Stability Analysis of the Particle Swarm Optimization Without Stagnation Assumption

In this letter, we study the first-and second-order stabilities of a stochastic recurrence relation that represents a class of particle swarm optimization (PSO) algorithms. We assume that the personal and global best vectors in that relation are random variables (with arbitrary means and variances) that are updated during the run so that our calculations do not require the stagnation assumption. We prove that the convergence of expectation and variance of the positions generated by that relation is independent of the mean and variance of the distribution of the personal and global best vectors. We also provide convergence boundaries for that relation and compare them with those of standard PSO algorithms (as a specific case of the stochastic recurrence relation) provided in earlier studies.