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 Mat. Sb., 1990, Volume 181, Number 12, Pages 1694–1709 (Mi msb1255)

A formula for the optimal value in the Monge–Kantorovich problem with a smooth cost function, and a characterization of cyclically monotone mappings

V. L. Levin

Central Economics and Mathematics Institute, USSR Academy of Sciences

Abstract: The general Monge–Kantorovich problem consists in the computation of the optimal value
$$\mathscr A(c,\rho):=\inf\{\int_{X\times X}c(x,y)\mu(d(x,y))\colon\mu\in V_+(X\times X), (\pi_1-\pi_2)\mu=\rho\},$$
where the cost function $c\colon X\times X\to \mathbf R^1$ and the measure $\rho$ on $X$ with $\rho X=0$ are assumed to be given, $V_+(X\times X)$ is the cone of finite positive Borel measures on $X\times X$, and $\pi_1$ and $\pi_2$ are the projections on the first and second coordinates, which assign to a measure $\mu$ the corresponding marginal measures.
An explicit formula is obtained for $\mathscr A(c,\rho)$ in the case when $X$ is a domain in $\mathbf R^n$ and $c$ is bounded, vanishes on the diagonal, and is continuously differentiable in a neighborhood of the diagonal.
Conditions for the set
$$Q_0(c):=\{u\colon X\to\mathbf R^1:u(x)-u(y)\leqslant c(x,y) \forall x,y\in X\}$$
to be nonempty are investigated, and with their help new characterizations of cyclically monotone mappings are obtained.

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English version:
Mathematics of the USSR-Sbornik, 1992, 71:2, 533–548

Bibliographic databases:

UDC: 517.9
MSC: Primary 46N05, 90C08; Secondary 28B20, 54C60

Citation: V. L. Levin, “A formula for the optimal value in the Monge–Kantorovich problem with a smooth cost function, and a characterization of cyclically monotone mappings”, Mat. Sb., 181:12 (1990), 1694–1709; Math. USSR-Sb., 71:2 (1992), 533–548

Citation in format AMSBIB
\Bibitem{Lev90} \by V.~L.~Levin \paper A~formula for the optimal value in the Monge--Kantorovich problem with a~smooth cost function, and a~characterization of cyclically monotone mappings \jour Mat. Sb. \yr 1990 \vol 181 \issue 12 \pages 1694--1709 \mathnet{http://mi.mathnet.ru/msb1255} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1099522} \zmath{https://zbmath.org/?q=an:0776.90086} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..71..533L} \transl \jour Math. USSR-Sb. \yr 1992 \vol 71 \issue 2 \pages 533--548 \crossref{https://doi.org/10.1070/SM1992v071n02ABEH002136} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1992HU58600017} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Levin V., “Duality Theorems for a Nontopological Version of the MASS Transfer Problem”, Dokl. Akad. Nauk, 350:5 (1996), 588–591
2. Levin V., “A Superlinear Multifunction Arising in Connection with MASS Transfer Problems”, Set-Valued Anal., 4:1 (1996), 41–65
3. V. L. Levin, “On duality theory for non-topological variants of the mass transfer problem”, Sb. Math., 188:4 (1997), 571–602
4. Vladimir L. Levin, “Reduced cost functions and their applications”, Journal of Mathematical Economics, 28:2 (1997), 155
5. V. L. Levin, “Existence and Uniqueness of a Measure-Preserving Optimal Mapping in a General Monge–Kantorovich Problem”, Funct. Anal. Appl., 32:3 (1998), 205–208
6. V. L. Levin, “Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem”, Funct. Anal. Appl., 36:2 (2002), 114–119
7. Levin, VL, “Solving the Monge and Monge-Kantorovich problems: Theory and examples”, Doklady Mathematics, 67:1 (2003), 1
8. V. L. Levin, “Optimality conditions and exact solutions to the two-dimensional Monge–Kantorovich problem”, J. Math. Sci. (N. Y.), 133:4 (2006), 1456–1463
9. Levin, VL, “A method in mathematical demand theory connected with the Monge-Kantorovich duality”, Doklady Mathematics, 70:2 (2004), 770
10. V. L. Levin, “Best approximation problems relating to Monge–Kantorovich duality”, Sb. Math., 197:9 (2006), 1353–1364
11. Levin, VL, “Smooth feasible solutions to a dual Monge-Kantorovich problem and their application to the best approximation and mathematical economics problems”, Doklady Mathematics, 77:2 (2008), 281
12. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890
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