Abstract
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.
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Original Russian Text © D. A. Zaev, 2015, published in Matematicheskie Zametki, 2015, Vol. 98, No. 5, pp. 664–683.
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Zaev, D.A. On the Monge–Kantorovich problem with additional linear constraints. Math Notes 98, 725–741 (2015). https://doi.org/10.1134/S0001434615110036
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DOI: https://doi.org/10.1134/S0001434615110036