Duality for Borel Measurable Cost Functions
نویسندگان
چکیده
We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if c : X × Y → [0,∞) is an arbitrary Borel measurable cost function on the product of Polish spaces X,Y . In the course of the proof we show how to relate a non-optimal transport plan to the optimal transport costs via a “subsidy” function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations.
منابع مشابه
On the Duality Theory for the Monge–Kantorovich Transport Problem
This article, which is an accompanying paper to [BLS09], consists of two parts: In section 2 we present a version of Fenchel’s perturbation method for the duality theory of the Monge– Kantorovich problem of optimal transport. The treatment is elementary as we suppose that the spaces (X,μ), (Y, ν), on which the optimal transport problem [Vil03, Vil09] is defined, simply equal the finite set {1, ...
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