Constrained Optimal Transport

TL;DR

This paper generalizes the duality theory of classical optimal transport to an abstract Banach lattice framework, providing new duality characterizations and extending classical results to broader settings.

Contribution

It introduces a generalized duality framework for optimal transport within Banach lattices, broadening the theoretical foundation of the field.

Findings

Established duality results in an abstract Banach lattice setting.

Provided characterizations of dual elements in the generalized framework.

Extended classical optimal transport results to new mathematical structures.

Abstract

The classical duality theory of Kantorovich and Kellerer for the classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice $\cal{X}$ with a order unit. The primal problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of $\cal{X}$ and the dual problem is defined on the bi-dual of $\cal{X}$. These results are then applied to several extensions of the classical optimal transport.