An optimal matching problem

Ivar Ekeland

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • 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 (2005): 57-71. <http://eudml.org/doc/246109>.

@article{Ekeland2005,
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},
language = {eng},
number = {1},
pages = {57-71},
publisher = {EDP-Sciences},
title = {An optimal matching problem},
url = {http://eudml.org/doc/246109},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Ekeland, Ivar
TI - An optimal matching problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
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
UR - http://eudml.org/doc/246109
ER -

References

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  1. [1] Y. Brenier, Polar factorization and monotone rearrangements of vector-valued functions. Comm. Pure App. Math. 44 (1991) 375–417. Zbl0738.46011
  2. [2] G. Carlier, Duality and existence for a class of mass transportation problems and economic applications, Adv. Math. Economics 5 (2003) 1–21. Zbl1176.90409
  3. [3] I. Ekeland and R. Temam, Convex analysis and variational problems. North-Holland Elsevier (1974) new edition, SIAM Classics in Appl. Math. (1999). Zbl0322.90046MR1727362
  4. [4] W. Gangbo and R. McCann, The geometry of optimal transportation. Acta Math. 177 (1996) 113–161. Zbl0887.49017
  5. [5] I. Ekeland, J. Heckman and L. Nesheim, Identification and estimation of hedonic models. J. Political Economy 112 (2004) 60–109. 
  6. [6] L. Kantorovitch, On the transfer of masses, Dokl. Ak. Nauk USSR 37 (1942) 7–8. Zbl0061.09705
  7. [7] S. Rachev and A. Ruschendorf, Mass transportation problems. Springer-Verlag (1998). Zbl0990.60500
  8. [8] C. Villani, Topics in mass transportation. Grad. Stud. Math. 58 (2003) Zbl1106.90001MR1964483

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