How can I solve the following saddle point optimization problem

Question

The function $f(\mathbf{x},\mathbf{y}):\mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R}$ is defined as $f(\mathbf{x},\mathbf{y})=\mathbf{x}^T\mathbf{A}\mathbf{y}+\mathbf{Bx}$, where $\mathbf{x}\in \mathcal{P}, \mathbf{y}\in \mathcal{Q}$. $\mathcal{P},\mathcal{Q}$ are convex polytopes. I want to find a saddle point $(\mathbf{x}^*,\mathbf{y}^*)$ such that $f (\mathbf{x}^*,\mathbf{y})\geq f(\mathbf{x}^*,\mathbf{y}^*)\geq f(\mathbf{x},\mathbf{y}^*)$, where $\forall \mathbf{x}\in \mathcal{P}, \forall \mathbf{y}\in \mathcal{Q}$.

Can I solve such problem by just solving the following min max or max min problem?

$\min\limits_{\mathbf{y}\in \mathcal{Q}}\max\limits_{\mathbf{x}\in \mathcal{P}}f(\mathbf{x},\mathbf{y})$ and $\max\limits_{\mathbf{x}\in \mathcal{P}}\min\limits_{\mathbf{y}\in \mathcal{Q}}f(\mathbf{x},\mathbf{y})$.

Suppose that such saddle point must exists for our problem.

Answer 1

A polytope is bounded. Your function $f$ is convex-concave (convex in $x$ for fixed $y$ and vice versa). The function therefore has a (unique) saddle point (by, e.g., Lemma 37.3.2 in Convex Analysis by Rockafellar). Both your min-max formulations will find it.

Answer 2

$\min \max = \max \min$ by Sion's minimax theorem, since $\mathcal P$ is compact. In particular, you don't need $\mathcal Q$ to be compact.