Optimal transportation with capacity constraints

Korman, Jonathan; McCann, Robert

doi:10.1090/S0002-9947-2014-06032-7

Optimal transportation with capacity constraints
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by Jonathan Korman and Robert J. McCann PDF

Trans. Amer. Math. Soc. 367 (2015), 1501-1521

Abstract:

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 optimal one. For this variant, we show that 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.

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