Improved chaotic particle swarm optimization with a perturbation-based chaotic system for a virtual quartic function

In this paper, we discuss the particle swarm optimization method (PSO). In particular, we focus on the CPSO-VQO, a PSO with a perturbation-based chaotic system derived from the steepest descent method for a virtual quartic function having global minima at the lbest and gbest, which selects the updating system of the particle's position from the standard system of the original PSO and the chaotic one on the basis of a threshold distance between two bests. Although the good performance of CPSO-VQO is reported, it is not so easy to select appropriate parameter values of its chaotic system for each problem because the bifurcation structure of the chaotic system depends on the distance of two bests, and, moreover, it is required an appropriate threshold for selecting the updating system. Therefore, we improve the CPSO-VQO by proposing a modified chaotic system having the bifurcation structure irrelevant to the distance of two bests, and a new stochastic selection of the updating system. In addition, we theoretically show the desirable properties of the modified chaotic system and evaluate the improved CPSO-VQOs called CPSO-TSV and CPSO-SSV.