I am given a convex set $K \subset \mathbb{R}^n$ and a convex function $f : K \to \mathbb{R}$. This function is very complicated, but I know how to evaluate it (to arbitrary precision) at any given point $x \in K$. I also know that this function is Lipschitz, but I do not know what the best Lipschitz constant for the function is. Is there any way that we can compute the best Lipschitz constant of $f$?
I know that it sounds hopeless, and I did research on it with essentially nothing of interest showing up. The most relevant paper I found was this one. One of their main results says that for my function $f$ it is sufficient to examine the sub-gradients of boundary points. I however do not know how to evaluate a sub-gradient of this function, and potentially there are uncountably many boundary points of $K$.