Funktsional'nyi Analiz i ego Prilozheniya
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2002, Volume 36, Issue 2, Pages 38–44 (Mi faa189)

Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem

V. L. Levin

Central Economics and Mathematics Institute, RAS

Abstract: The Monge–Kantorovich problem (MKP) with given marginals defined on closed domains $X\subset\mathbb{R}^n$, $Y\subset\mathbb{R}^m$ and a smooth cost function $c\colon X\times Y\to\mathbb{R}$ is considered. Conditions are obtained (both necessary ones and sufficient ones) for the optimality of a Monge solution generated by a smooth measure-preserving map $f\colon X\to Y$. The proofs are based on an optimality criterion for a general MKP in terms of nonemptiness of the sets $Q_0(\zeta)=\{u\in\mathbb{R}^X:u(x)-u(z)\le\zeta(x,z)$ for all $x,z\in X\}$ for special functions $\zeta$ on $X\times X$ generated by $c$ and $f$. Also, earlier results by the author are used when considering the above-mentioned nonemptiness conditions for the case of smooth $\zeta$.

Keywords: Monge–Kantorovich problem, marginal, Monge solution

DOI: https://doi.org/10.4213/faa189

Full text: PDF file (129 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2002, 36:2, 114–119

Bibliographic databases:

UDC: 517.9

Citation: V. L. Levin, “Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem”, Funktsional. Anal. i Prilozhen., 36:2 (2002), 38–44; Funct. Anal. Appl., 36:2 (2002), 114–119

Citation in format AMSBIB
\Bibitem{Lev02} \by V.~L.~Levin \paper Optimality Conditions for Smooth Monge Solutions of the Monge--Kantorovich problem \jour Funktsional. Anal. i Prilozhen. \yr 2002 \vol 36 \issue 2 \pages 38--44 \mathnet{http://mi.mathnet.ru/faa189} \crossref{https://doi.org/10.4213/faa189} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1922017} \zmath{https://zbmath.org/?q=an:1021.49029} \elib{https://elibrary.ru/item.asp?id=14146864} \transl \jour Funct. Anal. Appl. \yr 2002 \vol 36 \issue 2 \pages 114--119 \crossref{https://doi.org/10.1023/A:1015666422861} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000176341200004} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0036273401} 

• http://mi.mathnet.ru/eng/faa189
• https://doi.org/10.4213/faa189
• http://mi.mathnet.ru/eng/faa/v36/i2/p38

 SHARE:

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. 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
•  Number of views: This page: 424 Full text: 157 References: 51