Numerical enclosures of the optimal cost of the Kantorovitch's mass transportation problem

The problem of optimal transportation was formalized by the French mathematician Gaspard Monge in 1781. Since Kantorovitch, this (generalized) problem is formulated with measure theory. Based on Interval Arithmetic, we propose a guaranteed discretization of the Kantorovitch’s mass transportation problem. Our discretization is spatial: supports of the two mass densities are partitioned into finite families. The problem is relaxed to a finite dimensional linear programming problem whose optimum is a lower bound to the optimum of the initial one. Based on Kantorovitch duality and Interval Arithmetic, a method to obtain an upper bound to the optimum is also provided. Preliminary results show that good approximations are obtained.