An efficient eigenvector-based crossover for differential evolution: Simplifying with rank-one updates

We propose a new approach to enhancing the efficiency of the differential evolution (DE) algorithm, specifically targeting rotational invariance. The performance of the DE algorithm can be hampered by the crossover's dependency on the coordinate system, particularly in optimization problems involving strongly correlated variables. Previous attempts to achieve rotational invariance in the DE algorithm have involved estimating the covariance matrix using the population's distribution information and executing the crossover operation in an eigen coordinate system. However, these methods are computationally intensive. Our approach exclusively employs the rank-one update method, estimating the covariance matrix using the means of the current and previous generations' populations. This lightweight technique reduces the computational costs from $O(Np \cdot D^{2})$ to $O(D^{2})$ (where $ Np $ is the population size and $ D $ is the dimension) operations, yet still preserves the critical rotational invariance property. Experiments conducted on 57 benchmark functions demonstrated that our method finds quicker and more accurate solutions than previous methods. This represents a substantial improvement in achieving rotational invariance in the DE algorithm.