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Mat. Zametki, 2015, Volume 98, Issue 5, Pages 664–683 (Mi mz10896)  

This article is cited in 14 scientific papers (total in 14 papers)

On the Monge–Kantorovich Problem with Additional Linear Constraints

D. Zaev

National Research University "Higher School of Economics" (HSE), Moscow

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.

Keywords: Monge–Kantorovich problem, optimal transportation plan, Kantorovich duality.


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English version:
Mathematical Notes, 2015, 98:5, 725–741

Bibliographic databases:

UDC: 519.2+517.98
Received: 17.06.2015

Citation: D. Zaev, “On the Monge–Kantorovich Problem with Additional Linear Constraints”, Mat. Zametki, 98:5 (2015), 664–683; Math. Notes, 98:5 (2015), 725–741

Citation in format AMSBIB
\by D.~Zaev
\paper On the Monge--Kantorovich Problem with Additional Linear Constraints
\jour Mat. Zametki
\yr 2015
\vol 98
\issue 5
\pages 664--683
\jour Math. Notes
\yr 2015
\vol 98
\issue 5
\pages 725--741

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    2. Alexander V. Kolesnikov, Danila A. Zaev, “Exchangeable optimal transportation and log-concavity”, Theory Stoch. Process., 20(36):2 (2015), 54–62  mathnet  mathscinet  zmath
    3. M. Beiglbock, M. Nutz, N. Touzi, “Complete duality for martingale optimal transport on the line”, Ann. Probab., 45:5 (2017), 3038–3074  crossref  mathscinet  zmath  isi  scopus
    4. A. V. Kolesnikov, D. A. Zaev, “Optimal transportation of processes with infinite Kantorovich distance: independence and symmetry”, Kyoto J. Math., 57:2 (2017), 293–324  crossref  mathscinet  zmath  isi  scopus
    5. Nikolay Lysenko, “Maximization of functionals depending on the terminal value and the running maximum of a martingale: a mass transport approach”, Theory Stoch. Process., 22(38):1 (2017), 30–40  mathnet  mathscinet  zmath
    6. I. Ekren, H. M. Soner, “Constrained optimal transport”, Arch. Ration. Mech. Anal., 227:3 (2018), 929–965  crossref  mathscinet  zmath  isi  scopus
    7. A. N. Doledenok, “On a Kantorovich Problem with a Density Constraint”, Math. Notes, 104:1 (2018), 39–47  mathnet  crossref  crossref  isi  elib
    8. M. Nutz, F. Stebegg, “Canonical supermartingale couplings”, Ann. Probab., 46:6 (2018), 3351–3398  crossref  mathscinet  zmath  isi  scopus
    9. C. Griessler, “$C$-cyclical monotonicity as a sufficient criterion for optimality in the multimarginal Monge-Kantorovich problem”, Proc. Amer. Math. Soc., 146:11 (2018), 4735–4740  crossref  mathscinet  zmath  isi  scopus
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