Abstract
The paper deals with some theoretical and numerical aspects for an optimal matching problem with constraints. It is known that the uniqueness of the optimal matching measure does not hold even with $L^p$ sources and targets. In this paper, the uniqueness is proven under geometric conditions. On the other hand, we also introduce a dual formulation with a linear cost functional on a convex set and show that its Fenchel--Rockafellar dual formulation gives the right solution to the optimal matching problem. Basing on our formulations, a numerical approximation is given via an augmented Lagrangian method. We compute at the same time the optimal matching measure, optimal flows, and Kantorovich potentials. The convergence of the discretization is also studied in detail.
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