Characterization of optimal transport plans for the Monge-Kantorovich problem

By Walter Schachermayer, Josef Teichmann. 2007

64 citations recorded by the source corpus.

We prove that c-cyclically monotone transport plansoptimize the Monge-Kantorovich transportation problem under an additional measur- ability condition. This measurability condition is always satisfied for finitely valued, lower semi-continuous cost functions. In particular, this yields a pos- itive answer to Problem 2.25 in C. Villani's book. We emphasize that we do not need any regularity conditions as were imposed in the previous literature.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e65527e400276b4e95f96e60e97e590d251a750c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures (2015, unknown): “We use a clever truncation argument of [40], based on the maps Tn : R → R, Tn(φ) := −n ∨ φ ∧ n, (4.”
The Monge problem for strictly convex norms in Rd (2010, background): “Duality and sufficiency of cyclical monotonicity may be pursued in very general settings [23, 24, 25, 3, 26, 22, 8]; however for the purpose of this paper duality may be obtained more easily [1, 29].”
Optimal transport and Skorokhod embedding (2013, methodology): “Following contributions of Ambrosio and Pratelli [3], this problem was resolved by Pratelli [42] and Schachermayer and Teichmann [52] who established the clear-cut characterization stated in Theorem 3.2.”
Optimal transportation on non-compact manifolds (2007, background): “The reader should consult the books and surveys given above to have a better view of the history of the subject, in particular Villani’s second book on the subject [42].”
Duality for Borel measurable cost functions (2008, unknown): “Similarly as in [ST08] we deﬁne Additionally, ξ dπ ≤ c dπ and ξ d ˜ π ≤ c d ˜ π exist, possibly taking the value −∞ , since ξ ≤ c holds π -almost surely as well as ˜ π -almost surely, and we assume that (cid:5) c dπ, (cid:5) c d ˜ π ∞ .”
Optimal and better transport plans (2008, methodology): “We consider the Monge-Kantorovich transport problem (µ, ν, c) for Borel probability measures µ, ν on Polish spaces X, Y and a Borel measurable cost func-”