Given $f_1(x)$, $f_2(x)$, $x\in \mathbb{R}^d$, two convex functions, we define the following problem:
$\underset{x\in C}{{\rm minimize}}\,{\rm max}\left(f_{1}\left(x\right),f_{2}\left(x\right)\right)$,
where $C$ is a convex set. This problem should be easy for d=1 because even in the case that the solution is in the set where $f_1(x)=f_2(x)$, this set is convex for sure. However this set could not be convex for $d>1$.
Could anyone point me to any clue about how to solve the problem for $d>1$ (e.g. alternating-like algorithm)?
Thanks.