A discrete-time switched linear model of the particle swarm optimization algorithm

In this paper, the convergence issue of the Particle Swarm Optimization (PSO) algorithm is investigated. Most of the models of PSO algorithms are time-invariant linear models with the assumption the local and global best solutions do not change, i.e., the stagnation assumption. However, in this paper, a discrete-time switched linear model is introduced to study the stability and convergence of the PSO algorithm without the stagnation assumption. By considering the updates of local best positions and global best solutions, a sequence of state transform matrixes is generated during the searching process. The semistability of the proposed switched linear system is studied. The conditions of the convergence in mean and convergence in probability are derived by using the recently developed results in paracontraction. Moreover, numerical examples are provided to verify the results proposed in this paper.