusted, and its high computational efficiency, PSO has successfully attracted the extensive attention of engineers and scholars in recent decades. The classical PSO constructs a learning mechanism in which each particle realizes its own evolution through self-cognition and social cognition. Self-cognition makes it maintain its own historical experience, and social cognition makes the individuals cooperate and share information. This learning mechanism enables particles to continuously mine better positions from the search space and guide the subsequent search behavior of particles, which gives PSO a strong ability to solve complex problems. The learning model of PSO, that is, the information sharing mode among particles, largely determines the ability of the algorithm to solve complex problems. Generally, it is an effective way to enhance the algorithm ability to fully mine and use the effective information in the population to build an efficient learning model. In the classical PSO, particles perform the learning process by measuring the fitness and selecting the global best-so-far particles or the personal best experience as a learning model [8; 9]. When optimizing the single-mode function, this learning model can help particles quickly find the global optimum. However, PSO may lose its diversity and fall into a premature convergence because the overall learning mode makes particles gradually move to one dominating candidate in the population until convergence, especially when solving nonseparable and high-dimensional optimization problems or optimization problems with many local optimizations [10]. For the population-based evolutionary algorithm, maintaining the diversity of the population gathers increasingly effective evolutionary information. According to the characteristics of population evolution, different evolutionary information is selected as a particle learning example, which balances the exploration and development and guides the population to evolve effectively. For instance, dynamic multiswarm PSO (DMS-PSO) [11] allocates all particles into multiple subswarms. The individuals in each subswarm receive different social information so that each subswarm has different evolutionary paths, and information among the subswarms is shared through population reorganization, which effectively maintains the diversity and makes it perform well on complex multimode problems.