Modified Jacobian smoothing method for nonsmooth complementarity problems

  • Pin-Bo Chen
  • Peng Zhang
  • Xide Zhu
  • Gui-Hua LinEmail author


This paper is devoted to solving a nonsmooth complementarity problem where the mapping is locally Lipschitz continuous but not continuously differentiable everywhere. We reformulate this nonsmooth complementarity problem as a system of nonsmooth equations with the max function and then propose an approximation to the reformulation by simultaneously smoothing the mapping and the max function. Based on the approximation, we present a modified Jacobian smoothing method for the nonsmooth complementarity problem. We show the Jacobian consistency of the function associated with the approximation, under which we establish the global and fast local convergence for the method under suitable assumptions. Finally, to show the effectiveness of the proposed method, we report our numerical experiments on some examples based on MCPLIB/GAMSLIB libraries or network Nash–Cournot game is proposed.


Nonsmooth complementarity problem Jacobian consistency Jacobian smoothing method Convergence Network Nash–Cournot game 

Mathematics Subject Classification

90C30 90C33 90C56 



This research was supported in part by National Natural Science Foundation of China (Nos. 11671250, 11901380, 11431004, 71831008) and Humanity and Social Science Foundation of Ministry of Education of China (No. 15YJA630034). The authors are grateful to two anonymous referees for their helpful comments and suggestions, which have led to much improvement of the paper.


  1. 1.
    Chen, C.H., Mangasarian, O.L.: A class of smoothing function for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5(2), 97–138 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Chen, C.H., Mangasarian, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Program. 71(1), 51–69 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chen, X., Qi, L.Q., Sun, D.F.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67(222), 519–540 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chen, X., Wets, R.J.-B., Zhang, Y.F.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22(2), 649–673 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chen, Y.Y., Gao, Y.: Two new Levenberg–Marquardt methods for nonsmooth nonlinear complementarity problems. Scienceasia 40(1), 89–93 (2014)CrossRefGoogle Scholar
  6. 6.
    Chu, A.J., Du, S.Q., Su, Y.X.: A new smoothing conjugate gradient method for solving nonlinear nonsmooth complementarity problems. Algorithms 8(4), 1195–1209 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  8. 8.
    Dirkse, S.P., Ferris, M.C.: MCPLIB: a collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5(4), 319–345 (1995)CrossRefGoogle Scholar
  9. 9.
    Ermoliev, Y.M., Norkin, V.I., Wets, R.J.-B.: The minimization of semicontinuous functions: mollifier subgradients. SIAM J. Control Optim. 33(1), 149–167 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175(1), 177–211 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
  12. 12.
    Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669–713 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Program. 76(3), 513–532 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fischer, A., Jeyakumar, V., Luc, D.T.: Solution point characterizations and convergence analysis of a descent algorithm for nonsmooth continuous complementarity problems. J. Optim. Theory Appl. 110(3), 493–513 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gao, Y.: A Newton method for a nonsmooth nonlinear complementarity problem. Oper. Res. Trans. 15(2), 53–58 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hobbs, B.F., Pang, J.S.: Nash–Cournot equilibrium in electric power markets with piecewise linear demand functions and joint constraints. Oper. Res. 55(1), 113–127 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Izmailov, A.F., Solodov, M.V.: Superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems. SIAM J. Optim. 13(2), 386–405 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Jiang, H.Y.: Smoothed Fischer–Burmeister equation methods for the complementarity problem. Technical Report, Department of Mathematics, University of Melbourne, Parkville, Australia (1997)Google Scholar
  19. 19.
    Jiang, H.Y.: Unconstrained minimization approaches to nonlinear complementarity problems. J. Glob. Optim. 9(2), 169–181 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kanzow, C., Pieper, H.: Jacobian smoothing methods for nonlinear complementarity problems. SIAM J. Optim. 9(2), 342–373 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kanzow, C., Qi, H.D.: A QP-free constrained Newton-type method for variational inequality problems. Math. Program. 85(1), 81–106 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Li, G.Y., Ng, K.F.: Error bound of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems. SIAM J. Optim. 20(2), 667–690 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Metzler, C.B., Hobbs, B.F., Pang, J.S.: Nash–Cournot equilibria in power markets on a linearized dc network with arbitrage: formulations and properties. Netw. Spat. Econ. 3(2), 123–150 (2003)CrossRefGoogle Scholar
  24. 24.
    Ng, K.F., Tan, L.L.: D-gap functions for nonsmooth variational inequality problems. J. Optim. Theory Appl. 133(1), 77–97 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ng, K.F., Tan, L.L.: Error bounds for regularized gap function for nonsmooth variational inequality problem. Math. Program. 110(2), 405–429 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Qi, H.D., Liao, L.Z.: A smoothing Newton method for general nonlinear complementarity problems. Comput. Optim. Appl. 17(2–3), 231–253 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Qi, L.Q., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58(1–3), 353–367 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Ralph, D., Xu, H.F.: Implicit smoothing and its application to optimization with piecewise smooth equality constraints. J. Optim. Theory Appl. 124(3), 673–699 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)zbMATHCrossRefGoogle Scholar
  30. 30.
    Song, L.S., Gao, Y.: On the local convergence of a Levenberg–Marquardt method for nonsmooth nonlinear complementarity problems. Scienceasia 43(6), 377–382 (2017)CrossRefGoogle Scholar
  31. 31.
    Sun, D.F., Qi, L.Q.: Solving variational inequality problems via smoothing–nonsmooth reformulations. J. Comput. Appl. Math. 129(1–2), 37–62 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Xu, H.F.: Adaptive smoothing method, deterministically computable generalized Jacobians, and the Newton method. J. Optim. Theory Appl. 109(1), 215–224 (2001)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Pin-Bo Chen
    • 1
  • Peng Zhang
    • 2
  • Xide Zhu
    • 1
  • Gui-Hua Lin
    • 1
    Email author
  1. 1.School of ManagementShanghai UniversityShanghaiChina
  2. 2.School of Economics and ManagementChongqing University of Posts and TelecommunicationsChongqingChina

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