Duality for rectified cost functions
Beiglböck M, Pratelli A (2012)
Publication Language: English
Publication Status: Published
Publication Type: Journal article, Review article
Publication year: 2012
Journal
Publisher: Springer Verlag (Germany)
Book Volume: 45
Pages Range: 27-41
Journal Issue: 1-2
DOI: 10.1007/s00526-011-0449-0
Abstract
It is well-known that duality in the Monge-Kantorovich transport problem holds true provided that the cost function c : X x Y -> [0, a] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification c (r) of the cost c, so that the Monge-Kantorovich duality holds true replacing c by c (r) . In particular, passing from c to c (r) only changes the value of the primal Monge-Kantorovich problem. Finally, the rectified function c (r) 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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APA:
Beiglböck, M., & Pratelli, A. (2012). Duality for rectified cost functions. Calculus of Variations and Partial Differential Equations, 45(1-2), 27-41. https://dx.doi.org/10.1007/s00526-011-0449-0
MLA:
Beiglböck, Mathias, and Aldo Pratelli. "Duality for rectified cost functions." Calculus of Variations and Partial Differential Equations 45.1-2 (2012): 27-41.
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