Kantorovich Dual Problem (for general costs)

Introduction

The main advantage of Kantorovich Problem, in comparison to Monge problem, is in the convex constraint property. It is possible to formulate the dual problem.

Statement of Theorem

(Kantorovich Duality) Let X and Y be Polish spaces, let $\mu \in {\mathcal {P}}(X)$ and $\nu \in {\mathcal {P}}(Y)$, and let a cost function $c:X\times Y\rightarrow [0,+\infty ]$ be lower semi-continuous. Whenever $\pi \in {\mathcal {P}}(X\times Y)$ and $(\varphi ,\psi )\in L^{1}(d\mu )\times L^{1}(d\nu )$, define $I[\pi ]=\int _{X\times Y}c(x,y)d\pi (x,y),\quad J(\varphi ,\psi )=\int _{X}\varphi (x)d\mu (x)+\int _{Y}\psi (y)d\nu (y)$ ...

References

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