Kantorovich Dual Problem (for general costs): Difference between revisions
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==Statement of Theorem== | ==Statement of Theorem== | ||
(Kantorovich Duality) Let X and Y be Polish spaces, let <math>\mu \in \mathcal{P}(X)</math> and <math>\nu \in \mathcal{P}(Y)</math>, and let a cost function <math> c:X \times Y \rightarrow[0,+\infty] </math> be lower semi-continuous | (Kantorovich Duality) Let X and Y be Polish spaces, let <math>\mu \in \mathcal{P}(X)</math> and <math>\nu \in \mathcal{P}(Y)</math>, and let a cost function <math> c:X \times Y \rightarrow[0,+\infty] </math> be lower semi-continuous. | ||
==Proof of Theorem== | ==Proof of Theorem== | ||
Revision as of 21:45, 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.
- ↑ 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.]