Impacts of Coefficients on Movement Patterns in the Particle Swarm Optimization Algorithm

In this paper, we investigate movement patterns of a particle in the particle swarm optimization (PSO) algorithm. We characterize movement patterns of the particle by two factors: 1) the correlation between its consecutive positions and 2) its range of movement. We introduce the base frequency of movement as a measure for the correlation between positions and the variance of movement as a measure for the range of movement. We determine the base frequency and the variance of movement theoretically and we show how they change with the values of coefficients. We extract a system of equations that enables practitioners to find coefficients' values to guarantee achieving a given base frequency and variance of movement, i.e., control the movement pattern of particles. We also show that if the base frequency of movement for a particle is small, mid range, or large then the particle's position at each iteration is positively correlated (smooth movement), uncorrelated (chaotic movement), or negatively correlated (jumping at each iteration) with its previous positions, respectively. We test the effects of the base frequency and variance of movement on the search ability of particles and we show that small base frequencies (i.e., smooth movement) are more effective when the maximum number of function evaluations is large. We found that the most frequently-used coefficient values in PSO literature impose mid-range base frequencies that correspond with a chaotic movement. We also provide new sets of coefficients that outperform existing ones on a set of benchmark functions.