Influence of Binomial Crossover on Approximation Error of Evolutionary Algorithms

Cong Wang, Jun He, Yu Chen () and Xiufen Zou
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Cong Wang: School of Science, Wuhan University of Technology, Wuhan 430070, China
Jun He: Department of Computer Science, Nottingham Trent University, Clifton Campus, Nottingham NG11 8NS, UK
Yu Chen: School of Science, Wuhan University of Technology, Wuhan 430070, China
Xiufen Zou: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Mathematics, 2022, vol. 10, issue 16, 1-23

Abstract: 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.

Keywords: binomial crossover; differential evolution; fixed-budget analysis; evolutionary computation; approximation error (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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