This paper studies a class of minimax (saddle‑point) problems of the form max_{c in C} min_{β in S} g(c;β). The authors focus on the special but common case when the outcome is bilinear: g(c;β)=c^T A β, where C and S are compact convex sets. They show that for this matrix‑game structure the difficult step inside the Alternating Direction Method of Multipliers (ADMM) — the so‑called proximal operator — simplifies exactly to a generalized projection onto the confidence region S. No approximation or linearization is required.

At a high level, ADMM solves the problem by splitting it into easier subproblems and iterating. In general one substep requires solving a proximal minimax problem which can be as hard as the original problem. When g is bilinear the authors exchange the min and max and carry out the inner minimization in closed form. That algebra reduces the proximal step to solving a convex quadratic program: find β in S that minimizes ||Aβ−w||^2 for an explicit vector w. The remaining ADMM updates are then simple: form y = v + (1/ρ)Aβ, project y+u onto C (a standard Euclidean projection), and update the dual variable u.

The paper gives a concrete algorithm (Algorithm 1). Each iteration does a generalized projection onto S (line 4), an auxiliary update y (line 5), a Euclidean projection onto C (line 6), and a dual update (line 7). The generalized projection is a convex quadratic minimization over S. When A is the identity map this reduces to the familiar Euclidean projection. The algorithm ends by recovering the worst‑case parameter β* = argmin_{β∈S} g(c_k;β) for the final decision c_k.

Why this matters: many practical decisions are made under parameter uncertainty. The paper motivates the method with a marketing example. If β encodes conversion rates from randomized lift studies and c is a budget allocation, then the linearized incremental outcome can be written as c^T A β. A confidence set S for β (for example an ellipsoid or a likelihood‑ratio region for binomial counts) expresses uncertainty. Solving the resulting maximin problem finds a robust decision that performs best in the worst‑case β in S. The reduction shown here makes such robust solutions tractable for general convex confidence regions S.