# Effects of Random Values for Particle Swarm Optimization Algorithm
## 1 Introduction  
Based on the intelligent collective behaviors of some animals such as fish schooling and bird flocking, particle swarm optimization (PSO) algorithm was first introduced by [PERSON] and [PERSON] [1]. This algorithm is a stochastic population based heuristic global optimization technology, and it has advantages of simple implementation and rapid convergence capability [2; 3; 4]. Therefore, PSO algorithm has been widely utilized in function optimization [5], neural network training [6; 7; 8; 9], parameters optimization of fuzzy system [10; 11; 12], and control system [13; 14; 15; 16; 17], etc. However, the PSO algorithm is easily trapped in local optima when it is used to solve complex problems [18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. This disadvantage seriously limits the application of the PSO algorithm. In order to deal with this issue, many modifications or improvements are proposed to improve the performance of the PSO algorithm. Generally, the improved methods include changing the parameter values [19; 20; 21], tuning the inertia weight or population size [22; 23; 24; 25], and combining with other evolution algorithms [26; 27; 28; 29; 30; 31], etc. In recent years, the PSO algorithms for dynamic optimization problems are developed. The multi-swarm PSO algorithms, such as, multi-swarm charged/quantumPSO [32], species-based PSO [33], clustering-based PSO [34], child and parent swarms PSO [35], multi-strategy ensemble PSO [36], chaos mutation-based PSO [37], distributed adaptive PSO [38], and stochastic diffusion search--aided PSO [39], are developed to improve their performances. Furthermore, some dynamic neighborhood topology-based PSO algorithms are developed to deal with dynamic optimization problems [40; 41]. These improvements or modifications have improved the global optimization ability of the PSO algorithm to some extent. However, these methods cannot effectively prevent the stagnation of optimization and premature convergence. This is because the velocity of particle becomes very small in the position of the local optima, which renders the particle unable to escape from the local optimum. Therefore, it is necessary to propose an effective way to make the particle jump out of the local optimum. In the PSO algorithm, the velocity of particle is updated according to its previous velocity and the distances of its current position from its own best position and the group's best position [20]. The coefficients of previous velocity and two distances are inertia weight and random values, respectively. In previous research, a variety of inertia weight strategies were proposed and developed to improve the performance of the PSO algorithm. However, the random values for most modified PSO algorithms are always generated by uniform distribution in the range of [0, 1]. Obviously, the random values represent the weights of two distances for updating the particle velocity. If the range of random values is small, these two distances have little influence on the new particle velocity, which means that the velocity cannot be effectively increased or changed to escape from local optima. In order to improve the global optimization ability of the PSO algorithm, it is necessary to expand the range of random values. In this paper, the random values generated by different probability distributions are utilized to investigate their effects on the PSO algorithms. In addition, the deterministic PSO algorithm, in which the random values are set as 0.5, is investigated for comparison. The performances of PSO algorithms with different types of random values are tested and compared by the experiments of benchmark functions in three space dimensions. The rest of the paper is organized as follow. Section 2 presents the standard PSO algorithm and its modification strategies. The different types of random values are provided in Section 3.