In this study, we consider a continuous min--max optimization problem $\min_{x \in \mathbb{X} \max_{y \in \mathbb{Y}}}f(x,y)$ whose objective function is a black-box. We propose a novel approach to minimize the worst-case objective function $F(x) = \max_{y} f(x,y)$ directly using a covariance matrix adaptation evolution strategy (CMA-ES) in which the rankings of solution candidates are approximated by our proposed worst-case ranking approximation (WRA) mechanism. We develop two variants of WRA combined with CMA-ES and approximate gradient ascent as numerical solvers for the inner maximization problem. Numerical experiments show that our proposed approach outperforms several existing approaches when the objective function is a smooth strongly convex--concave function and the interaction between $x$ and $y$ is strong. We investigate the advantages of the proposed approach for problems where the objective function is not limited to smooth strongly convex--concave functions. The effectiveness of the proposed approach is demonstrated in the robust berthing control problem with uncertainty.ngly convex--concave functions. The effectiveness of the proposed approach is demonstrated in the robust berthing control problem with uncertainty.

翻译：在本研究中，我们考虑一个连续极小极大优化问题 $\min_{x \in \mathbb{X}} \max_{y \in \mathbb{Y}} f(x,y)$，其目标函数为黑箱函数。我们提出一种新颖方法，直接利用协方差矩阵自适应进化策略（CMA-ES）最小化最坏情况目标函数 $F(x) = \max_{y} f(x,y)$，其中解候选者的排序通过我们提出的最坏情况排序近似（WRA）机制进行近似。我们开发了WRA与CMA-ES及近似梯度上升法相结合的两种变体，作为内层最大化问题的数值求解器。数值实验表明，当目标函数为光滑强凸-凹函数且 $x$ 与 $y$ 之间交互作用较强时，所提方法优于若干现有方法。我们进一步探究了该方法在目标函数不限于光滑强凸-凹函数情形下的优势。所提方法的有效性在具有不确定性的鲁棒靠泊控制问题中得到了验证。