2013/07/30 by Jonathan Korman, Robert J. McCann, Korman, Jonathan +3 · 1 citation
Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #math.OC
paper · pdf · doi:10.48550/arxiv.1307.7774
Revised version
arxiv created 2014/03/03 · arxiv updated 2014/03/05
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 ≤ h∈ 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 (arXiv:1309.3022), this amounts to a new and elementary proof of Kantorovich's duality.