Influence of crossover on the behavior of Differential Evolution Algorithms

In Differential Evolution Algorithms the crossover operator allows the construction of a new trial element starting from the current and mutant elements. Thus it controls which and how many components are mutated in each element of the current population. This work aims to analyze the impact the crossover operator and its parameter, the crossover rate, has on the behavior of Differential Evolution. The influence of the crossover rate on the distribution of the number of mutated components and on the probability for a component to be taken from the mutant vector (mutation probability) is theoretically analyzed for several variants of crossover, including classical binomial and exponential strategies. For each crossover variant the relationship between the crossover rate and the mutation probability is identified and its impact on the choice and adaptation of control parameters is analyzed theoretically and numerically. The numerical experiments illustrate the fact that the difference between binomial and exponential crossover variants is mainly due to different distributions of the number of mutated components. On the other hand, the behavior of exponential crossover variants was found to be more sensitive to the problem size than the behavior of variants based on binomial crossover. (C) 2009 Elsevier B. V. All rights reserved.