A saddle-point approach to the Monge-Kantorovich optimal transport problem

TL;DR

This paper introduces a saddle-point method for the Monge-Kantorovich optimal transport problem, providing new optimality conditions, a proof of Kantorovich duality, and improved convergence of solutions, especially with infinite cost functions.

Contribution

It presents a novel saddle-point approach that avoids c-conjugates, offering new characterizations and conditions for optimal plans in optimal transport.

Findings

New explicit optimality conditions for transport plans.

A novel proof of Kantorovich dual equality.

Enhanced convergence results for minimizing sequences.

Abstract

The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to $c$-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.