Augmented Lagrangian Method for Optimal Partial Transportation

The use of augmented Lagrangian algorithm for optimal transport problems goes back to Benamou & Brenier (2000, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math., 84, 375-393), in the case where the cost corresponds to the square of the Euclidean distance. It was recently extended in Benamou & Carlier (2015, Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations. J. Optim. Theory Appl., 167, 1-26), to the optimal transport with the Euclidean distance and Mean-Field Games theory and in Benamou et al. (2017, A numerical solution to Monge's problem with a Finsler distance cost ESAIM: M2AN), to the optimal transportation with Finsler distances. Our aim here is to show how one can use this method to study the optimal partial transport problem with Finsler distance costs. To this aim, we introduce a suitable dual formulation of the optimal partial transport, which contains all the information on the active regions and the associated flow. Then, we use a finite element discretization with the FreeFem++ software to provide numerical simulations for the optimal partial transportation. A convergence study for the potential together with the flux and the active regions is given to validate the approach.