  The covariance matrix adaptation evolution strategy (CMA-ES) is a powerful
optimization method for continuous black-box optimization problems. Several
noise-handling methods have been proposed to bring out the optimization
performance of the CMA-ES on noisy objective functions. The adaptations of the
population size and the learning rate are two major approaches that perform
well under additive Gaussian noise. The reevaluation technique is another
technique that evaluates each solution multiple times. In this paper, we
discuss the difference between those methods from the perspective of stochastic
relaxation that considers the maximization of the expected utility function. We
derive that the set of maximizers of the noise-independent utility, which is
used in the reevaluation technique, certainly contains the optimal solution,
while the noise-dependent utility, which is used in the population size and
leaning rate adaptations, does not satisfy it under multiplicative noise. Based
on the discussion, we develop the reevaluation adaptation CMA-ES (RA-CMA-ES),
which computes two update directions using half of the evaluations and adapts
the number of reevaluations based on the estimated correlation of those two
update directions. The numerical simulation shows that the RA-CMA-ES
outperforms the comparative method under multiplicative noise, maintaining
competitive performance under additive noise.
