# Dual potentials for capacity constrained optimal transport

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.

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### 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....

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

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^{1}, Pomona College

^{2}, University of Hawaii at Hilo

^{3}, University of Georgia

^{4}

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### "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→ ∞....

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...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)....

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

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...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....

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

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791 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....

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

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69 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= +∞....

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...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....

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