Can I convert $\min_x\max_y: f(x,y)$ to $\max_x\min_y: -f(x,y)$?

Question

Can I convert $\min_x\max_y: f(x,y)$ to $\max_x\min_y: -f(x,y)$? There are lots of methods for minmax problem. So I think there should be a way to convert maxmin problem to minmax problem.

Answer 1

Use the fact that $\min(A)=-\max(-A)$ for any set $A$ of real numbers (where, obviously, $-A=\left\{-a:a\in A\right\}$. [If you don't know that already, try to prove it yourself.]

\begin{align*}\min_x\max_y f(x,y)&=-\max_x(-\max_y f(x,y))\\&=-\max_x(\min_y-f(x,y))\end{align*}

so the process of "maximizing $f(x,y)$ for $y$ and then minimizing for $x$" is the same as "minizing $-f(x,y)$ for $y$ and then maximizing for $x$".

The only deal is that the results are opposites.

More formally, a point $(x_0,y_0)$ such that $f(x_0,y_0)=\min_x\max_y f(x,y)$ will also yield $-f(x_0,y_0)=\max_x\min_y-f(x,y)$ and vice-versa.