Kantorovich Dual Problem (for general costs)

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

The Shipper's Problem

Type of this problem is stated by Caffarelli. We will provide the modern one.

During the pandemic, people are in the lockdown and it is the best time to enjoy a coffee time. All in all, it costs Amazon dollars to ship one box of necessary espresso capsules from place to place , 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 they will charge dollars and dollars to deliver at place . However, mathematicians will not reveal their shipping routes. Of course, in order for Amazon to accept this offer, the price 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!

Statement of Theorem

Theorem.[1] Let X and Y be Polish spaces, let and , and let a cost function be lower semi-continuous.

Whenever and , define

.

Define to be the set of Borel probability measures on such that for all measurable sets and ,

, ,

and define to be the set of all measurable functions satisfying for almost everywhere in X and almost everywhere in Y.

Then .

Moreover, the infimum is attained. In addition it is possible to restrict and to be continuous and bounded.

Outline of the Proof

First, we assume that our spaces and are compact and that the cost function 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.)

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

C-concave functions

In Kantorovich Duality Theorem, the left-hand side of the last equality, the infimum is attained. We do not know something similar about the right-hand side. However, when cost function is bounded we can restrict to pairs where is bounded and

The pair is called a pair of conjugate c-concave functions. It is known that and that 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 such that
This should lead to much faster proof.

References

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  1. C. Villani, Topics in Optimal Transportation, Chapter 1. (pages 17-21)
  2. https://link-springer-com.proxy.library.ucsb.edu:9443/book/10.1007/978-3-319-20828-2 F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 1.] (pages 9-16)