Optimal transportation with capacity constraints

Jonathan Korman, Robert J. McCann
2014 Transactions of the American Mathematical Society  
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function. Here we consider a natural but largely unexplored variant of this problem by imposing a pointwise constraint on the joint (absolutely continuous) measures: among all joint densities with fixed marginals and which are dominated by a given density, find the
more » ... imal one. For this variant, we show local non-degeneracy of the cost function implies every minimizer is extremal in the convex set of competitors, hence unique. An appendix develops rudiments of a duality theory for this problem, which allows us to compute several suggestive examples.
doi:10.1090/s0002-9947-2014-06032-7 fatcat:yyrrqqmwdva6xbvmru2ygwp2l4