Consider the Kantorovich formulation of optimal transport: $$ \inf_{\pi\in\Pi(\mu,\nu)} \int c(x,y)d\pi $$ whose dual problem is $$ \sup_{\phi} \int \phi(x)d\mu + \int \phi^c(y) d\nu. $$
Now, suppose I have a dual optimizer $\phi$. My question: how can I construct a optimal transport plan $\pi$ from it(without any regularity assumptions on $\mu$ and $\nu$? Thanks in advance!