Theoretical analysis of particle swarm trajectories through a mechanical analogy

Particle swarm optimization (PSO) is an evolutionary computational technique used for optimization. It is motivated by the social behavior of individuals in large groups in nature. In this paper we demonstrate that PSO is a particular discretization of a damped mass-spring system. Using this physical analogy we derive the differential and the discrete PSO models, generalizing standard PSO for any iteration time and discretization step. We present a systematic study of discrete PSO trajectories, showing analogies to their continuous counterparts. As a result of this analysis, we identify criteria such as trajectory attenuation, trajectory oscillation and center attraction potential, which can be used to select promising PSO parameters and explain the success of some parameter sets proposed in the literature. These results are validated by numerical experiments using well-known benchmark functions which have a progressive degree of difficulty. PSO parameters from the complex stability region, with inertia values from 0.5 to 0.9, and medium to high total accelerations (1.7 to 2.2) seem to give systematically good results. This zone is also the best for cost functions with elongated flat valleys. In case of multiple local minima, the real stability zone of negative inertia values (−0.6 to −0.5) and total accelerations (0.6 to 0.7) seems to be the best with respect to the percentage of success.