论文标题
非传统成本的摇滚类型定理
A Rockafellar-type theorem for non-traditional costs
论文作者
论文摘要
在本说明中，我们提出了一种统一的方法，即有关非传统成本函数的最佳运输问题的存在问题，即假定无限值的成本。我们建立了一种新方法，该方法依赖于证明线性不平等的特殊（可能是无限）家族的可溶性。当该家族的索引集可计数时，我们就确保解决方案存在的系数给出了必要和充分的条件，并且在运输理论的设置中，我们称之为$ c $ path的态度。如果是无数索引集，则需要一个额外的假设来解决。在这种情况下，我们提出了足够的条件。我们注意到，任何承认潜力的集合都必须是$ c $ path的，并且这种情况取代了经典理论的$ c $ cyclic单调性，即何时实现成本。我们的方法还为Rockafellar，Rochet和Rüschendorf的经典结果提供了新的基本证明。
In this note, we present a unified approach to the problem of existence of a potential for the optimal transport problem with respect to non-traditional cost functions, that is, costs that assume infinite values. We establish a new method that relies on proving solvability of a special (possibly infinite) family of linear inequalities. When the index set of this family is countable, we give a necessary and sufficient condition on the coefficients that assures the existence of a solution, and which, in the setting of transport theory, we call $c$-path-boundedness. In the case of an uncountable index set, one needs an additional assumption for solvability. We propose a sufficient condition in this case. We note that any set admitting a potential must be $c$-path-bounded, and this condition replaces $c$-cyclic monotonicity from the classical theory, i.e. when the cost is real-valued. Our method also gives a new and elementary proof for the classical results of Rockafellar, Rochet and Rüschendorf.
