Optimal transportation for a quadratic cost with convex constraints and applications

Chloé Jimenez (1) , Filippo Santambrogio (2)

1 LM - Laboratoire de mathématiques de Brest
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We prove existence of an optimal transport map in the Monge-Kantorovich problem associated to a cost $c(x,y)$ which is not finite everywhere, but coincides with $|x-y|^2$ if the displacement $y-x$ belongs to a given convex set $C$ and it is $+\infty$ otherwise. The result is proven for $C$ satisfying some technical assumptions allowing any convex body in $\R^2$ and any convex polyhedron in $\R^d$, $d>2$. The tools are inspired by the recent Champion-DePascale-Juutinen technique. Their idea, based on density points and avoiding disintegrations and dual formulations, allowed to deal with $L^\infty$ problems and, later on, with the Monge problem for arbitrary norms.

Domaines: Optimisation et contrôle [math.OC]

Chloé Jimenez, Filippo Santambrogio. Optimal transportation for a quadratic cost with convex constraints and applications. 2011. ⟨hal-00632570⟩