A saddle-point approach to the Monge-Kantorovich optimal transport problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 682-704.

The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.

DOI : 10.1051/cocv/2010013
Classification : 46N10, 49J45, 28A35
Mots-clés : convex optimization, saddle-point, conjugate duality, optimal transport 
@article{COCV_2011__17_3_682_0,
     author = {L\'eonard, Christian},
     title = {A saddle-point approach to the {Monge-Kantorovich} optimal transport problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {682--704},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {3},
     year = {2011},
     doi = {10.1051/cocv/2010013},
     mrnumber = {2826975},
     zbl = {1234.46058},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2010013/}
}
TY  - JOUR
AU  - Léonard, Christian
TI  - A saddle-point approach to the Monge-Kantorovich optimal transport problem
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2011
SP  - 682
EP  - 704
VL  - 17
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2010013/
DO  - 10.1051/cocv/2010013
LA  - en
ID  - COCV_2011__17_3_682_0
ER  - 
%0 Journal Article
%A Léonard, Christian
%T A saddle-point approach to the Monge-Kantorovich optimal transport problem
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 682-704
%V 17
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2010013/
%R 10.1051/cocv/2010013
%G en
%F COCV_2011__17_3_682_0
Léonard, Christian. A saddle-point approach to the Monge-Kantorovich optimal transport problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 682-704. doi : 10.1051/cocv/2010013. http://www.numdam.org/articles/10.1051/cocv/2010013/

[1] L. Ambrosio and A. Pratelli, Existence and stability results in the L1-theory of optimal transportation - CIME Course, in Lecture Notes in Mathematics 1813. Springer Verlag (2003) 123-160. | MR | Zbl

[2] M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. (to appear). | MR | Zbl

[3] M. Beiglböck, M. Goldstern, G. Maresh and W. Schachermayer, Optimal and better transport plans. J. Funct. Anal. 256 (2009) 1907-1927. | MR | Zbl

[4] M. Beiglböck, C. Léonard and W. Schachermayer, A general duality theorem for the Monge-Kantorovich transport problem. Preprint (2009). | Zbl

[5] J.M. Borwein and A.S. Lewis, Decomposition of multivariate functions. Can. J. Math. 44 (1992) 463-482. | MR | Zbl

[6] H. Brezis, Analyse fonctionnelle - Théorie et applications. Masson, Paris (1987). | MR | Zbl

[7] G. Dal Maso, An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and Their Applications 8. Birkhäuser (1993). | MR | Zbl

[8] L. Decreusefond, Wasserstein distance on configuration space. Potential Anal. 28 (2008) 283-300. | MR | Zbl

[9] L. Decreusefond, A. Joulin and N. Savy, Upper bounds on Rubinstein distances on configuration spaces and applications. Communications on Stochastic Analysis (to appear). | MR

[10] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28. SIAM (1999). | MR | Zbl

[11] D. Feyel and A.S. Üstünel, Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theory Relat. Fields 128 (2004) 347-385. | MR | Zbl

[12] C. Léonard, Convex minimization problems with weak constraint qualifications. Journal of Convex Analysis 17 (2010) 312-348. | MR | Zbl

[13] J. Neveu, Bases mathématiques du calcul des probabilités. Masson, Paris (1970). | Zbl

[14] A. Pratelli, On the sufficiency of the c-cyclical monotonicity for optimality of transport plans. Math. Z. 258 (2008) 677-690. | MR | Zbl

[15] S. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory, Vol. II: Applications. Springer-Verlag, New York (1998). | MR | Zbl

[16] L. Rüschendorf, On c-optimal random variables. Statist. Probab. Lett. 27 (1996) 267-270. | MR | Zbl

[17] W. Schachermayer and J. Teichman, Characterization of optimal transport plans for the Monge-Kantorovich problem. Proc. Amer. Math. Soc. 137 (2009) 519-529. | MR | Zbl

[18] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence (2003). | MR | Zbl

[19] C. Villani, Optimal Transport - Old and New, Grundlehren der mathematischen Wissenschaften 338. Springer (2009). | MR | Zbl

Cité par Sources :