Verification of a hypothesis about unification and simplification for position updating formulas in particle swarm optimization

Particle swarm optimization (PSO) has been a popular research area in artificial intelligence technology, where the two issues of theoretical analysis and premature convergence have been the focus of attention. However, due to complex dynamics in particle swarm, the former has been conducted only in simplified systems. And the latter has been dealt with only by introducing some additional operations, which inevitably increases the complexity of PSO and makes the theoretical analysis more difficult. To handle the above problems, we already proposed a unified and simplified formula for position updating in the existing algorithms, but that formula depends heavily on the hypothesis of that new positions of particles are centered on their weighted experience. In this paper, we selected ten algorithms that were widely used or more novel, and generated a large number of data samples to test their frequency histograms. The experiment results verified this hypothesis, and further proved the correctness of the unification and simplification for position updating formulas. Introduction Particle swarm optimization (PSO), first introduced by Kennedy and Eberhart , has been a popular research area in artificial intelligence technology, which has been used to solve a range of optimization problems. In PSO, the two issues of theoretical analysis and premature convergence have been the focus of attention. However, due to complex dynamics in particle swarm, the former has been conducted only in simplified systems, for example, not considering the random number in the position updating formulas , and keeping the previous best position unchanged . And the latter has been dealt with only by introducing some additional operations, for example, modifying the coefficients’ values , changing the communication topologies , and combining genetic operators , which inevitably increases the complexity of PSO and makes the theoretical analysis more difficult. In order to handle the above problems, reference [7] proposes a unified and simplified formula for position updating in the existing PSO algorithms. That formula simplifies the multiple-order stochastic difference equation to a first-order stochastic difference equation, reduces the multiple parameters to only one parameter, and makes it is easier to analyze the convergence and to control the search behavior of particles. However, in this reference, the process of unification and simplification depends heavily on the hypothesis of that new positions of particles are centered on their weighted experience. In this paper, we will verify this hypothesis, which further proves the correctness of the unification and simplification for position updating formulas. The rest of this paper is organized as follows. Section 2 describes standard PSO and introduces the unified and simplified position updating formula. This is followed in Section 3 by the methodology to verify the hypothesis. Section 4 is about experimental configuration, results and discussions. Conclusions are given in Section 5. 2nd Workshop on Advanced Research and Technology in Industry Applications (WARTIA 2016) © 2016. The authors Published by Atlantis Press 1841 Standard Particle Swarm Optimization and a Unified and Simplified Position Updating Formula To build a baseline for performance testing of improvements to PSO, in Proceedings of the 2007 IEEE Swarm Intelligence Symposium, D. Bratton and J. Kennedy defined the constriction PSO as the standard PSO (abbreviated as SPSO-SIS). In SPSO-SIS, a particle’s position updating formula is often described as the following Eq. 1.     1 1 2 2 ( ) ( 1) * ( -1) * * ( 1) * * ( 1) i n X t X t V t C rand P X t C rand P X t           (1) In this notation, V and X are the velocity and position of the current particle; Pi and Pn are the previous best positions of the current particle and of the current particle’s neighborhood;  , C1 and C2 are called a inertia weight, a confidence parameter and a social parameter, respectively; rand1 and rand2 are independent uniform random numbers; t is the generation number. So far, there have been hundreds of modified PSO. Reference [7] unified most of these PSO’s position updating formulas into the following formula: ( ) * ( ) * j k k j k X t a X t j b P      , (2) where X(t) and X(t-j) are the current particle’s positions in the t-th and the (t-j)-th generation, Pk are is the k-th particle’s previous best position, and aj and bk are the coefficients. The following equation 1 j k j k a b     (3) was proven in reference [7] . Bringing Eq. 3 into Eq. 2, it is gotten that   ( ) * ( ) 1 * * ( ) j j j j j j X t a X t j a Q Q a X t j Q                  , (4)