  Particle Swarm Optimization (PSO) is a meta-heuristic for continuous
black-box optimization problems. In this paper we focus on the convergence of
the particle swarm, i.e., the exploitation phase of the algorithm. We introduce
a new convergence indicator that can be used to calculate whether the particles
will finally converge to a single point or diverge. Using this convergence
indicator we provide the actual bounds completely characterizing parameter
regions that lead to a converging swarm. Our bounds extend the parameter
regions where convergence is guaranteed compared to bounds induced by
converging variance which are usually used in the literature. To evaluate our
criterion we describe a numerical approximation using cubic spline
interpolation. Finally we provide experiments showing that our concept,
formulas and the resulting convergence bounds represent the actual behavior of
PSO.
