Applied Mathematics & Optimization

, Volume 74, Issue 3, pp 535–578 | Cite as

Mean Field Type Control with Congestion (II): An Augmented Lagrangian Method



This work deals with a numerical method for solving a mean-field type control problem with congestion. It is the continuation of an article by the same authors, in which suitably defined weak solutions of the system of partial differential equations arising from the model were discussed and existence and uniqueness were proved. Here, the focus is put on numerical methods: a monotone finite difference scheme is proposed and shown to have a variational interpretation. Then an Alternating Direction Method of Multipliers for solving the variational problem is addressed. It is based on an augmented Lagrangian. Two kinds of boundary conditions are considered: periodic conditions and more realistic boundary conditions associated to state constrained problems. Various test cases and numerical results are presented.


Mean Field Type Control Congestion Augmented Lagrangian Finite Difference Schemes 

Mathematics Subject Classification

49K20 49M20 65K10 65M06 


  1. 1.
    Achdou, Y.: Hamilton–Jacobi equations: approximations, numerical analysis and applications, Lecture Notes in Math. In: Loreti, P., Tchou, N.A. (eds.) Finite Difference Methods for Mean Field Games, vol. 2074, pp. 1–47. Springer, Heidelberg (2013)Google Scholar
  2. 2.
    Achdou, Y., Camilli, F., Capuzzo Dolcetta, I.: Mean field games: numerical methods for the planning problem. SIAM J. Control Optim. 50(1), 77–109 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Achdou, Y., Camilli, F., Capuzzo-Dolcetta, I.: Mean field games: convergence of a finite difference method. SIAM J. Numer. Anal. 51(5), 2585–2612 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Achdou, Y., Capuzzo-Dolcetta, I.: Mean field games: numerical methods. SIAM J. Numer. Anal. 48(3), 1136–1162 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Achdou, Y., Lasry, J.-M.: Mean field games for modeling crowd motion, preprintGoogle Scholar
  6. 6.
    Achdou, Y., Laurière, M.: On the system of partial differential equations arising in mean field type control. Discret. Contin. Dyn. Syst. 35(9), 3879–3900 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Achdou, Y., Laurière, M.: Mean field type control with congestion. Appl. Math. Optim. 73(3), 393–418 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Achdou, Y., Porretta, A.: Mean field games with congestion, in preparationGoogle Scholar
  9. 9.
    Achdou, Y., Porretta, A.: Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games. SIAM J. Numer. Anal. 54(1), 161–186 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Andreev, R.: Preconditioning the augmented lagrangian method for instationary mean field games with diffusion. (2016)Google Scholar
  11. 11.
    Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the monge-kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Benamou, J.-D., Carlier, G.: Augmented lagrangian methods for transport optimization, mean field games and degenerate elliptic equations. J. Optim. Theory Appl. 167(1), 1–26 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bensoussan, A., Frehse, J.: Control and Nash games with mean field effect. Chin. Ann. Math. Ser. B 34(2), 161–192 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bensoussan, A., Frehse, J., Yam, P.: Mean field games and mean field type control theory, Springer Briefs in Mathematics. Springer, New York (2013)CrossRefMATHGoogle Scholar
  15. 15.
    Cardaliaguet, P., Carlier, G., Nazaret, B.: Geodesics for a class of distances in the space of probability measures. Calc. Var. Part. Differ. Equ. 48(3–4), 395–420 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cardaliaguet, P., Graber, J., Porretta, A., Tonon, D.: Second order mean field games with degenerate diffusion and local coupling. NoDEA Nonlinear Differ. Equ. Appl. 22(5), 1287–1317 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Carmona, R., Delarue, F.: Mean field forward-backward stochastic differential equations. Electron. Commun. Probab. 18(68), 15 (2013)MathSciNetMATHGoogle Scholar
  18. 18.
    Carmona, R., Delarue, F., Lachapelle, A.: Control of McKean–Vlasov dynamics versus mean field games. Math. Financ. Econ. 7(2), 131–166 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Eckstein, J., Bertsekas, D.P.: On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1–3), 293–318 (1992)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Eckstein, J., Yao, W.: Augmented lagrangian and alternating direction methods for convex optimization: a tutorial and some illustrative computational results. RUTCOR Res. Rep. 32 (2012)Google Scholar
  21. 21.
    Fortin, M., Glowinski, R.: Augmented lagrangian methods: applications to the numerical solution of boundary-value problems. Elsevier, Amsterdam (2000)MATHGoogle Scholar
  22. 22.
    Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Compu. Math. Appl. 2(1), 17–40 (1976)CrossRefMATHGoogle Scholar
  23. 23.
    Gomes, D.A., Saúde, J.: Mean field games models—a brief survey. Dyn. Games Appl. 4(2), 110–154 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lions, P.-L.: Cours du Collège de France. (2007–2011)
  29. 29.
    Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    McKean Jr., H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. USA 56, 1907–1911 (1966)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1(3), 127–239 (2014)CrossRefGoogle Scholar
  32. 32.
    Porretta, A.: Weak solutions to Fokker–Planck equations and mean field games. Arch. Ration. Mech. Anal. 216, 1–62 (2014)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Powell, M.J.D.: A method for nonlinear constraints in minimization problems, Optimization (Sympos., Univ. Keele, Keele), vol. 1969, pp. 283–298. Academic Press. London (1968)Google Scholar
  34. 34.
    Rockafellar, R.T.: Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, Reprint of the 1970 original, Princeton Paperbacks (1997)Google Scholar
  35. 35.
    Sznitman, A.-S.: Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464, pp. 165–251. Springer, Berlin (1991)Google Scholar
  36. 36.
    van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 13(2), 631–644 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRSUniv. Paris Diderot, Sorbonne Paris CitéParisFrance

Personalised recommendations