The Particle Swarm Optimiser (PSO) is a population based stochastic optimisation algorithm,
empirically shown to be efficient and robust. This paper provides a proof to show that the
original PSO does not have guaranteed convergence to a local optimum. A flaw in the original PSO
is identified which causes stagnation of the swarm. Correction of this flaw results in a PSO algorithm
with guaranteed convergence to a local minimum. Further extensions with provable global
convergence are also described. Experimental results are provided to elucidate the behavior of the
modified PSO as well as PSO variations with global convergence.