Mixed-integer extensions of evolution strategies (ES) that discretize selected coordinates of sampled continuous vectors often impose a lower bound on the standard deviation of integer variables to prevent premature convergence. While these methods show promising empirical results, this handling can slow the convergence of continuous variables, and its impact has lacked a clear theoretical account. In this paper, we provide a convergence analysis of evolution strategies for mixed-integer optimization, inspired by the drift analysis of the (1+1)-ES in the continuous domain. Specifically, we consider two (1+1)-ES variants for mixed-integer domains: (1+1)-LB-ES, which introduces a lower bound on the standard deviation for integer variables, and (1+1)-LUB-ES, which combines both lower and upper bounds to enhance the convergence of the continuous variables. Focusing on the optimization phase after the integer variables have been optimized, we rigorously analyze their convergence behavior on a benchmark function designed for mixed-integer domains. Our results show that (1+1)-LB-ES can suffer from premature convergence when the number of integer variables is large, while (1+1)-LUB-ES achieves linear convergence under suitable parameter settings. These findings provide theoretical insights into the impact of integer handling on convergence performance and guidance for the design of mixed-integer ES.

翻译：混合整数进化策略（ES）通常会对采样连续向量中的选定坐标进行离散化处理，并对整数变量的标准差设置下界以防止过早收敛。尽管这些方法在实验上表现良好，但这种处理可能会减缓连续变量的收敛速度，且其影响缺乏清晰的理论解释。本文受连续域中(1+1)-ES漂移分析的启发，对混合整数优化中的进化策略进行了收敛性分析。具体而言，我们考虑了两种适用于混合整数域的(1+1)-ES变体：(1+1)-LB-ES（对整数变量标准差引入下界）和(1+1)-LUB-ES（同时引入上界和下界以增强连续变量的收敛性能）。在整数变量已优化的阶段，我们严格分析了它们在面向混合整数域的基准函数上的收敛行为。结果表明，当整数变量数量较大时，(1+1)-LB-ES可能遭遇过早收敛，而(1+1)-LUB-ES在合适的参数设置下可实现线性收敛。这些发现为整数处理对收敛性能的影响提供了理论视角，并为混合整数进化策略的设计提供了指导。