A Rockafellar-type theorem for non-traditional costs

TL;DR

This paper introduces a unified approach to establish the existence of potentials in optimal transport problems with non-traditional, possibly infinite-valued costs, extending classical theory and providing new solvability conditions.

Contribution

It develops a new method based on solving linear inequalities, generalizes Rockafellar's classical results, and introduces the concept of $c$-path-boundedness for non-traditional costs.

Findings

Provides necessary and sufficient conditions for countable index sets.

Proposes a sufficient condition for uncountable index sets.

Offers a new elementary proof for classical optimal transport results.

Abstract

In this note, we present a unified approach to the problem of existence of a potential for the optimal transport problem with respect to non-traditional cost functions, that is, costs that assume infinite values. We establish a new method that relies on proving solvability of a special (possibly infinite) family of linear inequalities. When the index set of this family is countable, we give a necessary and sufficient condition on the coefficients that assures the existence of a solution, and which, in the setting of transport theory, we call $c$-path-boundedness. In the case of an uncountable index set, one needs an additional assumption for solvability. We propose a sufficient condition in this case. We note that any set admitting a potential must be $c$-path-bounded, and this condition replaces $c$-cyclic monotonicity from the classical theory, i.e. when the cost is real-valued. Our method also gives a new and elementary proof for the classical results of Rockafellar, Rochet and R\"uschendorf.