Necessary and sufficient conditions for globally stable convex minimax theorems

In this paper, we establish a necessary and sufficient condition for a globally stable minimax equality, where the minimax equality holds for each convex perturbation of the convex-concave bi-function involved. The necessary and sufficient condition is expressed as a closedness condition using conjugate functions of the bi-function. As an application, we obtain a necessary and sufficient condition for a globally stable Lagrangian duality theorem, and also a constraint qualification which completely characterizes the strong Lagrangian duality theorem for convex minimization problems. As a consequence of these results, we obtain a globally stable Farkas' lemma for cone-convex systems.