A general duality theorem for the Monge–Kantorovich transport problem

Mathias Beiglböck¹ ; Christian Léonard² ; Walter Schachermayer¹

¹Faculty of Mathematics University of Vienna Nordbergstrasse 15 1090 Wien, Austria
²Modal-X, Université Paris Ouest Bât. G, 200 av. de la République 92001 Nanterre, France

Studia Mathematica, Tome 209 (2012) no. 2, pp. 151-167

Résumé

The duality theory for the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be Polish and equipped with Borel probability measures $\mu $ and $\nu $. The transport cost function $c:X\times Y \to [0,\infty ]$ is assumed to be Borel. Our main result states that in this setting there is no duality gap provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses $1-\varepsilon $ from $(X,\mu )$ to $(Y, \nu )$ as $\varepsilon >0$ tends to zero. The classical duality theorems of H. Kellerer, where $c$ is lower semicontinuous or uniformly bounded, quickly follow from these general results.

Détail

DOI : 10.4064/sm209-2-4

Keywords: duality theory monge kantorovich transport problem analyzed general setting spaces assumed polish equipped borel probability measures transport cost function times infty assumed borel main result states setting there duality gap provided optimal transport problem formulated suitably relaxed relaxed transport problem defined limiting cost partial transport masses varepsilon varepsilon tends zero classical duality theorems kellerer where lower semicontinuous uniformly bounded quickly follow these general results