I have a min-max problem: $$ \min_x\{\max\{f_1(x),f_2(x)\}\} $$
Can this min-max be solved (or somehome equivalent to): $$ \max\{\min_xf_1(x),\min_xf_2(x)\} $$
when both $f_1(\cdot)$ and $f_2(\cdot)$ are convex?
My initial thought is to rewrite the original problem by introducing binary variable $a$: $$ \min_x\max_{a\in Z}af_1(x) + (1-a)f_2(x) $$
I know that if $a\in[0,1]$, then we can use Von Neumann's Minimax theorem. However, I am not sure there is a version of Von Neumann's that can be applied to integer variable or will the relaxation from $a\in\{0,1\}$ to $a\in[0,1]$ result in the same solution to the original problem.
Any suggestions or references are appreciated!
The two forms are not equivalent. Take $f_1(x):=(x-1)^2$ and $f_2(x):=(x+1)^2$. Each function is convex, and you can check that $$\min_x\{\max\{f_1(x),f_2(x)\}\} = 1,$$ with the minimum occurring at $x=0$, while $\min_x f_1(x) =0$, occurring at $x=1$, and $\min_x f_2(x)=0$, occurring at $x=-1$.