On the Existence of Monge Solutions to Multi-marginal Optimal Transport with Quadratic Cost and Uniform Discrete Marginals

Pedram Emami () and Brendan Pass ()
Additional contact information
Pedram Emami: University of Alberta
Brendan Pass: University of Alberta

SN Operations Research Forum, 2025, vol. 6, issue 2, 1-8

Abstract: Abstract A natural and important question in multi-marginal optimal transport is whether the Monge ansatz is justified; does there exist a solution of Monge, or deterministic, form? We address this question for the quadratic cost when each marginal measure is m-empirical (that is, uniformly supported on m points). By direct computation, we provide an example showing that the ansatz can fail when the underlying dimension d is 2, the number of marginals N to be matched is 3, and the size m of their supports is 3. As a consequence, the set of m-empirical measures is not barycentrically convex when $$N \ge 3$$ N ≥ 3 , $$d \ge 2$$ d ≥ 2 , and $$m \ge 3$$ m ≥ 3 . It is a well-known consequence of the Birkhoff-von Neumann theorem that the Monge ansatz holds for $$N=2$$ N = 2 , standard techniques show it holds when $$d=1$$ d = 1 , and we provide a simple proof here that it holds whenever $$m=2$$ m = 2 . Therefore, the N, d, and m in our counterexample are as small as possible.

Keywords: Multi-marginal optimal transport; Discrete marginals; Monge solutions; Wasserstein barycenter; 49Q22 (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s43069-025-00437-w Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:snopef:v:6:y:2025:i:2:d:10.1007_s43069-025-00437-w

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/43069

DOI: 10.1007/s43069-025-00437-w

Access Statistics for this article

SN Operations Research Forum is currently edited by Marco Lübbecke

More articles in SN Operations Research Forum from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().