Let $f(x,y)$ be a saddle function (convex in $x$ and concave in $y$), where $f \colon X \times Y \rightarrow \mathbb{R}$, and $X \subset \mathbb R^n$, $Y \subset \mathbb R^m$ are convex sets.

I consider a constrained optimization problem of the form:

$$\min_{x \in X} \max_{y \in Y} f(x,y)$$

subject to:

$$A x + B y = c$$

where $A \in \mathbb R^{r\times n}, B \in \mathbb R^{r\times m}, c \in \mathbb R^r$.

I have many questions about these kind of optimization programs. Reggretably I have not found the appropriate references. Is there a name for this kind of optimization programs? Under what conditions can one exchange min. with max. and obtain the same solution?

Let $f(x,y)$ be a smooth (twice differentiable) saddle function (convex in $x$ and concave in $y$), where $f \colon X \times Y \rightarrow \mathbb{R}$, and $X \subset \mathbb R^n$, $Y \subset \mathbb R^m$ are convex compact sets.

I consider a constrained optimization problem of the form:

$$\min_{x \in X} \max_{y \in Y} f(x,y)$$

subject to:

$$A x + B y = c$$

where $A \in \mathbb R^{r\times n}, B \in \mathbb R^{r\times m}, c \in \mathbb R^r$.

Can $\min\max$ be exchanged with $\max\min$? Is the following equality true or false?

$$\min_{x \in X} \max_{y \in Y} f(x,y) = \max_{y \in Y} \min_{x \in X} f(x,y)$$