Dual potentials for capacity constrained optimal transport

TL;DR: In this article, the existence of a pair of Lagrange multipliers for optimal transport with capacity constraints was proved under mild assumptions on the given data, which can be used to characterize the solution of the primal problem.

Abstract: Optimal transportation with capacity constraints, a variant of the well-known optimal transportation problem, is concerned with transporting one probability density $$f \in L^1(\mathbf {R}^m)$$ onto another one $$g \in L^1(\mathbf {R}^n)$$ so as to optimize a cost function $$c \in L^1(\mathbf {R}^{m+n})$$ while respecting the capacity constraints $$0\le h \le \bar{h}\in L^\infty (\mathbf {R}^{m+n})$$ . A linear programming duality for this problem was first proposed by Levin. In this note, we prove under mild assumptions on the given data, the existence of a pair of $$L^1$$ -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 [Korman et al. J Convex Anal arXiv:1309.3022 [8] (in press)], this amounts to a new and elementary proof of Kantorovich’s duality.