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A Convergence Proof for the Particle Swarm Optimiser

Bergh F. van den, Engelbrecht A.P.

Fundamenta Informaticae

2011

Vol. 105, nr 4

341-374

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.

Particle Swarms for Linearly Constrained Optimisation

Paquet U., Engelbrecht A.P.

Fundamenta Informaticae

2007

Vol. 76, nr 1-2

147-170

Particle Swarm Optimisation (PSO) has proved to be a very useful algorithm to optimise unconstrained functions. This paper extends PSO to a Linear PSO (LPSO) to optimise functions constrained by a set of equality constraints of the form Ax = b. By initialising particles within a constrained hyperplane, the LPSO is guaranteed to `fly' only through this hyperplane. A criterion on the initial swarm stipulates when the optimum solution can possibly be reached. The Linear PSO is modified to the Converging Linear PSO, for which it is proved to always find at least a local minimum. Experimental results are given, which compare the LPSO and CLPSO with Genocop II.