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].
This work presents an improvement of the approximation scheme for the Monge-Kantorovich (MK) mass transfer problem on compact spaces, which is studied by Gabriel et al. (2010), whose scheme discretizes the MK problem, reduced to solve a sequence of finite transport problems. The improvement presented in this work uses a metaheuristic algorithm inspired by scatter search in order to reduce the dimensionality of each transport problem. The new scheme solves a sequence of linear programming problems similar to the transport ones but with a lower dimension. The proposed metaheuristic is supported by a convergence theorem. Finally, examples with an exact solution are used to illustrate the performance of our proposal.
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