# Particle Swarm Optimization Based on a Novel Evaluation of Diversity
## 1 Introduction  
Inspired by the emergent motion of the foraging behavior of a flock of birds in nature, the particle swarm optimization (PSO) was first proposed by [PERSON] and [PERSON] [1, 2] in 1995 to solve the continuous nonlinear optimization problems. Compared with other evolutionary algorithms, the PSO has attracted much attention since it was proposed because of its fewer control parameters and better convergence. Nowadays, PSO has been widely used in many fields, such as communication networks [3, 4], medicine engineering [5, 6, 7], task scheduling [8, 9], energy management [10], linguistics studies [11], supply chain management [12] and neural networks [13, 14]. Despite its robustness for solving complex optimization problems, rapid convergence may cause the swarm to be easily trapped into some local optima when solving multimodal problems via PSO [15, 16, 17, 18, 19, 20, 21, 22]. Therefore, reasonably using the swarm's exploration ability (global investigation of the search place) and exploitation ability (finer search around a local optimum) is a crucial factor for PSO's success, especially for complex multimodal problems having a large number of local minima. For this purpose, several PSO variants have been reported. In the following, we shall discuss three main types: PSO with adjusted parameters, PSO with various topology structures, and PSO with hybrid strategies. PSO with adjusted parameters: It is evident that appropriate control parameters, such as the inertia weight \(\omega\) and the acceleration coefficients \(c_{1}\) and \(c_{2}\), have significant effects on the exploration and exploitation abilities of the swarm. By increasing the value of \(\omega\) significantly diversity of the swarm increases while increasing the values of \(c_{1}\) and \(c_{2}\) accelerates the particles towards the historical optimal position or the optimal position of the whole swarm. [PERSON] [15] proposed an adaptive particle swarm optimization (APSO) algorithm in 2009, which updated control parameters (\(\omega\), \(c_{1}\) and \(c_{2}\)) adaptively based on the distribution of positions and fitness of the swarm. [PERSON] et al. [23] proposed an inertia weight adjustment scheme based on Bayesian techniques to enhance the swarm's exploitation ability. [PERSON] et al. [24] proposed a PSO algorithm (SRPSO) that employed a self-regulating inertia weight strategy to the best particle to enhance the exploration ability. [PERSON] [25] proposed an adaptive approach which determines the inertia weight in different dimensions for each particle, based on its performance and distance from its best position. PSO with various topology structures: PSO with different topology structures, namely: improved fixed topology structures, dynamic topology structures, and multi-swarms, have been shown to be efficient in controlling the exploration and exploitation capabilities [26; 27; 28]. [PERSON] [29] proposed a small-world social network and studied different topologies' influences on PSO algorithms' performances. It was found that sparsely connected networks were suitable for complex functions, and densely connected networks were useful for simple functions. [PERSON] [30] first introduced the concept of dynamic topologies for PSO, where sub-swarms (each was a particle initially) were gradually merged as the evolution progressed. [PERSON] et al. [28] proposed an adaptive PSO called TRIBES, which were multiple sub-swarms with independent topological structures changing over time. [PERSON] et al. [31] presented dynamic topologies by growing the sub-swarms' sizes, merging the sub-swarms. [PERSON] [32] proposed DEPSO, which generated a weighted search center based on top-\(k\) elite particles to guide the swarm. [PERSON] and [PERSON] [33] divided a d-dimension swarm into k (k < d) sub-swarms and made sub-swarms cooperated by exchanging their information (e.g., the best particle). [PERSON] and [PERSON] [34] proposed a multi-swarm PSO for dynamic functions with the optimal values changing over time.