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

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

1 EDP - Equations aux Dérivées Partielles
2 IMB - Institut de Mathématiques de Bordeaux

In the beginning of the 2000 years, J. D. Benamou and Y. Brenier have proposed a dynamical formulation of the optimal transport problem, corresponding to the time-space search of a density and a momentum minimizing a transport energy between two densities. They proposed, in order to solve this problem in practice, to deal with it by looking for a saddle point of some Lagrangian by an augmented Lagrangian algorithm. Using the theory of non-expansive operators, 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 cancel on some areas of the transportation domain. The principal difficulty of our study will consist of the proof, in 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: that is why an important part of our works will have for object a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space.

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. 2017. ⟨hal-01128793v1⟩