https://hal-univ-tln.archives-ouvertes.fr/hal-03353658Alibert, Jean-JacquesJean-JacquesAlibertIMATH - Institut de Mathématiques de Toulon - EA 2134 - UTLN - Université de ToulonBouchitté, GuyGuyBouchittéIMATH - Institut de Mathématiques de Toulon - EA 2134 - UTLN - Université de ToulonChampion, ThierryThierryChampionIMATH - Institut de Mathématiques de Toulon - EA 2134 - UTLN - Université de ToulonA new class of cost for optimal transport planningHAL CCSD2019Optimal transportKantorovich-Rubinstein dualityMartingale constraint[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]Bouchitte, Guy2021-09-24 11:24:262022-01-04 09:57:082021-09-24 11:24:26enJournal articles10.1017/s09567925180006691We 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.