The Convergence Basis of Particle Swarm Optimization

The general mathematical description of the Particle Swarm Optimization is given in this paper. Also has proved that the solution space of PSO is Normed Space, the iterative relations of PSO are contraction mapping. Using the theorem of Banach space and contraction mapping principle prove the existence and uniqueness of the convergence position. And summary the mathematical description of the convergence, using the theory of probability and stochastic process give the parameter condition that influences the stability of PSO. At last proves that if parameter satisfying the condition the probability that Particle Swarm Optimization converging the best position is one.