# Dual potentials for capacity constrained optimal transport

##### Citations

8 citations

8 citations

### Cites background or result from "Dual potentials for capacity constr..."

...Combining the techniques presented in the following with some of the results derived by the authors in a companion paper [7], we also provide a new elementary proof of Kantorovich’s duality....

[...]

...In [7], the authors prove (under some additional assumptions) that the supremum on the right is attained by triple of functions....

[...]

3 citations

^{1}, Pomona College

^{2}, University of Hawaii at Hilo

^{3}, University of Georgia

^{4}

2 citations

2 citations

##### References

5,524 citations

1,046 citations

### "Dual potentials for capacity constr..." refers background in this paper

...As a consequence of our analysis, we are also able to deduce Kantorovich’s duality theorem for the unconstrained optimal transportation problem as a singular limit h̄→ ∞....

[...]

...In the optimal transport problem of Monge [10] and Kantorovich [2], one is given distributions f(x) of sources and g(y) of sinks, and is asked which pairing h(x, y) ≥ 0 of sources with sinks minimizes a given transportation cost c(x, y)....

[...]

...The duality theory initiated by Kantorovich provides a key tool for the analysis of this question....

[...]

...The remainder of this paper is organized as follows: In Section 2, we derive weak duality and complementary slackness conditions; Section 3 contains the key (coercivity) estimates; Section 4 contains the main result; in Section 5, we give a new elementary proof of the Kantorovich duality; we finally conclude this paper with a discussion on future work in Section 6....

[...]

...In tandem with the results obtained in the companion paper [7], this amounts to a new and elementary proof of the Kantorovich duality theorem....

[...]

823 citations

### "Dual potentials for capacity constr..." refers background in this paper

...14 of [11], for which a new proof is given in [7]), it has not been clear until now whether the dual problem admits solutions....

[...]

...14 of [11] to a handwritten manuscript of Levin....

[...]

83 citations

### "Dual potentials for capacity constr..." refers methods in this paper

...However, the compactification techniques used to find them in the unconstrained problem [9] [13] fail miserably when h̄ 6= +∞....

[...]

...As in the theory for the unconstrained Monge-Kantorovich problem [9] [13] which has developed since the work of Brenier [1], we expect our characterization of primal optimizers using dual solutions will be the starting point for any future analysis of their analytic or geometric properties....

[...]