Kantorovich Dual Problem (for general costs): Difference between revisions
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<ref name="Villani">[https://people.math.gatech.edu/~gangbo/Cedric-Villani.pdf C. Villani, ''Topics in Optimal Transportation'', Chapter 1.] (pages 17-21)</ref> | <ref name="Villani">[https://people.math.gatech.edu/~gangbo/Cedric-Villani.pdf C. Villani, ''Topics in Optimal Transportation'', Chapter 1.] (pages 17-21)</ref> | ||
<ref name="Santambrogio">https://link-springer-com.proxy.library.ucsb.edu:9443/book/10.1007/978-3-319-20828-2 F. Santambrogio, ''Optimal Transport for Applied Mathematicians'', Chapter 1.]</ref> | <ref name="Santambrogio">https://link-springer-com.proxy.library.ucsb.edu:9443/book/10.1007/978-3-319-20828-2 F. Santambrogio, ''Optimal Transport for Applied Mathematicians'', Chapter 1.] (pages 9-16)</ref> | ||
</ references> | </ references> | ||
Revision as of 21:52, 16 May 2020
Introduction
The main advantage of Kantorovich Problem, in comparison to Monge problem, is in the convex constraint property. It is possible to formulate the dual problem.
Statement of Theorem
(Kantorovich Duality) Let X and Y be Polish spaces, let and , and let a cost function be lower semi-continuous.
![{\displaystyle c:X\times Y\rightarrow [0,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b86eb4eb62728fa63010b95eb2c285d7d6a18649)
Proof of Theorem
References
</ references>
- ↑ C. Villani, Topics in Optimal Transportation, Chapter 1. (pages 17-21)
- ↑ https://link-springer-com.proxy.library.ucsb.edu:9443/book/10.1007/978-3-319-20828-2 F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 1.] (pages 9-16)