An approximate gaussian model of Differential Evolution with spherical fitness functions

An analytical method is proposed to study the evolutionary stochastic properties of the population in differential evolution (DE) for a spherical function model. Properties of mutation and selection are developed, based on which a Gaussian approximate model of DE is introduced to facilitate mathematical derivations. The evolutionary dynamics and the convergence behavior of DE are investigated based on the derived analytical formulae and their appropriateness is verified by experimental results. It is shown that the lower limit of mutation factor should be as high as 0.68 to avoid premature convergence if the initial population is isotropically normally distributed and infinitely far from the optimum (i.e., the function landscape becomes a hyper-plane). The lower limit, however, may be decreased if the population becomes closer to the optimum and an accordingly smaller mutation factor is beneficial to speed up the convergence. This motivates future research to improve DE by dynamically adapting control parameters as evolution search proceeds.