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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 \(f \in L^1(\mathbf {R}^m)\) 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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  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.


  1. Bertrand, J., Puel, M.: The optimal mass transport problem for relativistic costs. Calc. Var. PDE 46, 353–374 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C.R. Acad. Sci. Paris Sér. I Math. 305, 805–808 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Kantorovich, L.: On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.) 37, 199–201 (1942)

    MathSciNet  MATH  Google Scholar 

  4. Kellerer, H.G.: Marginalprobleme für Funktionen. Math. Ann. 154, 147–156 (1964)

  5. Kellerer, H.G.: Maßtheoretische marginalprobleme. Math. Ann. 153, 168–198 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  6. Korman, J., McCann, R.J.: Optimal transportation with capacity constraints. Trans. Am. Math. Soc. (in press)

  7. Korman, J., McCann, R.J.: Insights into capacity constrained optimal transport. Proc. Natl. Acad. Sci 110, 10064–10067 (2013)

    Article  Google Scholar 

  8. Korman, J., McCann, R.J., Seis, C: An elementary approach to linear programming duality with application to capacity constrained transport. J. Convex Anal. arXiv:1309.3022 (in press)

  9. Levin, V.L.: The problem of mass transfer in a topological space, and probability measures having given marginal measures on the product of two spaces. Sov. Math. (Doklady) 29, 638–643 (1984)

    MATH  Google Scholar 

  10. McCann, R.J., Guillen, N.: Five lectures on optimal transportation: geometry, regularity, and applications. In: Dafni, G. et al. (ed.) Analysis and geometry of metric measure spaces: lecture notes of the séminaire de Mathématiques Supérieure (SMS) Montréal 2011, pp. 145–180. American Mathematical Society, Providence (2013)

  11. Monge, G.: Mémoire sur la théorie des déblais et de remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pp. 666–704 (1781)

  12. Rachev, S.T., Rüschendorf, L.: Mass transportation problems, vol. 1. Springer, New York (1998)

    MATH  Google Scholar 

  13. Reed, M., Simon, B.: Functional analysis.In: Methods of modern mathematical physics, vol. 1, Academic Press, San Diego (1980)

  14. Villani, C.: Optimal transport. Old and new. Springer, New York (2009)

    Book  MATH  Google Scholar 

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We would like to acknowledge Robert Jerrard for fruitful discussions.

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Correspondence to Robert J. McCann.

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Communicated by L. Ambrosio.

©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.

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Korman, J., McCann, R.J. & Seis, C. Dual potentials for capacity constrained optimal transport. Calc. Var. 54, 573–584 (2015).

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Mathematics Subject Classification

  • Primary 90C46
  • Secondary 49J45