Mathematical Sciences: On Monge-Kantorovich Problems

Monge.Kantorovich问题(MKP)处理抽象空间上给定的两个或多个相同体积的质量之间的“最优”对应，其中最优性取决于问题的上下文。自G.Monge(1781)和L.V.Kantorovich(1940)首次提出MKPS以来，MKPS在数学科学的不同领域有着悠久而丰富多彩的历史。尽管对特定的多个重点项目有不同的结果，但没有认真尝试探讨各种办法之间的相互作用。本项目的目的是系统地研究在MKPS“连续案例”中方法和结果之间的关系。概率中的边际问题和矩问题。以及“离散案例”中的案例。运输多面体上的优化和测量问题的公理方法。主要研究者相信，离散情形的算法和公理方法在获得连续MKP的显式解方面有很大的希望。对称地，概率方法、极限定理、正态模型和稳定模型。将导致在离散情况下识别新的数学规划和博弈论结构、算法和模型。

Monge.Kantorovich problems (MKPs) deal with "optimal" correspondences between two or more masses of equal volume given on an abstract space, where the optimality depends upon the context of the problem. MKPs have a long and colorful history in different areas of mathematical sciences since their original formulation by G. Monge (1781) and L.V. Kantorovich (1940). Despite the existence of separate results for particular MKPs no serious attempt has been made to explore the interplay among the various approaches. The objective of this project is to make a systematic study of the relationships between the methods and results in the "continuous case" of MKPs . marginal and moment problems in probability . and those in the "discrete case" . optimization over transportation polytopes and axiomatic approaches to measurement problems. It is the belief of the principal investigators that the algorithms and axiomatic approaches of the discrete case hold great promise in obtaining explicit solutions for continuous MKPs. Symmetrically, the methods of probability .limit theorems, normal and stable models . will lead to the identification of new mathematical programming and game theoretic structures, algorithms and models in the discrete case.