Optimal and better transport plans

By Mathias Beiglbock, Martin Goldstern, G. Maresch, Walter Schachermayer. 2008

75 citations recorded by the source corpus.

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and nite, or continuous and may possibly attain the value1. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided thatfc =1g is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a \better" notion of optimality called robust optimality.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/87e40684022be4cf1e222a9ba62e50eaf0ae557f

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On a problem of optimal transport under marginal martingale constraints (2012, background): “8This approach is inspired by [1] where c-cyclical monotonicity is linked to optimality with the help of Kellerer’s result.”
Optimal transport from Lebesgue to Poisson (2010, background): “In fact, if the cost function is sufficiently nice (continuous is much more than needed, see [BGMS09]) also the reverse direction holds.”
Optimal transport and Skorokhod embedding (2013, background): “Lower semi-continuity of the cost function can also be relaxed, as shown in [6] and [8].”
On the extremality, uniqueness and optimality of transference plans (2009, methodology): “The problem of checking whether a measure μ can be perturbed by cycles has been considered in several different contexts, see for example [2, 3, 12].”
A General Duality Theorem for the Monge--Kantorovich Transport Problem (2009, methodology): “e time) that, similarly to the situation in Theorem 1.2, duality in the form Drel = Pholds without any additional assumption. For instance this is the case in Example 1.1 and combining the methods of [BGMS09] and [BS09] one may prove that Drel = P provided that the Borel 8By saying that hn increases”