Geodesic
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

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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
Keywords: convex optimization, saddle-point, conjugate duality, optimal transport 
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     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://geodesic.mathdoc.fr/articles/10.1051/cocv/2010013/}
}
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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

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