I'm trying to figure out the Fenchel dual of the following expectation minimization problem:
$$ \min_f E[l(x,f(z))] $$
where $x\sim S$ and $z\sim T$ ($S$ and $T$ are two distributions here).
I'm not very sure if I'm proceeding properly here because I'm trying to use a technique similar to the last few slides on http://homepages.wmich.edu/~ledyaev/zhu-talk2-sp2016.pdf when he derives Kantorovich duals.
Is this even possible? And how would I do this? I'm honestly stuck because it seemed like the constraint qualifications for the Kantorovich duals came out of nowhere.