Abstract
We study the existence of solutions to the Kantorovich optimal transportation problem with a nonlinear cost functional generated by a cost function depending on the transport plan. We also consider the case of a cost function depending on the conditional measures of the transport plan. Broad sufficient conditions are obtained for the existence of optimal plans for Radon marginal distributions on completely regular spaces and a lower semicontinuous cost function.
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Funding
This work was supported by the Russian Foundation for Basic Research under grant 20-01-00432, the project “Kantorovich’s parametric problem of optimal transportation”, with the support of PSTGU and the “Living Tradition” foundation, and also supported by the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No. 075-15-2022-284 (results in Sec. 2).
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Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 360–370 https://doi.org/10.4213/mzm13545.
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Bogachev, V.I., Rezbaev, A.V. Existence of Solutions to the Nonlinear Kantorovich Transportation Problem. Math Notes 112, 369–377 (2022). https://doi.org/10.1134/S0001434622090048
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DOI: https://doi.org/10.1134/S0001434622090048