Optimal and better transport plans

TL;DR

This paper extends the theory of optimal transport plans in a measure-theoretic setting, showing that c-monotonicity implies optimality under broader conditions, and introduces the concept of robust optimality.

Contribution

It generalizes the conditions under which c-monotone transport plans are optimal and introduces the notion of robust optimality, linking it to strong c-monotonicity.

Findings

c-monotone plans are optimal under broader measurable conditions

robust optimality is equivalent to strong c-monotonicity

extends optimal transport theory to more general cost functions

Abstract

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value infty. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=infty} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a "better" notion of optimality called robust optimality.