# A saddle-point approach to the Monge-Kantorovich optimal transport problem

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 3, page 682-704
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topLéonard, Christian. "A saddle-point approach to the Monge-Kantorovich optimal transport problem." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 682-704. <http://eudml.org/doc/221937>.

@article{Léonard2011,

abstract = {
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.
},

author = {Léonard, Christian},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Convex optimization; saddle-point; conjugate duality;
optimal transport; convex optimization; optimal transport},

language = {eng},

month = {8},

number = {3},

pages = {682-704},

publisher = {EDP Sciences},

title = {A saddle-point approach to the Monge-Kantorovich optimal transport problem},

url = {http://eudml.org/doc/221937},

volume = {17},

year = {2011},

}

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

DA - 2011/8//

PB - EDP Sciences

VL - 17

IS - 3

SP - 682

EP - 704

AB -
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.

LA - eng

KW - Convex optimization; saddle-point; conjugate duality;
optimal transport; convex optimization; optimal transport

UR - http://eudml.org/doc/221937

ER -

## References

top- L. Ambrosio and A. Pratelli, Existence and stability results in the L1-theory of optimal transportation – CIME Course, in Lecture Notes in Mathematics1813. Springer Verlag (2003) 123–160. Zbl1065.49026
- M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. (to appear). Zbl1228.49046
- M. Beiglböck, M. Goldstern, G. Maresh and W. Schachermayer, Optimal and better transport plans. J. Funct. Anal.256 (2009) 1907–1927. Zbl1157.49019
- M. Beiglböck, C. Léonard and W. Schachermayer, A general duality theorem for the Monge-Kantorovich transport problem. Preprint (2009). Zbl1270.49045
- J.M. Borwein and A.S. Lewis, Decomposition of multivariate functions. Can. J. Math.44 (1992) 463–482. Zbl0789.54012
- H. Brezis, Analyse fonctionnelle – Théorie et applications. Masson, Paris (1987).
- G. Dal Maso, An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and Their Applications8. Birkhäuser (1993).
- L. Decreusefond, Wasserstein distance on configuration space. Potential Anal.28 (2008) 283–300. Zbl1144.60004
- L. Decreusefond, A. Joulin and N. Savy, Upper bounds on Rubinstein distances on configuration spaces and applications. Communications on Stochastic Analysis (to appear).
- I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics28. SIAM (1999). Zbl0939.49002
- D. Feyel and A.S. Üstünel, Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theory Relat. Fields128 (2004) 347–385. Zbl1055.60052
- C. Léonard, Convex minimization problems with weak constraint qualifications. Journal of Convex Analysis17 (2010) 312–348. Zbl1193.49042
- J. Neveu, Bases mathématiques du calcul des probabilités. Masson, Paris (1970). Zbl0203.49901
- A. Pratelli, On the sufficiency of the c-cyclical monotonicity for optimality of transport plans. Math. Z.258 (2008) 677–690. Zbl1293.49110
- S. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory, Vol. II: Applications. Springer-Verlag, New York (1998). Zbl0990.60500
- L. Rüschendorf, On c-optimal random variables. Statist. Probab. Lett.27 (1996) 267–270. Zbl0847.62046
- W. Schachermayer and J. Teichman, Characterization of optimal transport plans for the Monge-Kantorovich problem. Proc. Amer. Math. Soc.137 (2009) 519–529. Zbl1165.49015
- C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics58. American Mathematical Society, Providence (2003). Zbl1106.90001
- C. Villani, Optimal Transport – Old and New, Grundlehren der mathematischen Wissenschaften338. Springer (2009).

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.