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The particle swarm is an algorithm for finding optimal regions of complex search spaces through the interaction of individuals in a population of particles. Even though the algorithm, which is based on a metaphor of social interaction, has been shown to perform well, researchers have not adequately explained how it works. Further, traditional versions of the algorithm have had some undesirable dynamical properties, notably the particles' velocities needed to be limited in order to control their trajectories. The present 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.

The particle swarm algorithm adjusts the trajectories of a population of "particles" through a problem space on the basis of information about each particle's previous best performance and the best previous performance of its neighbors. Previous versions of the particle swarm have operated in continuous space, where trajectories are defined as changes in position on some number of dimensions. The present paper reports a reworking of the algorithm to operate on discrete binary variables. In the binary version, trajectories are changes in the probability that a coordinate will take on a zero or one value. Examples, applications, and issues are discussed.

Particle swarm optimization (PSO) has undergone many changes since its introduction in 1995. As researchers have learned about the technique, they have derived new versions, developed new applications, and published theoretical studies of the effects of the various parameters and aspects of the algorithm. This paper comprises a snapshot of particle swarming from the authors' perspective, including variations in the algorithm, current and ongoing research, applications and open problems.