CMA evolution strategy assisted by kriging model and approximate ranking

The covariance matrix adaptation evolution strategy (CMA-ES) is a competitive evolutionary algorithm (EA) for difficult continuous optimization problems. However, expensive function evaluation of many real-world optimization problems poses a serious challenge to the application of CMA-ES (and other EAs) to these problems. To address this challenge, surrogate-assisted EAs has attracted increasing attention and become popular. In this paper, a new surrogate-assisted CMA-ES algorithm in which Kriging model is used to enhance CMA-ES via approximate ranking procedure is proposed. In the proposed algorithm, the approximate ranking procedure which estimates the rank of current population by using Kriging model and the exact fitness function together is adopted. In addition, the confidence interval method of training set selection is introduced for surrogate model construction. An initial sampling is performed before entering the evolution loop. In each iteration (generation), after the population sampling, the approximate ranking procedure is called instead of the original fitness evaluation, then, parameters of the sampling distribution are updated. This iterative search process continues until the target fitness is reached or the computational budget is exhausted. The proposed algorithm and confidence interval method of training set selection are analyzed through experimental study. The results demonstrate that the confidence interval method works well in Kriging-assisted CMA-ES, and that the proposed algorithm significantly reduces the number of function evaluations of CMA-ES and outperforms the Kriging-assisted CMA-ES using pre-selection and generation-based control on the tested problems.