A Generalized theoretical deterministic particle swarm model

Abstract

Particle swarm optimization (PSO) is a well known population-based search algorithm, originally developed by Kennedy and Eberhart in 1995. The PSO has been utilized in a variety of application domains, providing a wealth of empirical evidence for its effectiveness as an optimizer. The PSO itself has undergone many alterations subsequent to its inception, some of which are fundamental to the PSO's core behavior, others have been more application specific. The fundamental alterations to the PSO have to a large extent been a result of theoretical analysis of the PSO's particle's long term trajectory. The most obvious example, is the need for velocity clamping in the original PSO. While there were empirical fndings that suggested that each particle's velocity was increasing at a rapid rate, it was only once a solid theoretical study was performed that the reason for the velocity explosion was understood. There has been a large amount of theoretical research done on the PSO, both for the deterministic model, and more recently for the stochastic model. This thesis presents an extension to the theoretical deterministic PSO model. Under the extended model, conditions for particle convergence to a point are derived. At present all theoretical PSO research is done under the stagnation assumption, in some form or another. The analysis done under the stagnation assumption is one where the personal best and neighborhood best are assumed to be non-changing. While analysis under the stagnation assumption is very informative, it could never provide a complete description of a PSO's behavior. Furthermore, the assumption implicitly removes the notion of a social network structure from the analysis. The model used in this thesis greatly weakens the stagnation assumption, by instead assuming that each particle's personal best and neighborhood best can occupy an arbitrarily large number of unique positions. Empirical results are presented to support the theoretical fndings.