Sufficiency of $c$-cyclical monotonicity in a class of multimarginal optimal transport problems

TL;DR

This paper introduces a new approach to establish the sufficiency of $c$-cyclical monotonicity in multimarginal optimal transport problems, expanding the range of problems where optimality conditions are fully characterized.

Contribution

The paper develops a novel framework that simplifies proving sufficiency of $c$-cyclical monotonicity, enabling results in previously inaccessible problem settings.

Findings

New framework for proving sufficiency

Extended applicability to complex multimarginal problems

Unified derivation of known results

Abstract

$c$-cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.