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This article is cited in 13 scientific papers (total in 13 papers)
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
$$
\mathscr A(c,\rho):=\inf\biggl\{\int_{X\times X}c(x,y)\mu(d(x,y))\colon\mu\in V_+(X\times X),\ (\pi_1-\pi_2)\mu=\rho\biggr\},
$$
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.
Received: 13.03.1990
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
Linking options:
https://www.mathnet.ru/eng/sm1255https://doi.org/10.1070/SM1992v071n02ABEH002136 https://www.mathnet.ru/eng/sm/v181/i12/p1694
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Abstract page: | 684 | Russian version PDF: | 191 | English version PDF: | 19 | References: | 87 | First page: | 1 |
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