  This study targets the mixed-integer black-box optimization (MI-BBO) problem
where continuous and integer variables should be optimized simultaneously. The
CMA-ES, our focus in this study, is a population-based stochastic search method
that samples solution candidates from a multivariate Gaussian distribution
(MGD), which shows excellent performance in continuous BBO. The parameters of
MGD, mean and (co)variance, are updated based on the evaluation value of
candidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO with
straightforward discretization, however, the variance corresponding to the
integer variables becomes much smaller than the granularity of the
discretization before reaching the optimal solution, which leads to the
stagnation of the optimization. In particular, when binary variables are
included in the problem, this stagnation more likely occurs because the
granularity of the discretization becomes wider, and the existing modification
to the CMA-ES does not address this stagnation. To overcome these limitations,
we propose a simple modification of the CMA-ES based on lower-bounding the
marginal probabilities associated with the generation of integer variables in
the MGD. The numerical experiments on the MI-BBO benchmark problems demonstrate
the efficiency and robustness of the proposed method.
