Comptes Rendus
no. 12
Mathematical analysis/Partial differential equations
Multi-marginal Monge–Kantorovich transport problems: A characterization of solutions
[Problèmes de transport multi-marginal de Monge–Kantorovich : Une caractérisation des solutions]

Présenté par : the Editorial Board

Abbas Moameni1
1 School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada
Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 993-998.

Dans cet article, nous étudions le problème de transport optimal du point de vue de la théorie de la mesure, à l'aide de la dualité de Kantorovich. En particulier, nous étudions le support des plans optimaux où la fonction coût ne satisfait pas la condition de « twist » dans le problème à deux marginales, ainsi que dans le cas multi-marginales quand la condition « twist » est limitée à des sous-ensembles précis.

We shall present a measure theoretical approach that, together with the Kantorovich duality, provides an efficient tool to study the optimal transport problem. Specifically, we study the support of optimal plans where the cost function does not satisfy the classical twist condition in the two marginal problem as well as in the multi-marginal case when twistedness is limited to certain subsets.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.10.004

Abbas Moameni 1

1 School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada 
@article{CRMATH_2014__352_12_993_0,
     author = {Abbas Moameni},
     title = {Multi-marginal {Monge{\textendash}Kantorovich} transport problems: {A} characterization of solutions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {993--998},
     publisher = {Elsevier},
     volume = {352},
     number = {12},
     year = {2014},
     doi = {10.1016/j.crma.2014.10.004},
     language = {en},
}
TY  - JOUR
AU  - Abbas Moameni
TI  - Multi-marginal Monge–Kantorovich transport problems: A characterization of solutions
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 993
EP  - 998
VL  - 352
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2014.10.004
LA  - en
ID  - CRMATH_2014__352_12_993_0
ER  - 
%0 Journal Article
%A Abbas Moameni
%T Multi-marginal Monge–Kantorovich transport problems: A characterization of solutions
%J Comptes Rendus. Mathématique
%D 2014
%P 993-998
%V 352
%N 12
%I Elsevier
%R 10.1016/j.crma.2014.10.004
%G en
%F CRMATH_2014__352_12_993_0
Abbas Moameni. Multi-marginal Monge–Kantorovich transport problems: A characterization of solutions. Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 993-998. doi : 10.1016/j.crma.2014.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.10.004/

[1] K.P.S. Bhaskara Rao; M. Bhaskara Rao A remark on nonatomic measures, Ann. Math. Stat., Volume 43 (1972) no. 1, pp. 369-370

[2] V.I. Bogachev Measure Theory, vols. I, II, Springer-Verlag, Berlin, 2007

[3] G. Buttazzo; L. De Pascale; P. Gori-Giorgi Optimal transport formulation of electronic density-functional theory, Phys. Rev. A, Volume 85 (Jun 2012), p. 062502

[4] G. Carlier On a class of multidimensional optimal transportation problems, J. Convex Anal., Volume 10 (2003) no. 2, pp. 517-529

[5] G. Carlier; B. Nazaret Optimal transportation for the determinant, ESAIM Control Optim. Calc. Var., Volume 14 (2008) no. 4, pp. 678-698

[6] P.-A. Chiappori; R.J. McCann; L.P. Nesheim Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness, Econom. Theory, Volume 42 (2010), pp. 317-354

[7] W. Gangbo; A. Swiech Optimal maps for the multidimensional mongekantorovich problem, Commun. Pure Appl. Math., Volume 51 (1998) no. 1, pp. 23-45

[8] N. Ghoussoub; A. Moameni A self-dual polar factorization for vector fields, Commun. Pure Appl. Math., Volume 66 (2013) no. 6, pp. 905-933

[9] S. Graf Induced σ-homomorphisms and a parametrization of measurable sections via extremal preimage measures, Math. Ann., Volume 247 (1980) no. 1, pp. 67-80

[10] H. Heinich Probleme de Monge pour n probabilité, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002) no. 9, pp. 793-795

[11] Y.-H. Kim; B. Pass A general condition for Monge solutions in the multi-marginal optimal transport problem | arXiv

[12] V. Levin Abstract cyclical monotonicity and Monge solutions for the general Monge–Kantorovich problem, Set-Valued Anal., Volume 7 (1999) no. 1, pp. 7-32

[13] A. Moameni, A characterization for solutions of the Monge–Kantorovich mass transport problem, submitted for publication.

[14] A. Moameni, B. Pass, Solutions to multi-marginal optimal transport problems supported on several graphs, in preparation.

[15] B. Pass Uniqueness and Monge solutions in the multimarginal optimal transportation problem, SIAM J. Math. Anal., Volume 43 (2011) no. 6, pp. 2758-2775

[16] B. Pass On the local structure of optimal measures in the multimarginal optimal transportation problem, Calc. Var. Partial Differ. Equ., Volume 43 (2012), pp. 529-536

[17] C. Villani Optimal Transport, Old and New, Grundlehren Math. Wiss., Springer-Verlag, Berlin, 2009

[18] H. von Weizsäcker; G. Winkler Integral representation in the set of solutions of a generalized moment problem, Math. Ann., Volume 246 (1979/1980) no. 1, pp. 23-32

Cité par Sources :

Commentaires - Politique