Control-theoretic analysis of exploitation and exploration of the PSO algorithm

The behavior of the particle swarm optimization (PSO) algorithm is analyzed by regarding its dynamics as a system with multiplicative noise and applying control-theoretic analysis methods. In order to evaluate the convergence and diversity of the PSO algorithm, two measures related to the decay rate and l² gain of the PSO dynamics are used. These measures are characterized by linear matrix inequalities (LMIs) and are therefore efficiently computed by convex optimization tools. As for the measure related to the decay rate, moreover, an alternative condition in terms of nonlinear scalar inequalities is presented which can be checked more shortly and accurately than the LMI. Numerical experiments suggest that the measures are effective enough to evaluate the convergence and diversity of the PSO algorithm, which results in better understanding of the PSO algorithm from the viewpoints of exploitation and exploration abilities.