Geodesic
On the Monge--Kantorovich Problem with Additional Linear Constraints
Matematičeskie zametki, Tome 98 (2015) no. 5, pp. 664-683

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The Monge–Kantorovich problem with the following additional constraint is considered: the admissible transportation plan must become zero on a fixed subspace of functions. Different subspaces give rise to different additional conditions on transportation plans. The main results are stated in general form and can be carried over to a number of important special cases. They are also valid for the Monge–Kantorovich problem whose solution is sought for the class of invariant or martingale measures. We formulate and prove a criterion for the existence of an optimal solution, a duality assertion of Kantorovich type, and a necessary geometric condition on the support of the optimal measure similar to the standard condition for $c$-monotonicity.
Keywords: Monge–Kantorovich problem, Kantorovich duality.
Mots-clés : optimal transportation plan 
@article{MZM_2015_98_5_a2,
     author = {D. Zaev},
     title = {On the {Monge--Kantorovich} {Problem} with {Additional} {Linear} {Constraints}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {664--683},
     publisher = {mathdoc},
     volume = {98},
     number = {5},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_98_5_a2/}
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D. Zaev. On the Monge--Kantorovich Problem with Additional Linear Constraints. Matematičeskie zametki, Tome 98 (2015) no. 5, pp. 664-683. http://geodesic.mathdoc.fr/item/MZM_2015_98_5_a2/