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

$I[\pi ]=\int _{X\times Y}c(x,y)d\pi (x,y),\quad J(\varphi ,\psi )=\int _{X}\varphi (x)d\mu (x)+\int _{Y}\psi (y)d\nu (y)$.

Define $\Pi (\mu ,\nu )$ to be the set of Borel probability measures $\pi$ on $X\times Y$ such that for all measurable sets $A\subset X$ and $B\subset Y$,

$\pi [A\times Y]=\mu (A)$, $\pi [X\times B]=\nu (B)$,

and define $\Phi _{c}$ to be the set of all measurable functions $(\varphi ,\psi )\in L^{1}(d\mu )\times L^{1}(d\nu )$ satisfying $\varphi (x)+\psi (y)\leq c(x,y)$ for $d\mu$ almost everywhere in X and $d\nu$ almost everywhere in Y.

Then $inf_{\Pi (\mu ,\nu )}I[\pi ]=sup_{\Phi _{c}}J(\varphi ,\psi )$.

Moreover, the infimum $inf_{\Pi (\mu ,\nu )}I[\pi ]$ is attained. In addition it is possible to restrict $\varphi$ and $\psi$ to be continuous and bounded.

References

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