Upper Bounds for Particle Location Variance Convergence Measures in the Stochastic Model of Particle Swarm

Properties of the Particle Swarm Optimization method depend on the parameters configuring the behavior of single particles. In the stochastic particle stability analysis, there were proposed recurrent formulas of the expected value and the variance of particle location. Recently, a particle variance convergence time measure was defined, and for a selected set of configurations, a simple explicit formula for particle location variance under the stagnation assumption was derived. We propose an explicit formula for the upper bound of the particle variance convergence time for the same selected set of configurations. Furthermore, we propose an explicit formula for the upper bound for the earlier defined measure of the particle location variance stasis time. We visualize both upper bounds and verify them in simulations.