Duality for Borel measurable cost functions

Details Duality for Borel measurable cost functions Duality for Borel measurable cost functions

2008-07-09

arxiv.org/abs/0807.1468v1

Mathematics

Optimization and Control

Scientific paper

We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if $c:X\times Y\to [0,\infty)$ 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.

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