$c$-cyclical monotonicity as a sufficient criterion for optimality in the multi-marginal Monge-Kantorovich problem

TL;DR

This paper extends the concept of $c$-cyclical monotonicity to the multi-marginal Monge-Kantorovich problem, providing a sufficient condition for optimality under certain boundedness assumptions.

Contribution

It generalizes $c$-cyclical monotonicity as a sufficient optimality criterion to the multi-marginal setting, using martingale transport ideas.

Findings

$c$-cyclical monotonicity is sufficient for optimality in multi-marginal problems.

The result applies when the cost function is bounded by a sum of integrable functions.

The proof leverages techniques from martingale transport theory.

Abstract

This note establishes that a generalization of $c$-cyclical monotonicity from the Monge-Kantorovich problem with two marginals gives rise to a sufficient condition for optimality also in the multi-marginal version of that problem. To obtain the result, the cost function is assumed to be bounded by a sum of integrable functions. The proof rests on ideas from martingale transport.