Stochastic stability analysis of composite dynamic system for particle swarm optimization

The stochastic stability analysis for particle swarm optimization (PSO) has met formidable challenges due to the intricate influences on particles’ motion from collaboration capability, cognitive competence, selective randomness and environmental complexity. This paper firstly regards the above-mentioned factors as forces acting on particles of PSO: interaction, damping, randomness and external forces. Based on this mechanical analogy, a novel stochastic composite dynamic model (SCDM) for PSO is proposed which can be classified into two parts: diffusion process and drift process. The diffusion process can not be easily obtained by analytical analysis, but its approximate distribution can be experimentally justified to obey Gaussian model, and the drift process is partly determined by its optimization function.To analyze the influence on PSO from the external environment, linear and quadratic optimization functions are chose to simplify the analysis. And the SCMD’s stability is discussed via constructing the responding Lyapunov functions. We conclude that the gradient of linear function only decides the uniform ultimate bound while the gradient of quadratic function affects the stable conditions besides the uniform ultimate bound. Furthermore, the stable point of PSO is proved to be the optimum of quadratic function. The convergence time is also discussed to measure PSO’s performance.