Computational Optimization and Applications

, Volume 50, Issue 1, pp 49–73 | Cite as

A new class of penalized NCP-functions and its properties



In this paper, we consider a class of penalized NCP-functions, which includes several existing well-known NCP-functions as special cases. The merit function induced by this class of NCP-functions is shown to have bounded level sets and provide error bounds under mild conditions. A derivative free algorithm is also proposed, its global convergence is proved and numerical performance compared with those based on some existing NCP-functions is reported.


NCP-function Penalized Bounded level sets Error bounds 


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  1. 1.
    Billups, S.C., Dirkse, S.P., Soares, M.C.: A comparison of algorithms for large scale mixed complementarity problems. Comput. Optim. Appl. 7, 3–25 (1977) CrossRefGoogle Scholar
  2. 2.
    Chen, B., Chen, X., Kanzow, C.: A penalized Fischer-Burmeister NCP-function: theoretical investigation and numerical results. Math. Programm. 88, 211–216 (2000) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chen, J.-S.: The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem. J. Glob. Optim. 36, 565–580 (2006) MATHCrossRefGoogle Scholar
  4. 4.
    Chen, J.-S., Pan, S.: A family of NCP-functions and a descent method for the nonlinear complementarity problem. Comput. Optim. Appl. 40, 389–404 (2008) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chen, J.-S. Gao, H.-T., Pan, S.: A R-linearly convergent derivative-free algorithm for the NCPs based on the generalized Fischer-Burmeister merit function. J. Comput. Appl. Math. 232, 455–471 (2009) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cottle, R.W., Pang, J.-S., Stone, R.-E.: The Linear Complementarity Problem. Academic Press, New York (1992) MATHGoogle Scholar
  7. 7.
    Dafermos, S.: An iterative scheme for variational inequalities. Math. Programm. 26, 40–47 (1983) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Programm. 91, 201–213 (2002) MATHCrossRefGoogle Scholar
  9. 9.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
  10. 10.
    Facchinei, F., Soares, J.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225–247 (1997) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Fischer, A.: A special Newton-type optimization methods. Optimization 24, 269–284 (1992) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Geiger, C., Kanzow, C.: On the resolution of monotone complementarity problems. Comput. Optim. Appl. 5, 155–173 (1996) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Grippo, L., Lampariello, F., Lucidi, S.S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Harker, P.T., Pang, J.-S.: Finite dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications. Math. Programm. 48, 161–220 (1990) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hu, S.-L., Huang, Z.-H., Chen, J.-S.: Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems. J. Comput. Appl. Math. 230, 69–82 (2009) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Huang, Z.-H.: The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP. IMA J. Numer. Anal. 25, 670–684 (2005) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Huang, Z.-H., Gu, W.-Z.: A smoothing-type algorithm for solving linear complementarity problems with strong convergence properties. Appl. Math. Optim. 57, 17–29 (2008) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Huang, Z.-H., Qi, L., Sun, D.: Sub-quadratic convergence of a smoothing Newton algorithm for the P 0-and monotone LCP. Math. Programm. 99, 423–441 (2004) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Jiang, H.: Unconstrained minimization approaches to nonlinear complementarity problems. J. Glob. Optim. 9, 169–181 (1996) MATHCrossRefGoogle Scholar
  20. 20.
    Jiang, H., Fukushima, M., Qi, L.: A trust region method for solving generalized complementarity problems. SIAM J. Optim. 8, 140–157 (1998) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Kanzow, C.: Nonlinear complementarity as unconstrained optimization. J. Optim. Theory Appl. 88, 139–155 (1996) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Kanzow, C., Kleinmichel, H.: A new class of semismooth Newton method for nonlinear complementarity problems. Comput. Optim. Appl. 11, 227–251 (1998) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Kanzow, C., Yamashita, N., Fukushima, M.: New NCP-functions and their properties. J. Optim. Theory Appl. 94, 115–135 (1997) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Mangasarian, O.L.: Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J. Appl. Math. 31, 89–92 (1976) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Pang, J.-S.: Newton’s method for B-differentiable equations. Math. Oper. Res. 15, 311–341 (1990) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Pang, J.-S.: Complementarity problems. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, pp. 271–338. Kluwer Academic, Boston (1994) Google Scholar
  27. 27.
    Pang, J.-S., Chan, D.: Iterative methods for variational and complemantarity problems. Math. Programm. 27(99), 284–313 (1982) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sun, D., Qi, L.: On NCP-functions. Comput. Optim. Appl. 13, 201–220 (1999) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Yamashita, N., Fukushima, M.: On stationary points of the implicit Lagrangian for nonlinear complementarity problems. J. Optim. Theory Appl. 84, 653–663 (1995) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Yamashita, N., Fukushima, M.: Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems. Math. Programm. 76, 469–491 (1997) MathSciNetMATHGoogle Scholar
  31. 31.
    Yamada, K., Yamashita, N., Fukushima, M.: A new derivative-free descent method for the nonlinear complementarity problem. In: Di Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Related Topics, pp. 463–487. Kluwer Academic, Amsterdam (2000) Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan Normal UniversityTaipeiTaiwan
  2. 2.Department of MathematicsTianjin UniversityTianjinChina

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