On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities

The dynamical formulation of the optimal transport problem, introduced by J.D. Benamou and Y. Brenier [4], amounts to find a time dependent space density and velocity field minimizing a transport energy between two densities. In order to solve this problem, an algorithm has been proposed to estimate the saddle point of a Lagrangian. We study the convergence of this algorithm in the most general case where initial and final densities may vanish on regions of the transportation domain. Under these assumptions, the main difficulty of our study is the proof of existence of a saddle point and of uniqueness of the density-momentum component, as it leads to deal with non-regular optimal transportation maps. For these reasons, a detailed study of the regularity properties of the velocity field associated to an optimal transportation map is required. (C) 2020 Elsevier Inc. All rights reserved.