  Although differential evolution (DE) algorithms perform well on a large
variety of complicated optimization problems, only a few theoretical studies
are focused on the working principle of DE algorithms. To make the first
attempt to reveal the function of binomial crossover, this paper aims to answer
whether it can reduce the approximation error of evolutionary algorithms. By
investigating the expected approximation error and the probability of not
finding the optimum, we conduct a case study comparing two evolutionary
algorithms with and without binomial crossover on two classical benchmark
problems: OneMax and Deceptive. It is proven that using binomial crossover
leads to the dominance of transition matrices. As a result, the algorithm with
binomial crossover asymptotically outperforms that without crossover on both
OneMax and Deceptive, and outperforms on OneMax, however, not on Deceptive.
Furthermore, an adaptive parameter strategy is proposed which can strengthen
the superiority of binomial crossover on Deceptive.
