The particle swarm - explosion, stability, and convergence in a multidimensional complex space

TLDR

The particle swarm is an algorithm that locates optimal regions in complex search spaces by coordinating a population of interacting particles. This paper analyzes a particle’s trajectory in both discrete (algebraic) and continuous (analytical) time to understand its dynamics. The authors develop a complete five‑dimensional depiction that fully describes the particle swarm system and its trajectory behavior. The resulting generalized model introduces coefficients that control convergence, and the derived modifications enhance the optimizer’s ability to find optima on well‑studied test functions.

Abstract

The particle swarm is an algorithm for finding optimal regions of complex search spaces through the interaction of individuals in a population of particles. This paper analyzes a particle's trajectory as it moves in discrete time (the algebraic view), then progresses to the view of it in continuous time (the analytical view). A five-dimensional depiction is developed, which describes the system completely. These analyses lead to a generalized model of the algorithm, containing a set of coefficients to control the system's convergence tendencies. Some results of the particle swarm optimizer, implementing modifications derived from the analysis, suggest methods for altering the original algorithm in ways that eliminate problems and increase the ability of the particle swarm to find optima of some well-studied test functions.