A new class of cost for optimal transport planning

We study a class of optimal transport planning problems where the reference cost involves a non linear function G(x, p) representing the transport cost between the Dirac mesure x and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contra...

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Bibliographic Details
Published in:European journal of applied mathematics 2019-12, Vol.30 (6), p.1229-1263
Main Authors: Alibert, Jean-Jacques, Bouchitté, Guy, Champion, Thierry
Format: Article
Language:English
Subjects:
Mathematics
Optimization and Control
Online Access:Get full text
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Summary:We study a class of optimal transport planning problems where the reference cost involves a non linear function G(x, p) representing the transport cost between the Dirac mesure x and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case (G(x, p) = R c(x, y) dp) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich-Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with-convergence theory.
ISSN:0956-7925
1469-4425
DOI:10.1017/s0956792518000669