Particle Swarm Optimization – Stochastic Trajectory Analysis and Parameter Selection

Two important topics in Particle Swarm Optimization (PSO) research filed are trajectory analysis of particles and parameter selection method. Trajectory analysis is important because it can help to determine where the position of each particle is at each evolutionary step, and consequently it can help to clarify the running mechanism of PSO algorithm, so as to explain why and when PSO algorithm can be successful to solve optimization problems. Parameter selection of PSO algorithm is important because the performance of PSO algorithm is sensitive to the chosen parameters. Till now, some research works have been published in literatures to investigate both of these two topics, but unfortunately, the trajectory analysis is based on simplified deterministic algorithms, regardless of the randomness in real PSO algorithm; and the parameter selection is based on experimental results instead of theoretical results. This chapter is proposed to investigate both of these two important topics. In this chapter, the trajectory of particle in a general PSO algorithm is theoretically investigated, considering the randomness thoroughly. For arbitrary dimension d of an arbitrary particle i in the general particle swarm system, the update equations investigated in this chapter are given in Eqs. (1) and (2), where t is the evolutionary step, V is the velocity of particle i, X is the position of particle i, Pi is the history best position found by particle i, and Pg is the history best position found by the total swarm.