Comparison of linear and classical velocity update rules in particle swarm optimization: notes on scale and frame invariance

In this paper we investigate whether the particle swarm optimization (PSO) algorithm is invariant of the scale and frame (i.e. translation and rotation) in which an objective function is posed. To do so, we study the linear and classical velocity update rules. We will show that the linear velocity update rule is scale and frame invariant, but that the classical velocity update rule lacks rotational invariance. It is known that the linear velocity update rule lacks diversity, resulting in particle trajectories that collapse to line searches. In contrast, the classical velocity update rule maintains diverse (space filling) particle trajectories. To illustrate that diversity and invariance are not necessarily exclusive, we propose a new velocity update rule. This update rule, which is just one example of many possible formulations, is rotationally invariant and at the same time directionally diverse. This is achieved through consistent perturbation of the search directions. We quantify the (in)variance and performance of the three different implementations using a popular test set. The test problems are evaluated in both unrotated and arbitrarily rotated reference frames. Copyright q 2006 John Wiley & Sons, Ltd.