Comparison of linear and classical velocity update rules in particle swarm optimization: notes on diversity

In this study, we investigate the significance of diversity in the particle swarm optimization (PSO) algorithm. To do so, we study two different implementations of the PSO, being the so-called linear and classical PSO formulations. While the behaviour of these two implementations is markedly different, they only differ in the formulation of the velocity update rule. In fact, the differences are merely due to subtle differences in the introduction of randomness into the algorithm. In this paper, we show that in algorithms employing the linear PSO velocity update rule, particle trajectories collapse to line searches in n-dimensional space. The classical formulation does not suffer this drawback. Instead, directional diverse stochastic search trajectories are retained, which in turn helps to alleviate premature convergence. The performance of the classical implementation is therefore superior for all test problems considered, due to the presence of adequate diversity. Copyright q 2006 John Wiley & Sons, Ltd.