A new class of costs 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 measure δ x and a target probability p . This allows to consider interesting models which favour multi-valued transport maps in...

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Bibliographic Details
Published inEuropean journal of applied mathematics Vol. 30; no. 6; pp. 1229 - 1263
Main Authors ALIBERT, J.-J., BOUCHITTÉ, G., CHAMPION, T.
Format Journal Article
LanguageEnglish
Published Cambridge Cambridge University Press 01.12.2019
Subjects
Applied mathematics
Duality principle
Linear functions
Martingales
Transportation planning
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ISSN0956-7925
1469-4425
DOI10.1017/S0956792518000669

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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 measure δ 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 ( $G(x,p)=\int c(x,y)dp$ ) 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 Γ-convergence theory.
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ISSN:0956-7925
1469-4425
DOI:10.1017/S0956792518000669