On the rotational variance of the differential evolution algorithm

Abstract

In this study we examine the rotational (in)variance of the differential evolution (DE) algorithm. We show that the classic DE/rand/1/bin algorithm, which uses constant mutation and standard crossover, is rotationally variant. We then study a previously proposed rotationally invariant formulation in which the crossover operation takes place in an orthogonal base constructed using Gramm-Schmidt orthogonalization.We propose two new formulations by firstly considering a very simple rotationally invariant formulation using constant mutation and whole arithmetic crossover. This rudimentary formulation performs badly, due to lack of diversity. We introduce diversity into the formulation using two distinctly different strategies. The first adjusts the crossover step by perturbing the direction of the linear combination between the target vector and the mutant vector. This formulation is invariant in a stochastic sense only. The other formulation adds a self-scaling random vector with a standard normal distribution, sampled uniformly from the surface of an n-dimensional unit sphere to the unaltered whole arithmetic crossover vector. This formulation is strictly invariant, if in a stochastic sense only.We compare the four invariant formulations in terms of numerical efficiency for a modest set of test problems; the intention not being to propose yet another competitive and/or superior DE variant, but rather to present formulations that are both diverse and invariant, in the hope that this will stimulate additional future contributions, since rotational invariance in general is a desirable, salient feature for an optimization algorithm.