Convex duality in nonlinear optimal transport

TL;DR

This paper develops a convex duality framework for nonlinear optimal transport problems, unifying various formulations and providing explicit optimality conditions and existence results.

Contribution

It introduces a convex analytic approach to nonlinear optimal transport, generalizing classical formulations and deriving dual problems and optimality conditions.

Findings

Unified convex duality framework for nonlinear optimal transport

Explicit optimality conditions for convex integral functionals

Existence of solutions for relaxed problem formulations

Abstract

This article studies problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of a large class of related problems in probability theory and allows for generalizations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems. When the objective takes the form of a convex integral functional, we obtain more explicit optimality conditions and establish the existence of solutions for a relaxed formulation of the problem. This covers, in particular, the mass transportation problem and its nonlinear generalizations.