Dual potentials for capacity constrained optimal transport

Optimal transportation with capacity constraints, a variant of the well-known optimal transportation problem, is concerned with transporting one probability density f ∈ L1(Rm) onto another one g ∈ L1(Rn) so as to optimize a cost function c ∈ L1(Rm+n) while respecting the capacity constraints 0 ≤ h ≤ h̄ ∈ L∞(Rm+n). A linear programming duality theorem for this problem was first established by Levin. In this note, we prove under mild assumptions on the given data, the existence of a pair of L1-functions optimizing the dual problem. Using these functions, which can be viewed as Lagrange multipliers to the marginal constraints f and g, we characterize the solution h of the primal problem. We expect these potentials to play a key role in any further analysis of h. Moreover, starting from Levin’s duality, we derive the classical Kantorovich duality for unconstrained optimal transport. In tandem with results obtained in our companion paper [7], this amounts to a new and elementary proof of Kantorovich’s duality. ∗ c ©2014 by the authors. RJM is pleased to acknowledge the support of Natural Sciences and Engineering Research Council of Canada Grants 217006-08. This material is based in part upon work supported by the National Science Foundation under Grant No. 0932078 000, while the first two authors were in residence at the Mathematical Science Research Institute in Berkeley, California, during the Fall of 2013. Department of Mathematics, University of Toronto, Toronto Ontario M5S 2E4 Canada. [email protected], [email protected], [email protected]