# Trivial Cases for the Kantorovitch Problem

• Volume: 34, Issue: 1, page 49-59
• ISSN: 0399-0559

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## Abstract

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Let X and Y be two compact spaces endowed with respective measures μ and ν satisfying the condition µ(X) = v(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure ξ on X x Y whose marginals coincide with μ and ν, and such that the total cost ∫ ∫ c(x,y)dξ(x,y) be minimized. We first show that if the cost function c is decomposable, i.e., can be represented as the sum of two continuous functions defined on X and Y, respectively, then every feasible measure is optimal. Conversely, when X is the support of μ and Y the support of ν and when every feasible measure is optimal, we prove that the cost function is decomposable.

## How to cite

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Dubuc, Serge, Kagabo, Issa, and Marcotte, Patrice. "Trivial Cases for the Kantorovitch Problem." RAIRO - Operations Research 34.1 (2010): 49-59. <http://eudml.org/doc/197819>.

@article{Dubuc2010,
abstract = { Let X and Y be two compact spaces endowed with respective measures μ and ν satisfying the condition µ(X) = v(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure ξ on X x Y whose marginals coincide with μ and ν, and such that the total cost ∫ ∫ c(x,y)dξ(x,y) be minimized. We first show that if the cost function c is decomposable, i.e., can be represented as the sum of two continuous functions defined on X and Y, respectively, then every feasible measure is optimal. Conversely, when X is the support of μ and Y the support of ν and when every feasible measure is optimal, we prove that the cost function is decomposable. },
author = {Dubuc, Serge, Kagabo, Issa, Marcotte, Patrice},
journal = {RAIRO - Operations Research},
keywords = {Continuous programming; transportation.; Kantorovich problem; continuous programming; transportation},
language = {eng},
month = {3},
number = {1},
pages = {49-59},
publisher = {EDP Sciences},
title = {Trivial Cases for the Kantorovitch Problem},
url = {http://eudml.org/doc/197819},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Dubuc, Serge
AU - Kagabo, Issa
AU - Marcotte, Patrice
TI - Trivial Cases for the Kantorovitch Problem
JO - RAIRO - Operations Research
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 1
SP - 49
EP - 59
AB - Let X and Y be two compact spaces endowed with respective measures μ and ν satisfying the condition µ(X) = v(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure ξ on X x Y whose marginals coincide with μ and ν, and such that the total cost ∫ ∫ c(x,y)dξ(x,y) be minimized. We first show that if the cost function c is decomposable, i.e., can be represented as the sum of two continuous functions defined on X and Y, respectively, then every feasible measure is optimal. Conversely, when X is the support of μ and Y the support of ν and when every feasible measure is optimal, we prove that the cost function is decomposable.
LA - eng
KW - Continuous programming; transportation.; Kantorovich problem; continuous programming; transportation
UR - http://eudml.org/doc/197819
ER -

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