Reference request - Existence of a solution to an optimization problem in a Banach space

Question

Let $X$ be a Banach space, $C \subset X^*$ be a weak*-compact, convex subset, $M \subset X$ a closed subspace. For any $x \in X$, we may prove that \begin{equation}\inf_{y \in M} \sup_{x^* \in C} x^*(x - y) > -\infty. \end{equation}

How should I approach the question: what kind of conditions need to be imposed in order for the $\inf$ to be attained? Thank you.