Kantorovich Dual Problem (for general costs)

The Kantorovich Dual Problem is one of the minimization problems in Optimal Transport. It is a dual problem of the Kantorovich Problem.

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 in the very general metric spaces called Polish spaces, i.e. complete separable metric spaces.

The Shipper's Problem

One of the ways to understand this problem is stated by Caffarelli. We will provide the modern rephrase of his statement.

During the COVID-19 pandemic, people are in the lockdown and it is the best time to enjoy a coffee time. All in all, it costs Amazon $c(x,y)$ dollars to ship one box of necessary espresso capsules from place $x$ to place $y$, i.e. from warehouses to homes. We want to optimize this expensive habit and consequently to solve appropriate Monge-Kantorovich problem. The mathematicians come to Amazon and propose the new kind of payment. For every box at place $x$ they will charge $\varphi (x)$ dollars and $\psi (y)$ dollars to deliver at place $y$. However, mathematicians will not reveal their shipping routes. Of course, in order for Amazon to accept this offer, the price $\varphi (x)+\psi (y)\leq c(x,y).$ The moral is that if the mathematicians are smart enough, they will be capable to make this shipment cheaper. This is provided by Kantorovich duality theorem. Take care that in the same cases, mathematicians will also give negative prices, if it is necessary!

Statement of Theorem

This is the statement of the Theorem in the book "Topics in Optimal Transportation", by Cedric Villani.

Theorem. 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.

Ideas and the techniques used in the Proof

First, we assume that our spaces $X$ and $Y$ are compact and that the cost function $c(x,y)$ is continuous. The general case follows by an approximation argument.

The main idea is to use minimax principle, i.e. interchanging inf sup with sup inf in the proof. For this, we need some basic convex analysis techniques, namely Legendre-Fenchel transform (qoute needed) and Theorem on Fenchel-Rockafellar Duality (its proof is based on Hahn-Banach theorem consequence on separating convex sets). More information on the Convex analysis can be found in the book.

Take a note that at some point we use Arzela-Ascoli Theorem [1]. In a non-compact space this is not possible. In order to evade compactness property, we have to use Prokhorov's theorem.

Theorem. Let $\mu _{n}$ be a sequence of tight probability measures on Polish space $X$. Then, there exists $\mu \in P(X)$ and convergent subsequence $\mu _{n_{k}}$ such that $\mu _{n_{k}}\rightarrow \mu$ in the dual of $C_{b}(X)$. Conversely, every sequence $\mu _{n}\rightarrow \mu$ is tight.

The proof of the previous Theorem can be found in . For more information on $C_{b}(X)$ duality, take a look at Dual space of C_0(x) vs C_b(x).

C-concave functions

In Kantorovich Duality Theorem, the left-hand side of the last equality, the infimum $\inf _{\Pi (\mu ,\nu )}I[\pi ]$ is attained. We do not know something similar about the right-hand side. However, when cost function $c(x,y)$ is bounded we can restrict $\sup _{\Phi _{c}}J(\varphi ,\psi )$ to pairs $(\varphi ^{cc},\varphi ^{c})$ where $\varphi$ is bounded and

$\varphi ^{c}(y)=\inf _{y\in Y}[c(x,y)-\varphi (x)],\quad \varphi ^{cc}(x)=\inf _{y\in Y}[c(x,y)-\varphi ^{c}(y)].$

The pair $(\varphi ^{cc},\varphi ^{c})$ is called a pair of conjugate c-concave functions. It is known that $(\varphi ^{cc})^{c}=\varphi ^{c}$ and that $\varphi ^{c}$ is measurable.

In addition, it is possible to give a proof of Kantorovich Duality theorem using c-concave functions. Namely, we can find c-concave function $\varphi$ such that
$(x,y)\in \Gamma \implies \varphi (x)+\varphi ^{c}(y)=c(x,y).$ The proof of this fact could be found in the book