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 μ and ν. The transport cost function c: X × Y → [0,∞] 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 - ε from (X,μ) to (Y,ν) as ε > 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-2-4,
author = {Mathias Beiglb\"ock and Christian L\'eonard and Walter Schachermayer},
title = {A general duality theorem for the Monge-Kantorovich transport problem},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {151-167},
zbl = {1270.49045},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-2-4}
}
Mathias Beiglböck; Christian Léonard; Walter Schachermayer. A general duality theorem for the Monge-Kantorovich transport problem. Studia Mathematica, Tome 209 (2012) pp. 151-167. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-2-4/