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.
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] 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
and ,[2] Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. (to appear). | MR | Zbl
and ,[3] Optimal and better transport plans. J. Funct. Anal. 256 (2009) 1907-1927. | MR | Zbl
, , and ,[4] A general duality theorem for the Monge-Kantorovich transport problem. Preprint (2009). | Zbl
, and ,[5] Decomposition of multivariate functions. Can. J. Math. 44 (1992) 463-482. | MR | Zbl
and ,[6] Analyse fonctionnelle - Théorie et applications. Masson, Paris (1987). | MR | Zbl
,[7] An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and Their Applications 8. Birkhäuser (1993). | MR | Zbl
,[8] Wasserstein distance on configuration space. Potential Anal. 28 (2008) 283-300. | MR | Zbl
,[9] Upper bounds on Rubinstein distances on configuration spaces and applications. Communications on Stochastic Analysis (to appear). | MR
, and ,[10] Convex Analysis and Variational Problems, Classics in Applied Mathematics 28. SIAM (1999). | MR | Zbl
and ,[11] Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theory Relat. Fields 128 (2004) 347-385. | MR | Zbl
and ,[12] Convex minimization problems with weak constraint qualifications. Journal of Convex Analysis 17 (2010) 312-348. | MR | Zbl
,[13] Bases mathématiques du calcul des probabilités. Masson, Paris (1970). | Zbl
,[14] On the sufficiency of the c-cyclical monotonicity for optimality of transport plans. Math. Z. 258 (2008) 677-690. | MR | Zbl
,[15] Mass Transportation Problems. Vol. I: Theory, Vol. II: Applications. Springer-Verlag, New York (1998). | MR | Zbl
and ,[16] On c-optimal random variables. Statist. Probab. Lett. 27 (1996) 267-270. | MR | Zbl
,[17] Characterization of optimal transport plans for the Monge-Kantorovich problem. Proc. Amer. Math. Soc. 137 (2009) 519-529. | MR | Zbl
and ,[18] Topics in Optimal Transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence (2003). | MR | Zbl
,[19] Optimal Transport - Old and New, Grundlehren der mathematischen Wissenschaften 338. Springer (2009). | MR | Zbl
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