Topics About Optimal Transportation In Discrete Space

The origin of optimal transport comes from problem about moving sand which was proposed by French mathematician Monge.The problem considers about the minimization of total cost in mass transportation.Containing mapping hard to handle,Monge problem received much attention but little progress.Later Kantorovich proposed a new model which lead a new path to deal with the problem.From then on,theory of optimal transport developed rapidly,and showed great power in many fields.But there are lacks of it in discrete distance space,while discrete problems are quite common in real application.So it’s reasonable to give a research on discrete transport problems.On set of positive integers equipped with discrete distance,dual method plays no role anymore,due to the trival metric.Using some analytical method for discrete points,necessary and sufficient conditions of existence and uniqueness of the optimal mapping are concluded.On set of positve integers equipped with euclidean distance,dual method provides an excellent form of Kantorovich problem.Starting from some inequalities in transport problem,By constructing specific mapping and potential function,necessary and sufficient conditions of existence and uniqueness of the optimal mapping are concluded.On the set of real numbers equipped with euclidean distance,where measures are atomic,property of optimal mapping stays the same with that in the problem showed above.So by using similar methods,necessary and sufficient conditions of existence and uniqueness of the optimal mapping are concluded.Examples are given in each part to describe intentions and methods of the research in detail.