A non-convex relaxed version of minimax theorems

Given a subset $A\times B$ of a locally convex space $X\times Y$ (with $A$ compact) and a function $f:A\times B\to \bar{\mathbb{R}}$ such that $f(\cdot ,y),$ $y\in B,$ are concave and upper semicontinuous, the minimax inequality ${\max }_{x\in A}{\inf }_{y\in B}f(x,y)\ge {\inf }_{y\in B}{\sup }_{x\in A_0}f(x,y)$ is shown to hold provided that $A_0$ be the set of $x\in A$ such that $f(x,\cdot )$ is proper, convex and lower semi-contiuous. Moreover, if in addition $A\times B\subset f^{-1}(\mathbb{R})$, then we can take as $A_0$ the set of $x\in A$ such that $f(x,\cdot )$ is convex. The relation to Moreau’s biconjugate representation theorem is discussed, and some applications to\ convex duality are provided. Key words. Minimax theorem, Moreau theorem, conjugate function, convex optimization.