A saddle-point approach to the Monge-Kantorovich optimal transport problem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 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 : https://doi.org/10.1051/cocv/2010013
Classification:  46N10,  49J45,  28A35
Keywords: 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},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
pages = {682-704},
doi = {10.1051/cocv/2010013},
zbl = {1234.46058},
mrnumber = {2826975},
language = {en},
url = {http://www.numdam.org/item/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, Volume 17 (2011) no. 3, pp. 682-704. doi : 10.1051/cocv/2010013. http://www.numdam.org/item/COCV_2011__17_3_682_0/

[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 2006307 | Zbl 1065.49026

[2] M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. (to appear). | MR 2792985 | Zbl 1228.49046

[3] M. Beiglböck, M. Goldstern, G. Maresh and W. Schachermayer, Optimal and better transport plans. J. Funct. Anal. 256 (2009) 1907-1927. | MR 2498564 | Zbl 1157.49019

[4] M. Beiglböck, C. Léonard and W. Schachermayer, A general duality theorem for the Monge-Kantorovich transport problem. Preprint (2009). | Zbl 1270.49045

[5] J.M. Borwein and A.S. Lewis, Decomposition of multivariate functions. Can. J. Math. 44 (1992) 463-482. | MR 1176365 | Zbl 0789.54012

[6] H. Brezis, Analyse fonctionnelle - Théorie et applications. Masson, Paris (1987). | MR 697382 | Zbl 1147.46300

[7] G. Dal Maso, An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and Their Applications 8. Birkhäuser (1993). | MR 1201152 | Zbl 0816.49001

[8] L. Decreusefond, Wasserstein distance on configuration space. Potential Anal. 28 (2008) 283-300. | MR 2386101 | Zbl 1144.60004

[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 2677197

[10] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28. SIAM (1999). | MR 1727362 | Zbl 0939.49002

[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 2036490 | Zbl 1055.60052

[12] C. Léonard, Convex minimization problems with weak constraint qualifications. Journal of Convex Analysis 17 (2010) 312-348. | MR 2642734 | Zbl 1193.49042

[13] J. Neveu, Bases mathématiques du calcul des probabilités. Masson, Paris (1970). | Zbl 0203.49901 | Zbl 0137.11203

[14] A. Pratelli, On the sufficiency of the c-cyclical monotonicity for optimality of transport plans. Math. Z. 258 (2008) 677-690. | MR 2369050 | Zbl 1293.49110

[15] S. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory, Vol. II: Applications. Springer-Verlag, New York (1998). | MR 1619170 | Zbl 0990.60500

[16] L. Rüschendorf, On c-optimal random variables. Statist. Probab. Lett. 27 (1996) 267-270. | MR 1395577 | Zbl 0847.62046

[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 2448572 | Zbl 1165.49015

[18] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence (2003). | MR 1964483 | Zbl 1106.90001

[19] C. Villani, Optimal Transport - Old and New, Grundlehren der mathematischen Wissenschaften 338. Springer (2009). | MR 2459454 | Zbl 1156.53003