Contributions to Dynamic Analysis of Differential Evolution Algorithms

The Differential Evolution (DE) algorithm is one of the most successful evolutionary computation techniques. However, its structure is not trivially translatable in terms of mathematical transformations that describe its population dynamics. In this work, analytical expressions are developed for the probability of enhancement of individuals after each application of a mutation operator followed by a crossover operation, assuming a population distributed radially around the optimum for the sphere objective function, considering the DE/rand/1/bin and the DE/rand/1/exp algorithm versions. These expressions are validated by numerical experiments. Considering quadratic functions given by $f(x)=xTDTDx$ and populations distributed according to the linear transformation $D-1$ of a radially distributed population, it is also shown that the expressions still hold in the cases when $f(x)$ is separable (⁠$D$ is diagonal) and when $D$ is any nonsingular matrix and the crossover rate is $Cr=1.0$⁠. The expressions are employed for the analysis of DE population dynamics. The analysis is extended to more complex situations, reaching rather precise predictions of the effect of problem dimension and of the choice of algorithm parameters.