相关论文: 60 years of cyclic monotonicity: a survey

$c$-cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new…

This note establishes that a generalization of $c$-cyclical monotonicity from the Monge-Kantorovich problem with two marginals gives rise to a sufficient condition for optimality also in the multi-marginal version of that problem. To obtain…

This paper studies the geometry of the optimizer for the optimal transport problem with capacity constraints. We introduce the concept of c-capacity monotonicity, which is a generalization of c-cyclical monotonicity in optimal transport. We…

A fundamental concept in optimal transport is c-cyclical monotonicity: it allows to link the optimality of transport plans to the geometry of their support sets. Recently, related concepts have been successfully applied in the…

A measure theoretical approach is presented to study the Monge-Kantorovich optimal mass transport problem. This approach together with Kantorovich duality provide an effective tool to answer a long standing question about the support of…

We prove that $c$-cyclically monotone transport plans $\pi$ optimize the Monge-Kantorovich transportation problem under an additional measurability condition. This measurability condition is always satisfied for finitely valued, lower…

In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and…

We consider the problem to transport resources/mass while abiding by constraints on the flow through constrictions along their path between specified terminal distributions. Constrictions, conceptualized as toll stations at specified…

The aim of the present paper is to extend Kantorovich's mass transport problem to the framework of upper/lower continuous capacities and to prove the cyclic monotonicity of the supports of optimal supermodular plans. As in the probabilistic…

Inspired by the matching of supply to demand in logistical problems, the optimal transport (or Monge--Kantorovich) problem involves the matching of probability distributions defined over a geometric domain such as a surface or manifold. In…

In the classical Monge-Kantorovich problem, the transportation cost only depends on the amount of mass sent from sources to destinations and not on the paths followed by this mass. Thus, it does not allow for congestion effects. Using the…

We introduce and study a multi-marginal optimal partial transport problem. Under a natural and sharp condition on the dominating marginals, we establish uniqueness of the optimal plan. Our strategy of proof establishes and exploits a…

We consider the modified Monge-Kantorovich problem with additional restriction: admissible transport plans must vanish on some fixed functional subspace. Different choice of the subspace leads to different additional properties optimal…

We shall present a measure theoretical approach for which together with the Kantorovich duality provide an efficient tool to study the optimal transport problem. Specifically, we study the support of optimal plans where the cost function…

The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the convergence of the entropic values to the…

The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing…

This article is aimed at presenting the Schr\"odinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schr\"odinger problem. We also give a survey of the related literature.…

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are…

In the setting of optimal transport with $N\ge 2$ marginals, a necessary condition for transport plans to be optimal is that they are $c$-cyclically monotone. For $N=2$ there exist several proofs that in very general settings $c$-cyclical…

This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex…