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 $f \in L^1(\mathbb{R}^m)$ onto another one $g \in L^1(\mathbb{R}^n)$ so as to optimize a cost function $c \in L^1(\mathbb{R}^{m+n})$ while respecting the capacity constraints $0\le h \le \bar h\in L^\infty(\mathbb{R}^{m+n})$. 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 $L^1$-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 (arXiv:1309.3022), this amounts to a new and elementary proof of Kantorovich's duality.