An approximation scheme for the Kantorovich-Rubinstein problem on compact spaces

Abstract

This paper presents an approximation scheme for the Kantorovich-Rubinstein mass transshipment (KR) problem on compact spaces. A sequence of finite-dimensional linear programs, minimal cost network flow problems with bounds, are introduced and it is proven that the limit of the sequence of the optimal values of these problems is the optimal value of the KR problem. Numerical results are presented approximating the Kantorovich metric between distributions on [0,1].

Hausdorff distances between couplings and optimal transportation

“…The rate of approximation of µ and ν by discrete measures is, of course, a separate question (see [28]). If h is not Lipschitz, then a similar estimate can be obtained in terms of the modulus of continuity of h. Our result can be used in approximation schemes discussed in [6]. Finally, the case of noncompact spaces reduces in principle to the one we have considered, provided we have some information about compact sets on which marginals are ε-concentrated.…”

“…Kantorovich problems depending on a parameter were investigated in several papers; see [13]- [15], [18], [23], [30], [32], [43] and [46], where the questions of measurability were addressed. The continuity properties are also of great interest and fundamental importance; in particular, they can be useful in the study of differential equations and inclusions on spaces of measures (see [21]), in the regularization of optimal transportation (see [22] and [31]), in constructing approximations by discrete transportation problems (see [6]; a result of this kind is presented below), and in other applications (see, for instance, [1], [10], [41] and [45]). As already mentioned, the continuous dependence on marginals was considered in [9], [40] and [26].…”

Convergence Theorems for Varying Measures Under Convexity Conditions and Applications

“…We have the following Theorem 2.6. Let f n : Ω → [0, +∞], n ∈ N be a sequence of lower semicontinuous functions satisfying (2) and let m and m n , n ∈ N, belong to M + (Ω).Then there exists a measurable function…”

“…Convergence results for varying measures have significant applications to various fields of pure and applied sciences including stochastic processes, statistics, control and game theories, transportation problems, neural networks, signal and image processing (see, for example, [2,[5][6][7][8]15,20,24,28,34]). E.T.…”

A note on convergence results for varying interval valued multisubmeasures

“…Obviously, the interval valued multifunctions G n are discontinuous (see, e.g., [1,2,4,5]) hence a notion of convergence under a suitable definition of integrals of interval valued multifunctions is needed. Finally the convergence of measures is also used for example in [3] in order to obtain an approximation scheme for the Kantorovich-Rubinstein transportation problem.…”