Optimal measure transportation with respect to non-traditional costs

TL;DR

This paper investigates optimal mass transport problems with non-traditional costs that can be infinite, establishing conditions for the existence of optimal plans and potentials, and providing proofs under different assumptions including a novel approach for discrete measures.

Contribution

It introduces the notions of $c$-compatibility and strong-$c$-compatibility, and proves the existence of potentials for optimal plans under these conditions, including a new constructive method using Hall polytopes.

Findings

Finite-cost plans imply $c$-compatibility of measures.

Strong $c$-compatibility ensures the existence of a potential for optimal plans.

A new constructive proof using Hall polytopes is provided for discrete measures.

Abstract

We study optimal mass transport problems between two measures with respect to a non-traditional cost function, i.e. a cost $c$ which can attain the value $+\infty$. We define the notion of $c$-compatibility and strong-$c$-compatibility of two measures, and prove that if there is a finite-cost plan between the measures then the measures must be $c$-compatible, and if in addition the two measures are strongly $c$-compatible, then there is an optimal plan concentrated on a $c$-subgradient of a $c$-class function. This function is the so-called potential of the plan. We give two proofs of this theorem, under slightly different assumptions. In the first we utilize the notion of $c$-path-boundedness, showing that strong $c$-compatibility implies a strong connectivity result for a directed graph associated with an optimal map. Strong connectivity of the graph implies that the $c$-cyclic monotonicity of the support set (which follows from classical reasoning) guarantees its $c$-path-boundedness, implying, in turn, the existence of a potential. We also give a constructive proof, in the case when one of the measures is discrete. This approach adopts a new notion of `Hall polytopes', which we introduce and study in depth, to which we apply a version of Brouwer's fixed point theorem to prove the existence of a potential in this case.