The covariance matrix adaptation evolution strategy (CMA-ES) is an efficient continuous black-box optimization method. The CMA-ES possesses many attractive features, including invariance properties and a well-tuned default hyperparameter setting. Moreover, several components to specialize the CMA-ES have been proposed, such as noise handling and constraint handling. To utilize these advantages in mixed-integer optimization problems, the CMA-ES with margin has been proposed. The CMA-ES with margin prevents the premature convergence of discrete variables by the margin correction, in which the distribution parameters are modified to leave the generation probability for changing the discrete variable. The margin correction has been applied to ($\mu/\mu_\mathrm{w}$,$\lambda$)-CMA-ES, while this paper introduces the margin correction into (1+1)-CMA-ES, an elitist version of CMA-ES. The (1+1)-CMA-ES is often advantageous for unimodal functions and can be computationally less expensive. To tackle the performance deterioration on mixed-integer optimization, we use the discretized elitist solution as the mean of the sampling distribution and modify the margin correction not to move the elitist solution. The numerical simulation using benchmark functions on mixed-integer, integer, and binary domains shows that (1+1)-CMA-ES with margin outperforms the CMA-ES with margin and is better than or comparable with several specialized methods to a particular search domain.

翻译：协方差矩阵自适应进化策略（CMA-ES）是一种高效的连续黑箱优化方法。CMA-ES 具有许多吸引人的特性，包括不变性特性和经过良好调优的默认超参数设置。此外，已有多个针对CMA-ES的专用组件被提出，例如噪声处理和约束处理。为了在混合整数优化问题中利用这些优势，提出了带边界的CMA-ES。带边界的CMA-ES通过边界修正防止离散变量的早熟收敛，其中修改分布参数以保留改变离散变量的生成概率。边界修正已应用于($\mu/\mu_\mathrm{w}$,$\lambda$)-CMA-ES，而本文则将边界修正引入(1+1)-CMA-ES（CMA-ES的精英版本）。(1+1)-CMA-ES 通常对单峰函数具有优势，且计算成本较低。为了解决混合整数优化中的性能退化问题，我们使用离散化的精英解作为采样分布的均值，并修改边界修正使其不移动精英解。在混合整数、整数和二进制域上的基准函数数值仿真表明，带边界的(1+1)-CMA-ES 优于带边界的CMA-ES，并且优于或可比肩几种针对特定搜索域的专用方法。