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$f(x)=xTDTDx$$f(x)=xTDTDx$ and populations distributed according to the linear transformation D−1$D-1$$D-1$ of a radially distributed population, it is also shown that the expressions still hold in the cases when f(x)$f(x)$$f(x)$ is separable (⁠D$D$$D$ is diagonal) and when D$D$$D$ is any nonsingular matrix and the crossover rate is Cr=1.0$Cr=1.0$$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.

差分进化（DE）算法是最成功的进化计算技术之一。然而，它的结构并不能简单地转化为描述其种群动态的数学变换。在这项工作中，假设种群围绕球体目标函数的最佳值呈放射状分布，并考虑 DE/rand/1/，则在每次应用变异算子后进行交叉操作后，开发了个体增强概率的解析表达式。 bin 和 DE/rand/1/exp 算法版本。这些表达式通过数值实验得到验证。考虑由下式给出的二次函数F( x ) =X时间D时间dx _$F（X）=X时间D时间DX$$f(x)=xTDTDx$以及根据线性变换分布的种群D- 1$D-1$$D-1$对于径向分布的总体，还表明表达式在以下情况下仍然成立：F（X ）$F（X）$$f(x)$是可分离的（⁠D$D$$D$是对角线）并且当D$D$$D$是任意非奇异矩阵，交叉率为Cr= 1 。0$Cr=1。0$$Cr=1.0$⁠ . 该表达式用于 DE 种群动态分析。该分析扩展到更复杂的情况，对问题维度的影响和算法参数的选择达到相当精确的预测。