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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| Published in | European journal of applied mathematics Vol. 30; no. 6; pp. 1229 - 1263 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge
Cambridge University Press
01.12.2019
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0956-7925 1469-4425 |
| DOI | 10.1017/S0956792518000669 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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 δx 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. |
| Author | CHAMPION, T. ALIBERT, J.-J. BOUCHITTÉ, G. |
| Author_xml | – sequence: 1 givenname: J.-J. surname: ALIBERT fullname: ALIBERT, J.-J. – sequence: 2 givenname: G. orcidid: 0000-0002-8551-2640 surname: BOUCHITTÉ fullname: BOUCHITTÉ, G. – sequence: 3 givenname: T. surname: CHAMPION fullname: CHAMPION, T. |
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| Cites_doi | 10.1007/s00780-013-0205-8 10.1007/BFb0087685 10.1016/j.jfa.2017.08.015 10.1214/15-AIHP700 10.1111/j.1467-9965.2010.00473.x 10.1007/978-3-540-71050-9 10.1007/BF00387715 10.4213/mzm10896 10.1214/16-AOP1131 10.1007/BF02249263 10.1214/14-AOP966 10.1214/aoms/1177700153 10.1016/j.anihpb.2006.05.003 10.1007/978-3-319-20828-2 |
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| References | S0956792518000669_ref3 Zaev (S0956792518000669_ref20) 2015; 98 S0956792518000669_ref5 S0956792518000669_ref4 Villani (S0956792518000669_ref18) 2003; 58 Auslender (S0956792518000669_ref2) 2003 S0956792518000669_ref12 S0956792518000669_ref11 S0956792518000669_ref10 S0956792518000669_ref16 S0956792518000669_ref15 S0956792518000669_ref14 S0956792518000669_ref13 S0956792518000669_ref7 Attouch (S0956792518000669_ref1) 1984 S0956792518000669_ref19 S0956792518000669_ref6 S0956792518000669_ref9 S0956792518000669_ref8 S0956792518000669_ref17 |
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| Snippet | 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... 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... |
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| SubjectTerms | Applied mathematics Duality principle Linear functions Martingales Transportation planning |
| Title | A new class of costs for optimal transport planning |
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