Abstract
Particle swarm optimization (PSO) is a stochastic optimization algorithm, whose distribution is completely determined by its moments. Up to now, most of previous theoretical researches pay attention to the first and/or second moments only to give its stable condition, ignoring the analysis of third moment probably due to its complex third-order recurrence equation. To enrich theoretical cornerstones of PSO, this work analyzes its first, second and third moments, respectively. Built on first-order fixed point and two novel simplified strategies, we present the second and third order standardized recurrence equations for PSO. Their corresponding characteristic equations can be analyzed through the derivative functions to obtain the necessary and sufficient convergence condition. Besides, we show that the specific third-order stable region is slightly less than the second-order one in illustrations.
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