This study focuses on mixed-variable black-box optimization (MV-BBO), addressing continuous, integer, and categorical variables. Many real-world MV-BBO problems involve dependencies among these different types of variables, requiring efficient methods to optimize them simultaneously. Recently, stochastic optimization methods leveraging the mechanism of the covariance matrix adaptation evolution strategy have shown promising results in mixed-integer or mixed-category optimization. However, such methods cannot handle the three types of variables simultaneously. In this study, we propose CatCMA with Margin (CatCMAwM), a stochastic optimization method for MV-BBO that jointly optimizes continuous, integer, and categorical variables. CatCMAwM is developed by incorporating novel integer handling into CatCMA, a mixed-category black-box optimization method employing a joint distribution of multivariate Gaussian and categorical distributions. The proposed integer handling is carefully designed by reviewing existing integer handling and following the design principles of CatCMA. Furthermore, we extend CatCMAwM to multi-objective MV-BBO by instantiating it within the Sofomore framework, obtaining a multi-objective optimizer termed COMO-CatCMA with Margin (COMO-CatCMAwM). Numerical experiments on single-objective MV-BBO problems show that CatCMAwM effectively handles the three types of variables, outperforming state-of-the-art Bayesian optimization methods and baselines that simply incorporate existing integer handlings into CatCMA. Moreover, on bi-objective MV-BBO benchmarks, COMO-CatCMAwM achieves competitive or superior hypervolume compared to representative baselines.

翻译：本研究聚焦于混合变量黑盒优化问题，旨在同时处理连续变量、整数变量和类别变量。许多实际MV-BBO问题涉及不同类型变量间的依赖关系，需要高效方法对其进行同步优化。近年来，基于协方差矩阵自适应进化策略机制的随机优化方法在混合整数或混合类别优化中展现出良好性能，但此类方法无法同时处理三类变量。本研究提出CatCMA with Margin，这是一种面向MV-BBO的随机优化方法，可联合优化连续、整数及类别变量。该方法通过在CatCMA中引入新型整数处理机制而构建——CatCMA是一种采用多元高斯分布与类别分布联合分布的混合类别黑盒优化方法。所提出的整数处理机制在审视现有整数处理方法的基础上，遵循CatCMA的设计原则精心设计。进一步地，我们将CatCMAwM扩展至多目标MV-BBO领域，通过在Sofomore框架中实例化该方法，得到名为COMO-CatCMA with Margin的多目标优化器。在单目标MV-BBO问题上的数值实验表明，CatCMAwM能有效处理三类变量，其性能优于最先进的贝叶斯优化方法及简单整合现有整数处理机制的CatCMA基线方法。此外，在双目标MV-BBO基准测试中，COMO-CatCMAwM相较于代表性基线方法，获得了具有竞争力或更优的超体积指标。