Influence of Binomial Crossover on Approximation Error of Evolutionary Algorithms

TL;DR

This paper investigates how binomial crossover influences the approximation error in differential evolution algorithms, showing it can improve performance on certain benchmark problems and proposing an adaptive strategy to enhance this effect.

Contribution

It provides the first theoretical analysis of binomial crossover's role in reducing approximation error and introduces an adaptive parameter strategy to improve its effectiveness.

Findings

Binomial crossover leads to asymptotic dominance in transition matrices.

Algorithms with binomial crossover outperform those without on OneMax.

Adaptive strategies can enhance crossover performance on deceptive problems.

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.