We consider minimax (saddle-point) problems of the form max_{c \in C} min_{β\in S} g(c; β), where C and S are compact convex sets, and g is concave-convex. Applying the Alternating Direction Method of Multipliers (ADMM) requires evaluating a proximal operator that is, in general, as hard as the original problem. We show that when the outcome function g is bilinear, i.e. g(c; β) = c^T A β, the proximal operator reduces to a generalized projection onto the confidence region S. This reduction is exact -- it involves no approximation or linearization. The resulting ADMM algorithm alternates between (i) a generalized projection onto S and (ii) a Euclidean projection onto C. We describe the derivation, state the algorithm, and discuss convergence.

翻译：考虑形如 max_{c \in C} min_{β\in S} g(c; β) 的极小化极大（鞍点）问题，其中 C 和 S 为紧凸集，且 g 为凹-凸函数。应用交替方向乘子法（ADMM）需要评估近端算子，而该算子通常与原问题难度相当。我们证明：当目标函数 g 为双线性形式（即 g(c; β) = c^T A β），该近端算子可简化为置信域 S 上的广义投影。该简化是精确的——不涉及任何近似或线性化处理。由此得到的ADMM算法交替执行：(i) 对 S 的广义投影；(ii) 对 C 的欧几里得投影。本文阐述推导过程、给出算法公式并讨论收敛性。