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Dynamics and stability of the sampling distribution of particle swarm optimisers via moment analysis

For stochastic optimisation algorithms, knowing the probability distribution with which an algorithm allocates new samples in the search space is very important, since this explains how the algorithm really works and is a prerequisite to being able to match algorithms to problems. This is the only way to beat the limitations highlighted by the no-free lunch theory. Yet, the sampling distribution for velocity-based particle swarm optimisers has remained a mystery for the whole of the first decade of PSO research. In this paper, a method is presented that allows one to exactly determine all the characteristics of a PSO's sampling distribution and explain how it changes over time during stagnation (i.e., while particles are in search for a better personal best) for a large class of PSO's.