On rotationally invariant continuous-parameter genetic algorithms

We examine the rotational (in)variance of the continuous-parameter genetic algorithm (CPGA). We show that a standard CPGA, using blend crossover and standard mutation, is rotationally variant.To construct a rotationally invariant CPGA it is possible to modify the crossover operation to be rotationally invariant. This however results in a loss of diversity. Hence we introduce diversity in two ways: firstly using a modified mutation scheme, and secondly by adding a self-scaling random vector with a standard normal distribution, sampled uniformly from the surface of a n-dimensional unit sphere to the offspring vector. This formulation is strictly invariant, albeit in a stochastic sense only.We compare the three formulations in terms of numerical efficiency for a modest set of test problems; the intention not being the contribution of yet another competitive and/or superior CPGA 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.