Multi-marginal maximal monotonicity and convex analysis

  • Sedi BartzEmail author
  • Heinz H. Bauschke
  • Hung M. Phan
  • Xianfu Wang
Full Length Paper Series A


Monotonicity and convex analysis arise naturally in the framework of multi-marginal optimal transport theory. However, a comprehensive multi-marginal monotonicity and convex analysis theory is still missing. To this end we study extensions of classical monotone operator theory and convex analysis into the multi-marginal setting. We characterize multi-marginal c-monotonicity in terms of classical monotonicity and firmly nonexpansive mappings. We provide Minty type, continuity and conjugacy criteria for multi-marginal maximal monotonicity. We extend the partition of the identity into a sum of firmly nonexpansive mappings and Moreau’s decomposition of the quadratic function into envelopes and proximal mappings into the multi-marginal settings. We illustrate our discussion with examples and provide applications for the determination of multi-marginal maximal monotonicity and multi-marginal conjugacy. We also point out several open questions.


c-Convexity c-Monotonicity c-Splitting set Cyclic monotonicity Kantorovich duality Maximal monotonicity Minty theorem Moreau envelope Multi-marginal Optimal transport 

Mathematics Subject Classification

Primary 47H05 26B25 Secondary 49N15 49K30 52A01 91B68 



We thank three anonymous referees for their kind and useful remarks. Sedi Bartz was partially supported by a University of Massachusetts Lowell startup grant. Heinz Bauschke and Xianfu Wang were partially supported by the Natural Sciences and Engineering Research Council of Canada. Hung Phan was partially supported by Autodesk, Inc.


  1. 1.
    Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43, 904–924 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bartz, S., Reich, S.: Abstract convex optimal antiderivatives. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 29, 435–454 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bartz, S., Reich, S.: Optimal pricing for optimal transport. Set Valued Var. Anal. 22, 467–481 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bartz, S., Bauschke, H.H., Wang, X.: The resolvent order: a unification of the orders by Zarantonello, by Loewner, and by Moreau. SIAM J. Optim. 27, 466–477 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bartz, S., Bauschke, H.H., Wang, X.: A class of multi-marginal $c$-cyclically monotone sets with explicit $c$-splitting potentials. J. Math. Anal. Appl. 461, 333–348 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)CrossRefzbMATHGoogle Scholar
  7. 7.
    Beiglböck, M., Griessler, C.: An optimality principle with applications in optimal transport. arXiv preprint arXiv:1404.7054 (2014)
  8. 8.
    Brezis, H.: Liquid crystals and energy estimates for $S^2$-valued maps. In: Theory and Applications of Liquid Crystals (Minneapolis, Minn., 1985), The IMA Volumes in Mathematics and its Applications, vol. 5, pp. 31–52. Springer (1987)Google Scholar
  9. 9.
    Carlier, G.: On a class of multidimensional optimal transportation problems. J. Convex Anal. 10, 517–529 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Carlier, G., Nazaret, B.: Optimal transportation for the determinant. ESAIM Control Optim. Calc. Var. 14, 678–698 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Di Marino, S., De Pascale, L., Colombo, M.: Multimarginal optimal transport maps for 1-dimensional repulsive costs. Can. J. Math. 67, 350–368 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Di Marino, S., Gerolin, A., Nenna, L.: Optimal transportation theory with repulsive costs, topological optimization and optimal transport. Appl. Sci. 9, 204–256 (2017)zbMATHGoogle Scholar
  13. 13.
    Gangbo, W., McCann, R.: The geometry of optimal transportation. Acta Math. 177, 113–161 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gangbo, W., Swiech, A.: Optimal maps for the multidimensional Monge-Kantorovich problem. Commun. Pure Appl. Math. 51, 23–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ghoussoub, N., Maurey, B.: Remarks on multi-marginal symmetric Monge-Kantorovich problems. Discrete Contin. Dyn. Syst. 34, 1465–1480 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ghoussoub, N., Moameni, A.: Symmetric Monge-Kantorovich problems and polar decompositions of vector fields. Geom. Funct. Anal. 24, 1129–1166 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Griessler, C.: $c$-Cyclical monotonicity as a sufficient criterion for optimality in the multi-marginal Monge-Kantorovich problem. Proc. Am. Math. Soc. 146, 4735–4740 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kellerer, H.G.: Duality theorems for marginal problems. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 67, 399–432 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kim, Y.-H., Pass, B.: A general condition for Monge solutions in the multi-marginal optimal transport problem. SIAM J. Math. Anal. 46, 1538–1550 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Knott, M., Smith, C.S.: On a generalization of cyclic monotonicity and distances among random vectors. Linear Algebra Appl. 199, 363–371 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pass, B.: On the local structure of optimal measures in the multi-marginal optimal transportation problem. Calc. Var. Partial. Differ. Equ. 43, 529–536 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pass, B.: Multi-marginal optimal transport: theory and applications. ESAIM Math. Model. Numer. Anal. 49, 1771–1790 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rochet, J.-C.: A necessary and sufficient condition for rationalizability in a quasilinear context. J. Math. Econ. 16, 191–200 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Characterization of the subdifferentials of convex functions. Pac. J. Math. 17, 497–510 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  26. 26.
    Rüschendorf, L.: On $c$-optimal random variables. Stat. Probab. Lett. 27, 267–270 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rüschendorf, L., Uckelmann, L.: Distributions with given marginals and moment problems. In: On Optimal Multivariate Couplings, pp. 261–273. Springer (1997)Google Scholar
  28. 28.
    Santambrogio, F.: Optimal Transport for Applied Mathematicians. Birkhäuser, New York (2015)CrossRefzbMATHGoogle Scholar
  29. 29.
    Soloviov, V.: Duality for nonconvex optimization and its applications. Anal. Math. 19, 297–315 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Villani, C.: Optimal Transport: Old and New. Springer, New York (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.MathematicsUniversity of Massachusetts LowellLowellUSA
  2. 2.MathematicsUniversity of British ColumbiaKelownaCanada

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