Entropic Approximation of \(\infty \)-Optimal Transport Problems

We propose an entropic approximation approach for optimal transportation problems with a supremal cost. We establish \(\Gamma \) $\mathrm{\Gamma }$ -convergence for suitably chosen parameters for the entropic penalization and that this procedure selects \(\infty \) $\infty$ -cyclically monotone plans at the limit. We also present some numerical illustrations performed with Sinkhorn’s algorithm.