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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:485

Article

Hug, Romain¹  Maitre, Emmanuel²  Papadakis, Nicolas³ 

[1] Univ Artois, LML, F-62307 Lens, France

[2] Univ Grenoble Alpes, Grenoble INP, CNRS, LJK,Inst Engn, F-38000 Grenoble, France

[3] Univ Bordeaux, CNRS, IMB, UMR 5251, F-33400 Talence, France

关键词: Optimal transport;    Augmented Lagrangian method;    Existence and uniqueness problem;   

DOI  :  10.1016/j.jmaa.2019.123811

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

Free