Strong c-concavity and stability in optimal transport

Anatole Gallouët (1) , Quentin Mérigot (2) , Boris Thibert (1)

1 EDP - Equations aux Dérivées Partielles
2 LMO - Laboratoire de Mathématiques d'Orsay

The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong c-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions c. We then introduce a differential criterion for proving that a function is strongly c-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +∞ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.

Domaines: Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Anatole Gallouët, Quentin Mérigot, Boris Thibert. Strong c-concavity and stability in optimal transport. 2022. ⟨hal-03735775⟩