Dual potentials for capacity constrained optimal transport

TL;DR

This paper establishes the existence of dual potentials in capacity-constrained optimal transport, characterizes primal solutions via these potentials, and provides a new elementary proof of Kantorovich duality.

Contribution

It proves the existence of dual functions in capacity-constrained optimal transport and derives Kantorovich duality from Levin's duality, offering new insights and methods.

Findings

Existence of dual potentials under mild assumptions.

Characterization of primal solutions using dual potentials.

Elementary derivation of Kantorovich duality from Levin's duality.

Abstract

Optimal transportation with capacity constraints, a variant of the well-known optimal transportation problem, is concerned with transporting one probability density $f \in L^1(\mathbb{R}^m)$ 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.