On the uniqueness and numerical approximations for an optimal constrained matching problem

par Van Thanh Nguyen

vendredi 16 juin 2017, 10:30 → 11:30 Europe/Paris

XLIM Salle X.203

Optimal constrained matching problem, which is a variant from Ekeland's optimal matching problem, consists in transporting two kinds of goods and matching them into a target set with constraints on the amount of matter at the target. It is known that the uniqueness of optimal matching measure does not hold even with regular L^p sources and targets. In this talk, the uniqueness is proven under an additional geometric condition. Our proof will be based on the PDE of optimality conditions by using a deep result from optimal transport theory about the Lebesgue negligibility of endpoints of maximal transport rays. On the other hand, we also introduce a dual formulation with a linear cost functional on convex set (in an infinite-dimensional space) and show that its Fenchel-Rockafellar dual formulation gives right solution to the optimal constrained matching problem. Basing on our formulations, a numerical approximation is given via augmented Lagrangian methods (splitting methods). The convergence of the discretization is also provided. This is a joint work with Noureddine IGBIDA and Julian TOLEDO (Universitat de Valencia, Spain).