Conditions for Swapping min and mMax

Question

I have a two-step optimization problem now, which is \begin{equation} \max_{\boldsymbol{x}\in\mathcal{X}}\min_{\boldsymbol{y}\in\mathcal{Y}}f(\boldsymbol{x},\boldsymbol{y}) \;. \end{equation} I am curious under which conditions I can switch the order of $\min$ and $\max$ to \begin{equation} \min_{\boldsymbol{y}\in\mathcal{Y}}\max_{\boldsymbol{x}\in\mathcal{X}}f(\boldsymbol{x},\boldsymbol{y}) \;. \end{equation} Could anybody please refer me to some literature about this topic?

Thanks!

Answer 1

Theorems which provide these conditions are called minimax theorems. Two useful minimax theorems are Von Neumann's and Sion's.

Sion's theorem, which is more general, requires:

X to be compact and convex

Y to be convex

$f$ to be upper semi-continuous and quasi-concave in $y$ for all $x$

$f$ to be lower semi-continuous and quasi-convex in $x$ for all $y$