When $\min \max = \max \min$? Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets.
Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$.
Say under which condition we have
$$ \min_{x \in X}  \max_{y \in Y} f(x,y) = \max_{y \in Y} \min_{x \in X} f(x,y). $$
From this we have that $\max_{y \in Y} \min_{x \in X} f(x,y) \leq \min_{x \in X}  \max_{y \in Y} f(x,y)$. So here we are looking for conditions on $f$ such that we have the equality.

If you look at this problem as a Primal Problem and its Dual Problem then you're basically asking when Strong Duality Holds.  
Basically when $ f\left( x, y \right) $ is Convex in $ x $ and Concave in $ y $.
