A generalized theoretical deterministic particle swarm model

A number of theoretical studies of particle swarm optimization (PSO) have been done to gain a better understanding of the dynamics of the algorithm and the behavior of the particles under different conditions. These theoretical analyses have been performed for both the deterministic PSO model and more recently for the stochastic model. However, all current theoretical analyses of the PSO algorithm were based on the stagnation assumption, in some form or another. The analysis done under the stagnation assumption is one where the personal best and neighborhood best positions 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. This paper presents a generalization to the theoretical deterministic PSO model. Under the generalized model, conditions for particle convergence to a point are derived. The model used in this paper 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. It was found that the conditions derived in previous theoretical deterministic PSO research could be obtained as a specialization of the new generalized model proposed. Empirical results are presented to support the theoretical findings.