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This study targets the mixed-integer black-box optimization (MI-BBO) problemwhere continuous and integer variables should be optimized simultaneously. TheCMA-ES, our focus in this study, is a population-based stochastic search methodthat samples solution candidates from a multivariate Gaussian distribution(MGD), which shows excellent performance in continuous BBO. The parameters ofMGD, mean and (co)variance, are updated based on the evaluation value ofcandidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO withstraightforward discretization, however, the variance corresponding to theinteger variables becomes much smaller than the granularity of thediscretization before reaching the optimal solution, which leads to thestagnation of the optimization. In particular, when binary variables areincluded in the problem, this stagnation more likely occurs because thegranularity of the discretization becomes wider, and the existing integerhandling for the CMA-ES does not address this stagnation. To overcome theselimitations, we propose a simple integer handling for the CMA-ES based onlower-bounding the marginal probabilities associated with the generation ofinteger variables in the MGD. The numerical experiments on the MI-BBO benchmarkproblems demonstrate the efficiency and robustness of the proposed method.Furthermore, in order to demonstrate the generality of the idea of the proposedmethod, in addition to the single-objective optimization case, we incorporateit into multi-objective CMA-ES and verify its performance on bi-objectivemixed-integer benchmark problems.

Marginal Probability-Based Integer Handling for CMA-ES Tackling Single-and Multi-Objective Mixed-Integer Black-Box Optimization