Duality Theorems and Vector Measures in Optimal Transportation Theory

TL;DR

This paper generalizes the duality theorem in optimal transportation to an abstract setting without measures, enabling applications to vector measures and related problems in various fields.

Contribution

It introduces a broad abstract framework for optimal transport and duality, extending the theory to vector measures and other mathematical problems.

Findings

Generalized duality theorem for abstract optimal transport

Conditions for existence of transport plans and solutions

Application to vector measures and related problems

Abstract

The optimal transportation problem, first suggested by Gaspard Monge in the 18th century and later revived in the 1940s by Leonid Kantorovich, deals with the question of transporting a certain measure to another, using transport maps or transport plans that minimize the total cost of transportation. This problem is very popular since it has a variety of applications in economics, physics, computer science and more. One of the main tools in this theory is the duality theorem, which states that the optimal total cost equals the value of a different optimization problem called the dual problem. In this work, I show how the problem and duality theorem can be generalized to an abstract formulation, in which I omit the use of measures. I show how this generalization implies a wide range of different optimal transport problems, and even other problems from game theory, linear programming and functional analysis. In particular, I show how the optimal transport problem can be generalized to deal with vector measures as a result of the abstract theorem and discuss the properties of this problem: I present its corresponding duality theorem and formulate conditions for the existence of transport plans, transport maps and solutions to the dual problem.