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A saddle-point approach to the Monge-Kantorovichoptimal transport problem

Published online by Cambridge University Press:  31 March 2010

Christian Léonard
Christian Léonard*
Affiliation:
Modal-X, Université Paris Ouest, Bât. G, 200 av. de la République, 92001 Nanterre, France. christian.leonard@u-paris10.fr
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Abstract

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

Keywords

Convex optimizationsaddle-pointconjugate dualityoptimal transport
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 682 - 704
Copyright
© EDP Sciences, SMAI, 2010

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References

L. Ambrosio and A. Pratelli, Existence and stability results in the L 1-theory of optimal transportation – CIME Course, in Lecture Notes in Mathematics 1813. Springer Verlag (2003) 123–160.
M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. (to appear).
Beiglböck, M., Goldstern, M., Maresh, G. and Schachermayer, W., Optimal and better transport plans. J. Funct. Anal. 256 (2009) 1907–1927. CrossRef
M. Beiglböck, C. Léonard and W. Schachermayer, A general duality theorem for the Monge-Kantorovich transport problem. Preprint (2009).
Borwein, J.M. and Lewis, A.S., Decomposition of multivariate functions. Can. J. Math. 44 (1992) 463–482. CrossRef
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 Applications 8. Birkhäuser (1993).
Decreusefond, L., Wasserstein distance on configuration space. Potential Anal. 28 (2008) 283–300. CrossRef
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 Mathematics 28. SIAM (1999).
Feyel, D. and Üstünel, A.S., Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theory Relat. Fields 128 (2004) 347–385. CrossRef
Léonard, C., Convex minimization problems with weak constraint qualifications. Journal of Convex Analysis 17 (2010) 312–348.
J. Neveu, Bases mathématiques du calcul des probabilités. Masson, Paris (1970).
Pratelli, A., On the sufficiency of the c-cyclical monotonicity for optimality of transport plans. Math. Z. 258 (2008) 677–690. CrossRef
S. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory, Vol. II: Applications. Springer-Verlag, New York (1998).
Rüschendorf, L., On c-optimal random variables. Statist. Probab. Lett. 27 (1996) 267–270. CrossRef
Schachermayer, W. and Teichman, J., Characterization of optimal transport plans for the Monge-Kantorovich problem. Proc. Amer. Math. Soc. 137 (2009) 519–529. CrossRef
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence (2003).
C. Villani, Optimal Transport – Old and New, Grundlehren der mathematischen Wissenschaften 338. Springer (2009).