This article is cited in
14 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.
UDC:
517.9
MSC: Primary
46N05,
90C08; Secondary
28B20,
54C60 Received: 13.03.1990
Fulltext:
PDF file (1635 kB)
