Kantorovich Dual Problem (for general costs): Difference between revisions
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<math> I[\pi]= \int_{X\times Y} c(x,y) d\pi(x,y), \quad J(\varphi,\psi)=\int_{X}\varphi(x)d\mu(x)+\int_{Y}\psi(y) d\nu(y) </math>. | <math> I[\pi]= \int_{X\times Y} c(x,y) d\pi(x,y), \quad J(\varphi,\psi)=\int_{X}\varphi(x)d\mu(x)+\int_{Y}\psi(y) d\nu(y) </math>. | ||
Define <math> \Pi(\mu,\nu) </math> to be the set of Borel probability measures <math> \pi </math> on <math> X\times Y </math> such that for all measurable sets <math> A \subset X </math> and <math> B \subset Y </math>, <math> \pi[A\times Y]=\mu(A) </math>, <math> \pi[X\times B]=\nu(B) </math>, and define <math> \Phi_{c} </math> to be the set of all measurable functions <math> (\varphi, \psi) \in L^{1}(d\mu) \times L^{1}(d\nu) <\math> satisfying <math> \varphi(x)+\psi(y) | Define <math> \Pi(\mu,\nu) </math> to be the set of Borel probability measures <math> \pi </math> on <math> X\times Y </math> such that for all measurable sets <math> A \subset X </math> and <math> B \subset Y </math>, <math> \pi[A\times Y]=\mu(A) </math>, <math> \pi[X\times B]=\nu(B) </math>, and define <math> \Phi_{c} </math> to be the set of all measurable functions <math> (\varphi, \psi) \in L^{1}(d\mu) \times L^{1}(d\nu) <\math> satisfying <math> \varphi(x)+\psi(y) c(x,y) </math>. | ||
==Proof of Theorem== | ==Proof of Theorem== | ||
Revision as of 22:50, 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. Whenever and , define \newline .
![{\displaystyle c:X\times Y\rightarrow [0,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b86eb4eb62728fa63010b95eb2c285d7d6a18649)
![{\displaystyle I[\pi ]=\int _{X\times Y}c(x,y)d\pi (x,y),\quad J(\varphi ,\psi )=\int _{X}\varphi (x)d\mu (x)+\int _{Y}\psi (y)d\nu (y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe572f66aca1d528fc4ced0547d7fc298e00b502)
Define to be the set of Borel probability measures on such that for all measurable sets and , , , and define to be the set of all measurable functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\varphi, \psi) \in L^{1}(d\mu) \times L^{1}(d\nu) <\math> satisfying <math> \varphi(x)+\psi(y) c(x,y) } .
![{\displaystyle \pi [A\times Y]=\mu (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d027029dfcbb54d2291e6cc3849e36a9f971b19)
![{\displaystyle \pi [X\times B]=\nu (B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a697f291cc41bd1dde9d75e59465165aa8d7470)
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)