  Recently, much progress has been made on particle swarm optimization (PSO). A
number of works have been devoted to analyzing the convergence of the
underlying algorithms. Nevertheless, in most cases, rather simplified
hypotheses are used. For example, it often assumes that the swarm has only one
particle. In addition, more often than not, the variables and the points of
attraction are assumed to remain constant throughout the optimization process.
In reality, such assumptions are often violated. Moreover, there are no
rigorous rates of convergence results available to date for the particle swarm,
to the best of our knowledge. In this paper, we consider a general form of PSO
algorithms, and analyze asymptotic properties of the algorithms using
stochastic approximation methods. We introduce four coefficients and rewrite
the PSO procedure as a stochastic approximation type iterative algorithm. Then
we analyze its convergence using weak convergence method. It is proved that a
suitably scaled sequence of swarms converge to the solution of an ordinary
differential equation. We also establish certain stability results. Moreover,
convergence rates are ascertained by using weak convergence method. A centered
and scaled sequence of the estimation errors is shown to have a diffusion
limit.
