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On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities
Journal of Mathematical Analysis and Applications ( IF 1.3 ) Pub Date : 2020-05-01 , DOI: 10.1016/j.jmaa.2019.123811
Romain Hug , Emmanuel Maitre , Nicolas Papadakis

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.

中文翻译:

非负密度间最优输运的增广拉格朗日方法的收敛性

JD Benamou 和 Y. Brenier 介绍的最优输运问题的动力学公式对应于密度和动量的时空搜索,使两个密度之间的输运能量最小化。为了解决这个问题,已经提出了一种估计拉格朗日量的鞍点的算法。我们将在最一般的条件下研究该算法对拉格朗日的鞍点的收敛性,特别是在初始和最终密度在运输域的某些区域消失的情况下。我们研究的主要困难在于在这些条件下证明鞍点的存在,尤其是密度-动量分量的唯一性。事实上,这些条件意味着必须处理非常规的最优交通地图。由于这些原因,
更新日期:2020-05-01
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