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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Notes

  1. 1.

    In a private communication, Rachev and Rüschendorf attribute Theorem 4.6.14 of [12] to a handwritten manuscript of Levin which, so far as we know, has not been published.

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Acknowledgments

We would like to acknowledge Robert Jerrard for fruitful discussions.

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Correspondence to Robert J. McCann.

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©2014 by the authors. RJM is pleased to acknowledge the support of Natural Sciences and Engineering Research Council of Canada Grants 217006-08. This material is based in part upon work supported by the National Science Foundation under Grant No. 0932078 000, while the first two authors were in residence at the Mathematical Science Research Institute in Berkeley, California, during the Fall of 2013.

Communicated by L. Ambrosio.

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Korman, J., McCann, R.J. & Seis, C. Dual potentials for capacity constrained optimal transport. Calc. Var. 54, 573–584 (2015). https://doi.org/10.1007/s00526-014-0795-9

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Mathematics Subject Classification

  • Primary 90C46
  • Secondary 49J45