I have sufficiently smooth function $K(x, f)$, convex in $x$ and concave in $f$. I need to find $x$ such $$ x = \mbox{arg} \min_{x} \max_{f} K(x,f). $$
Does convexity of any help to me here? Can I just descend to the optimal solution? Or use some version alternating between $x$ and $f$?
Do I have minimax equality theorems like $$ \min_{x} \max_{f} K(x,f) = \min_{f} \max_{x} K(x,f). $$ ?