Convergence and Spectral Radius Analysis and Parameter Selection for the Particle Swarm Optimization Algorithm Based on the Stochastic Process

摘要

Randomness and parameter selection in the Particle Swarm Optimization (PSO) algorithm had great influence on its performance. This study presented a formal convergence and spectral radius analysis of the standard PSO algorithm model, where some of the parameters were stochastic. Based on the analysis of the relationship of {ω, c1, c2}, a sufficient condition was given to guarantee that the PSO algorithm was mean-square convergent, using the stochastic process theory. Then, the mean spectral radius was constructed. According to the relationship between the spectral radius and the convergent speed, it was shown that, a small spectral radius lead to a faster convergent speed than a big one. By optimizing the mean spectral radius of the PSO algorithm in the mean-square convergent region, a minimum spectral radius and corresponding parameter selection guidelines were derived to guarantee that the PSO algorithm was mean-square convergent and had a fast convergent speed in the stochastic sense. Finally, one parameter selection {c1 = c2 = 2, ω = 0.4222} was proposed. with the parameter, the study gave examples whose performance on benchmark functions were superior to previously published results.