Spectral radius of the canonical particle swarm optimization

Existing stability analysis of particle swarm optimization (PSO) algorithm, a class of widely used stochastic global optimization methods, is merely based on the constant transfer matrix, which is in fact the expectation of step-varying transfer matrices involving random variables, however, theoretically speaking, the stability of standard PSO algorithm involves one challenging yet long-term ignored problem of calculating spectral radius of the product of asymmetric transfer matrices at each step, whose mean and variance is carefully investigated in this contribution with the Monte Carlo approach. The extensive experimental studies conducted provides the guideline for parameter selection and the tradeoff between exploration ability and exploitation ability, and analyzes the relationship between the mean spectral radius and inertia weight as well as acceleration coefficients in PSO algorithm. Our results indicate that the existing stability analysis is essentially meaningless in sense that most sample trajectories of the system do not coincide with those analyzed in previous studies which simply utilize the constant transfer matrix.