Canonical Duality Method for Solving Kantorovich Mass Transfer Problem

  • Xiaojun Lu
  • David Yang Gao
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 37)


This paper addresses analytical solution to the Kantorovich mass transfer problem . Through an ingenious approximation mechanism, the Kantorovich problem is first reformulated as a variational form, which is equivalent to a nonlinear differential equation with Dirichlet boundary. The existence and uniqueness of the solution can be demonstrated by applying the canonical duality theory. Then, using the canonical dual transformation, a perfect dual maximization problem is obtained, which leads to an analytical solution to the primal problem . Its global extremality for both primal and dual problems can be identified by a triality theory. In addition, numerical maximizers for the Kantorovich problem are provided under different circumstances. Finally, the theoretical results are verified by applications to Monge’s problem. Although the problem is addressed in one-dimensional space, the theory and method can be generalized to solve high-dimensional problems.



The main results in this paper were obtained during a research collaboration in the Federation University Australia in August, 2015. The first author wishes to thank Professor David Gao for his hospitality and financial support. This project is partially supported by US Air Force Office of Scientific Research (AFOSR FA9550-10-1-0487 and FA9550-17-1-0151). This project is also supported by Jiangsu Planned Projects for Postdoctoral Research Funds (1601157B), Shanghai University Start-up Grant for Shanghai 1000-Talent Program Scholars, National Natural Science Foundation of China (NSFC 61673104, 71673043, 71273048, 71473036, 11471072), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Fundamental Research Funds for the Central Universities (2014B15214, 2242017K40086), Open Research Fund Program of Jiangsu Key Laboratory of Engineering Mechanics, Southeast University (LEM16B06). In particular, the authors also express their deep gratitude to the referees for their careful reading and useful remarks.


  1. 1.
    Ambrosio, L.: Lecture Notes on Optimal Transfer Problems, preprintGoogle Scholar
  2. 2.
    Ambrosio, L.: Optimal transport maps in Monge-Kantorovich problem. ICM 3, 1–3 (2002)zbMATHGoogle Scholar
  3. 3.
    Bourgain, J., Brezis, H.: Sur l’équation div \(u=f\). C. R. Acad. Sci. Paris, Ser. I334, 973–976 (2002)Google Scholar
  4. 4.
    Caffarelli, L.A.: Allocation maps with general cost functions, in partial differential equations and applications. Lect. Notes Pure Appl. Math. 177, 29–35 (1996)zbMATHGoogle Scholar
  5. 5.
    Caffarelli, L.A., Feldman, M., MCcann, R.J.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. AMS 15, 1–26 (2001)Google Scholar
  6. 6.
    Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré 7, 1–26 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Dunod, Paris (1976)zbMATHGoogle Scholar
  8. 8.
    Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc. 653 (1999)Google Scholar
  9. 9.
    Evans, L.C.: Partial differential equations and Monge-Kantorovich mass transfer(survey paper)Google Scholar
  10. 10.
    Evans, L.C.: Three singular variational problems, preprint (2002)Google Scholar
  11. 11.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (2002)Google Scholar
  12. 12.
    Gangbo, W., McCann, R.J.: Optimal maps in Monge’s transport problem, preprint (1995)Google Scholar
  13. 13.
    Gangbo, W., McCann, R.J.: Optimal maps in Monge’s mass transport problem. C. R. Acad. Sci. Paris Sér. I Math. 321, 1653–1658 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177, 113–161 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gangbo, W., Świȩch, A.: Optimal maps for the multidimensional Monge-Kantorovich problem, preprint (1996)Google Scholar
  16. 16.
    Gao, D.Y.: Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications. IMA J. Appl. Math. 61, 199–235 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory. Methods and Applications. Kluwer Academic Publishers, Boston (2000)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gao, D.Y.: Analytic solution and triality theory for nonconvex and nonsmooth variational problems with applications. Nonlinear Anal. 42(7), 1161–1193 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gao, D.Y., Lu, X.: Multiple solutions for non-convex variational boundary value problems in higher dimensions, preprint (2013)Google Scholar
  20. 20.
    Gao, D.Y., Ogden, R.W.: Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation. Q. Jl Mech. Appl. Math. 61(4), 497–522 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gao, D.Y., Strang, G.: Geometric nonlinearity: potential energy, complementary energy, and the gap function. Q. Appl. Math. 47(3), 487–504 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gao, D.Y., Ogden, R.W., Stravroulakis, G.: Nonsmooth and Nonconvex Mechanics: Modelling. Analysis and Numerical Methods. Kluwer Academic Publishers, Boston (2001)CrossRefGoogle Scholar
  23. 23.
    Kantorovich, L.V.: On the transfer of masses. Dokl. Akad. Nauk. SSSR 37, 227–229 (1942). (Russian)Google Scholar
  24. 24.
    Kantorovich, L.V.: On a problem of Monge. Uspekhi Mat. Nauk. 3, 225–226 (1948)Google Scholar
  25. 25.
    Li, Q.R., Santambrogio, F., Wang, X.J.: Regularity in Monge’s mass transfer problem. J. Math. Pures Appl. 102, 1015–1040 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications I-III, Dunod, Paris, 1968-1970Google Scholar
  27. 27.
    Ma, X.N., Trudinger, N.S., Wang, X.J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177, 151–183 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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)Google Scholar
  29. 29.
    Sudakov, V.N.: Geometric problems in the theory of infinite-dimensional probability distributions. Proc. Steklov Inst. 141, 1–178 (1979)MathSciNetGoogle Scholar
  30. 30.
    Trudinger, N.S., Wang, X.J.: On the Monge mass transfer problem. Calc. Var. 13, 19–31 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Vershik, A.M.: Some remarks on the infinite-dimensional problems of linear programming. Russian Math. Surv. 25, 117–124 (1970)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Mathematics and Jiangsu Key Laboratory of Engineering MechanicsSoutheast UniversityNanjingChina
  2. 2.Faculty of Science and TechnologyFederation University AustraliaBallaratAustralia

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