# A general duality theorem for the Monge-Kantorovich transport problem

Mathias Beiglböck; Christian Léonard; Walter Schachermayer

Studia Mathematica (2012)

- Volume: 209, Issue: 2, page 151-167
- ISSN: 0039-3223

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topMathias Beiglböck, Christian Léonard, and Walter Schachermayer. "A general duality theorem for the Monge-Kantorovich transport problem." Studia Mathematica 209.2 (2012): 151-167. <http://eudml.org/doc/285844>.

@article{MathiasBeiglböck2012,

abstract = {
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.
},

author = {Mathias Beiglböck, Christian Léonard, Walter Schachermayer},

journal = {Studia Mathematica},

keywords = {Monge-Kantorovich problem; duality; Borel probability measures; no duality gap; relaxed transport problem},

language = {eng},

number = {2},

pages = {151-167},

title = {A general duality theorem for the Monge-Kantorovich transport problem},

url = {http://eudml.org/doc/285844},

volume = {209},

year = {2012},

}

TY - JOUR

AU - Mathias Beiglböck

AU - Christian Léonard

AU - Walter Schachermayer

TI - A general duality theorem for the Monge-Kantorovich transport problem

JO - Studia Mathematica

PY - 2012

VL - 209

IS - 2

SP - 151

EP - 167

AB -
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.

LA - eng

KW - Monge-Kantorovich problem; duality; Borel probability measures; no duality gap; relaxed transport problem

UR - http://eudml.org/doc/285844

ER -

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