An optimal matching problem

Ivar Ekeland

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 1, page 57-71
  • ISSN: 1292-8119

Abstract

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Given two measured spaces ( X , μ ) and ( Y , ν ) , and a third space Z, given two functions u(x,z) and v(x,z), we study the problem of finding two maps s : X Z and t : Y Z such that the images s ( μ ) and t ( ν ) coincide, and the integral X u ( x , s ( x ) ) d μ - Y v ( y , t ( y ) ) d ν is maximal. We give condition on u and v for which there is a unique solution.

How to cite

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Ekeland, Ivar. "An optimal matching problem." ESAIM: Control, Optimisation and Calculus of Variations 11.1 (2010): 57-71. <http://eudml.org/doc/90756>.

@article{Ekeland2010,
abstract = { Given two measured spaces $(X,\mu)$ and $(Y,\nu)$, and a third space Z, given two functions u(x,z) and v(x,z), we study the problem of finding two maps $s:X\rightarrow Z$ and $t:Y\rightarrow Z$ such that the images $s(\mu)$ and $t(\nu)$ coincide, and the integral $\int_\{X\}u(x,s(x))d\mu-\int _\{Y\}v(y,t(y))d\nu$ is maximal. We give condition on u and v for which there is a unique solution. },
author = {Ekeland, Ivar},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal transportation; measure-preserving maps.; measure-preserving maps},
language = {eng},
month = {3},
number = {1},
pages = {57-71},
publisher = {EDP Sciences},
title = {An optimal matching problem},
url = {http://eudml.org/doc/90756},
volume = {11},
year = {2010},
}

TY - JOUR
AU - Ekeland, Ivar
TI - An optimal matching problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 1
SP - 57
EP - 71
AB - Given two measured spaces $(X,\mu)$ and $(Y,\nu)$, and a third space Z, given two functions u(x,z) and v(x,z), we study the problem of finding two maps $s:X\rightarrow Z$ and $t:Y\rightarrow Z$ such that the images $s(\mu)$ and $t(\nu)$ coincide, and the integral $\int_{X}u(x,s(x))d\mu-\int _{Y}v(y,t(y))d\nu$ is maximal. We give condition on u and v for which there is a unique solution.
LA - eng
KW - optimal transportation; measure-preserving maps.; measure-preserving maps
UR - http://eudml.org/doc/90756
ER -

References

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  1. Y. Brenier, Polar factorization and monotone rearrangements of vector-valued functions. Comm. Pure App. Math.44 (1991) 375–417.  Zbl0738.46011
  2. G. Carlier, Duality and existence for a class of mass transportation problems and economic applications, Adv. Math. Economics5 (2003) 1–21.  Zbl1176.90409
  3. I. Ekeland and R. Temam, Convex analysis and variational problems. North-Holland Elsevier (1974) new edition, SIAM Classics in Appl. Math. (1999).  Zbl0281.49001
  4. W. Gangbo and R. McCann, The geometry of optimal transportation. Acta Math.177 (1996) 113–161.  Zbl0887.49017
  5. I. Ekeland, J. Heckman and L. Nesheim, Identification and estimation of hedonic models. J. Political Economy112 (2004) 60–109.  
  6. L. Kantorovitch, On the transfer of masses, Dokl. Ak. Nauk USSR37 (1942) 7–8.  
  7. S. Rachev and A. Ruschendorf, Mass transportation problems. Springer-Verlag (1998).  Zbl0990.60500
  8. C. Villani, Topics in mass transportation. Grad. Stud. Math.58 (2003)  Zbl1106.90001

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