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 integer handling for the CMA-ES does not address this stagnation. To overcome these limitations, we propose a simple integer handling for 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. Furthermore, in order to demonstrate the generality of the idea of the proposed method, in addition to the single-objective optimization case, we incorporate it into multi-objective CMA-ES and verify its performance on bi-objective mixed-integer benchmark problems.

翻译：本研究针对混合整数黑箱优化（MI-BBO）问题，该问题需同时优化连续变量与整数变量。本文聚焦的CMA-ES算法是一种基于种群的随机搜索方法，通过多元高斯分布（MGD）采样候选解，在连续黑箱优化中表现优异。CMA-ES根据候选解的评估值更新MGD的均值与（协）方差参数。然而，当采用直接离散化方式将CMA-ES应用于MI-BBO时，整数变量对应的方差在达到最优解前会远小于离散化粒度，从而导致优化过程停滞。特别是当问题包含二元变量时，由于离散化粒度增大，此类停滞更易发生，而现有CMA-ES整数处理方法无法解决该问题。为克服上述局限，我们提出了一种基于MGD中整数变量生成过程相关边际概率下界的简易整数处理方法。MI-BBO基准问题的数值实验验证了所提方法的效率与鲁棒性。此外，为证明该方法的通用性，除单目标优化情形外，我们将其集成至多目标CMA-ES中，并在双目标混合整数基准问题上验证了其性能。