Entropic approximation of \(\infty \)-optimal transport problems (Q6589685)

The paper presents an approach to solving \(\infty\)-optimal transportation problems with a supremal cost of the form:\N\[\N\inf_{\gamma \in \Pi(\mu,\nu)} \gamma - \operatorname{ess sup} c = \|c\|_{L^{\infty}(\gamma)}.\N\]\NThe authors propose an entropic penalisation method for approximating \(L^\infty\) optimal transport problems.via introducing a functional \(J_{p,\epsilon}\) which combines the transport cost raised to the power of \(p\) and an entropic term. \(\Gamma\)-convergence of \(J_{p,\epsilon}\) to \(J_\infty\) under certain conditions on \(\epsilon\) and \(p\) is established, which demonstrates that, in the limit case, \(\infty\)-cyclic monotone plans are selected.\N\NThe authors also give error estimates in the form of an upper bound on the speed of convergence of \(v_p = \min_{\gamma \in \Pi(\mu,\nu)} J_p\) to \(v_\infty = \min_{\gamma \in \Pi(\mu,\nu)} J_p\).\N\NIn the discrete case, these error estimates are sharpened by also providing lower bounds on the speed of convergence.\N\NFinally, numerical illustrations using the Sinkhorn algorithm are presented based on the Sinkhorn algorithm to solve the entropically penalised transport problems. This demonstrates the convergence of the transport plans to \(\infty\)-cyclically monotone plans, and illustrates that the convergence speed of \(v_p\) to \(v_\infty\) adheres with its theoretical bounds.