Let $L(x,y,z)$ be a real-valued function and define $f(z) := \min_{y\in Y}\max_{x\in X} L(x,y,z)$.
When is $f(z)$ convex? In other words, what kind of restrictions on $L$ and the feasibility sets $X,Y$ allow to conclude $f$ is convex?
I'm sure people have studied this problem before, but after googling variants of "convex max-min" I couldn't find good references. Any pointers are much appreciated.
I'd really like to know if people study this kind of questions and if there is some machinery I can use. In particular, I have a few functions $L$ that I want to study, so a more general approach would be ideal.
A sufficient condition is that $g(y,z) = \max_{x\in X} L(x,y,z)$ is convex (which requires $Y$ and $Z$ to be convex), since partial minimization of a convex function yields a convex function.
A sufficient condition for $g(y,z)$ to be convex is if $L(x,y,z)$ is convex for each fixed $x$, since the maximum of a collection of convex functions is convex. There is no restriction on the set $X$.