Revisiting the analysis of population variance in Differential Evolution algorithms

Abstract

The performance of Differential Evolution (DE) algorithms is highly dependent on the trial population diversity and on the way the control parameter space is sampled. Therefore, identifying critical regions containing control parameters (e.g. scale factor, crossover rate) which can induce undesired behaviour (e.g. premature convergence) is useful. In this context, the aim of the paper is twofold. On one hand, the paper revisits some existing theoretical results on the expected variance of the trial population aiming to provide a comparative image on critical regions in the control parameter space for several DE variants: DE/rand/1/*, DE/best/1/*, DE/rand-to-best/*, DE/either-or. On the other hand, a new theoretical result on DE/rand/1/* population variance evolution is obtained under the assumption that the bound constraints are handled by random reinitialization of infeasible components. The relationship between the probability of violating the bound constraints and the value of the scale factor, F, is theoretically derived for DE/rand/1/* and empirically analyzed for other DE mutation operators.