On the Uniqueness and Numerical Approximations for a Matching Problem
研究了带约束最优匹配问题的唯一性条件，证明了几何条件下的唯一性，并基于对偶公式用增广拉格朗日方法同时计算最优匹配测度、最优流和Kantorovich势能，还分析了离散化的收敛性。
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.