Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem

Abstract

The Monge–Kantorovich problem (MKP) with given marginals defined on closed domains \(X \subset \mathbb{R}^n ,\;Y \subset \mathbb{R}^m \) and a smooth cost function \(c:X \times Y \to \mathbb{R}\) is considered. Conditions are obtained (both necessary ones and sufficient ones) for the optimality of a Monge solution generated by a smooth measure-preserving map \(f:X \to Y\). The proofs are based on an optimality criterion for a general MKP in terms of nonemptiness of the sets \(Q_0 {\zeta } = u \in \mathbb{R}^X :\left\{ {u(x) - u(z) \leqslant {\zeta} (x,z) for all x,z \in X} \right\}\) for special functions ζ on X × X generated by c and f. Also, earlier results by the author are used when considering the above-mentioned nonemptiness conditions for the case of smooth ζ.