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

Romain Hug (1, 2) , Emmanuel Maitre (1) , Nicolas Papadakis (3)

1 EDP - Equations aux Dérivées Partielles
2 RICAM - Johann Radon Institute for Computational and Applied Mathematics
3 IMB - Institut de Mathématiques de Bordeaux

The dynamical formulation of the optimal transport problem, introduced by J. D. Benamou and Y. Brenier, corresponds to the time-space search of a density and a momentum minimizing a transport energy between two densities. In order to solve this problem, an algorithm has been proposed to estimate a saddle point of a Lagrangian. We will study the convergence of this algorithm to a saddle point of the Lagrangian, in the most general conditions, particularly in cases where initial and final densities vanish on some areas of the transportation domain. The principal difficulty of our study will consist in the proof, under these conditions, of the existence of a saddle point, and especially in the uniqueness of the density-momentum component. Indeed, these conditions imply to have to deal with non-regular optimal transportation maps. For these reasons, a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required.

Domaines: Equations aux dérivées partielles [math.AP]; Traitement du signal et de l'image [eess.SP]

Romain Hug, Emmanuel Maitre, Nicolas Papadakis. On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities. Journal of Mathematical Analysis and Applications, 2020, 485 (2), pp.123811. ⟨10.1016/j.jmaa.2019.123811⟩. ⟨hal-01128793v5⟩