Suppose we have $$g(\mathbf{y}) = \sup_{\mathbf{x}\in \text{dom} f} \{\inf_{\mathbf{C} \in {\text{S}}} \{\mathbf{y}^T \mathbf{x} - \mathbf{x}^T \mathbf{C} \mathbf{\mathbf{x}}\}\} $$
where $\text{S} \subset\mathbb{S}_{++}^n$ ($S$ a compact convex subset of positive definite matrices). Could we interchange the $\sup$ and $\inf$ in this optimization problem like below:
$$g(\mathbf{y}) = \inf_{\mathbf{C} \in {\text{S}}} \{\sup_{\mathbf{x}\in \text{dom} f} \{\mathbf{y}^T \mathbf{x} - \mathbf{x}^T \mathbf{C} \mathbf{\mathbf{x}}\}\}\qquad *?*$$
Also, what is the general condition for this interchange its order? I mean under what conditions we have: $$\sup_{\mathbf{x}\in X} \{\inf_{\mathbf{C} \in {\text{S}}} \{f(\mathbf{x},\mathbf{C})\}\} = \inf_{\mathbf{C} \in {\text{S}}} \{\sup_{\mathbf{x}\in X} \{f(\mathbf{x},\mathbf{C})\}\}.$$