It is well-known that duality in the Monge-Kantorovich transport problem holds true provided that the cost function c:X×Y→[0,∞] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification cr of the cost c, so that the Monge-Kantorovich duality holds true replacing c by cr. In particular, passing from c to cr only changes the value of the primal Monge-Kantorovich problem. Finally, the rectified function cr is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport

Duality for rectified Cost Functions

PRATELLI, ALDO
2012-01-01

Abstract

It is well-known that duality in the Monge-Kantorovich transport problem holds true provided that the cost function c:X×Y→[0,∞] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification cr of the cost c, so that the Monge-Kantorovich duality holds true replacing c by cr. In particular, passing from c to cr only changes the value of the primal Monge-Kantorovich problem. Finally, the rectified function cr is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/275706
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