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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 

 

Optimal transportation with capacity constraints


Authors: Jonathan Korman and Robert J. McCann
Journal: Trans. Amer. Math. Soc. 367 (2015), 1501-1521
MSC (2010): Primary 90B06; Secondary 35R35, 49Q20, 58E17
DOI: https://doi.org/10.1090/S0002-9947-2014-06032-7
Published electronically: November 4, 2014
MathSciNet review: 3286490
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Abstract: The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function. Here we consider a natural but largely unexplored variant of this problem by imposing a pointwise constraint on the joint (absolutely continuous) measures: among all joint densities with fixed marginals and which are dominated by a given density, find the optimal one. For this variant, we show that local non-degeneracy of the cost function implies every minimizer is extremal in the convex set of competitors, hence unique. An appendix develops rudiments of a duality theory for this problem, which allows us to compute several suggestive examples.


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Additional Information

Jonathan Korman
Affiliation: Department of Mathematics, University of Toronto, Toronto Ontario M5S 2E4, Canada
Email: jkorman@math.toronto.edu

Robert J. McCann
Affiliation: Department of Mathematics, University of Toronto, Toronto Ontario M5S 2E4, Canada
Email: mccann@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06032-7
Keywords: Monge-Kantorovich mass transportation, resource allocation, optimal coupling, infinite dimensional linear programming, free boundary problems
Received by editor(s): June 5, 2012
Published electronically: November 4, 2014
Additional Notes: The second author is pleased to acknowledge the support of Natural Sciences and Engineering Research Council of Canada Grants 217006-08.
Article copyright: © Copyright 2014 by the authors