Computational Optimization and Applications

, Volume 52, Issue 3, pp 805–824 | Cite as

Properties and construction of NCP functions

  • Aurél Galántai


The nonlinear complementarity or NCP functions were introduced by Mangasarian and these functions are proved to be useful in constrained optimization and elsewhere. Interestingly enough there are only two general methods to derive such functions, while the known or used NCP functions are either individual constructions or modifications of the few individual NCP functions such as the Fischer-Burmeister function. In the paper we analyze the elementary properties of NCP functions and the various techniques used to obtain such functions from old ones. We also prove some new nonexistence results on the possible forms of NCP functions. Then we develop and analyze several new methods for the construction of nonlinear complementarity functions that are based on various geometric arguments or monotone transformations. The appendix of the paper contains the list and source of the known NCP functions.


Nonlinear complementarity problem NCP function Nonlinear equation Merit function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966) MATHGoogle Scholar
  2. 2.
    Alsina, C., Frank, M.J., Schweizer, B.: Associative Functions: Triangular Norms and Copulas. World Scientific, Singapore (2006) MATHCrossRefGoogle Scholar
  3. 3.
    Arnold, V.I.: On the representability of functions of two variables in the form χ(ϕ(x)+ψ(y)). Usp. Mat. Nauk 12, 119–121 (1957). (Russian) Google Scholar
  4. 4.
    Castillo, E., Iglesias, A., Ruíz-Cobo, R.: Functional Equations in Applied Sciences. Elsevier, Amsterdam (2005) MATHGoogle Scholar
  5. 5.
    Chen, J.-S.: On Some NCP-functions based on the generalized Fischer-Burmeister function. Asia-Pac. J. Oper. Res. 24, 401–420 (2007) MathSciNetCrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Chen, B., Chen, X., Kanzow, C.: A penalized Fischer-Burmeister NCP-function: theoretical investigation and numerical results. Hamburger Beiträge zur Angewandten Math. Reihe A, prepr. 126 (1997) Google Scholar
  8. 8.
    Chen, B., Chen, X., Kanzow, C.: A penalized Fischer-Burmeister NCP-function. Math. Program. 88, 211–216 (2000) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chen, X., Qi, L., Yang, Y.F.: Lagrangian globalization methods for nonlinear complementarity problems. J. Optim. Theory Appl. 112, 77–95 (2002) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Evtushenko, Yu.G., Purtov, V.A.: Sufficient conditions for a minimum for nonlinear programming problems. Sov. Math. Dokl. 30, 313–316 (1984) MATHGoogle Scholar
  11. 11.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I–II. Springer, Berlin (2003) Google Scholar
  12. 12.
    Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Fischer, A., Jiang, H.: Merit functions for complementarity and related problems: a survey. Comput. Optim. Appl. 17, 159–182 (2000) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization. Springer, Berlin (2009) MATHGoogle Scholar
  16. 16.
    Gibson, C.G.: Elementary Geometry of Algebraic Curves. Cambridge University Press, Cambridge (1998) MATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Jiménez-Fernández, V.M., Agustín-Rodríguez, J., Marcelo-Julián, P., Agamennoni, O.: Evaluation algorithm for a decomposed simplicial piecewise linear formulation. J. Appl. Res. Technol. 6, 159–169 (2008) Google Scholar
  19. 19.
    Kanzow, C., Kleinmichel, H.: A class of Newton-type methods for equality and inequality constrained optimization. Optim. Methods Softw. 5, 173–198 (1995) CrossRefGoogle Scholar
  20. 20.
    Kanzow, C., Kleinmichel, H.: A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput. Optim. Appl. 11, 227–251 (1998) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Kanzow, C., Yamashita, N., Fukushima, M.: New NCP-functions and their properties. J. Optim. Theory Appl. 94, 115–135 (1997) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Khavinson, S.Ya.: Best Approximation by Linear Superpositions (Approximate Nomography). AMS, Providence (1997) MATHGoogle Scholar
  23. 23.
    Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Kluwer Academic, Dordrecht (2002) MATHGoogle Scholar
  24. 24.
    Lorentz, G.G.: Approximation of Functions. AMS, Providence (1986) MATHGoogle Scholar
  25. 25.
    Luo, Z.Q., Tseng, P.: A new class of merit functions for the nonlinear complementarity problem. In: Ferris, M.C., Pang, J.S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 204–225. SIAM, Philadelphia (1997) Google Scholar
  26. 26.
    Mangasarian, O.L.: Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J. Appl. Math. 31, 89–92 (1976) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Magasarian, O.L., Solodov, M.V.: Nonlinear complementarity as unconstrained and constrained minimization. Math. Program. 62, 277–297 (1993) CrossRefGoogle Scholar
  28. 28.
    Mulholland, J., Monagan, J.: Algorithms for trigonometric polynomials. In: Kaltofen, E., Villard, G. (eds.) ISSAC’01: Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, pp. 245–252. ACM, New York (2001) CrossRefGoogle Scholar
  29. 29.
    Pang, J.-S.: A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems. Math. Program. 51, 101–131 (1991) MATHCrossRefGoogle Scholar
  30. 30.
    Pu, D., Zhou, Y.: Piecewise linear NCP function for QP free feasible methods. Appl. Math. J. Chin. Univ. 21, 289–301 (2006) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Qi, L.: Regular almost smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems. Technical Rep. AMR 97/14, University of New South Wales (1997) Google Scholar
  32. 32.
    Qi, L., Yang, Y.-F.: NCP functions applied to Lagrangian globalization for the nonlinear complementarity problem. J. Glob. Optim. 24, 261–283 (2002) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Sun, D.: A regularization Newton method for solving nonlinear complementarity problems. Appl. Math. Optim. 40, 315–339 (1999) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Sun, D., Qi, L.-Q.: On NCP-functions. Comput. Optim. Appl. 13, 201–220 (1999) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Sun, D., Womersley, R.S.: A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method. SIAM J. Optim. 9, 388–413 (1999) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. Technical Rep. 00-11, Department of Computational and Applied Mathematics. Rice University (2000) Google Scholar
  37. 37.
    Vitushkin, A.G., Khenkin, G.M.: Linear superpositions of functions. Russ. Math. Surv. 22, 77–125 (1967) MATHCrossRefGoogle Scholar
  38. 38.
    Walker, R.J.: Algebraic Curves. Springer, Berlin (1978) MATHCrossRefGoogle Scholar
  39. 39.
    Wierzbicki, A.P.: Note on the equivalence of Kuhn-Tucker complementarity conditions to an equation. J. Optim. Theory Appl. 37, 401–405 (1982) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Yamada, K., Yamashita, N., Fukushima, M.: A new derivative-free descent method for the nonlinear complementarity problems. In: Pillo, G.D., Gianessi, F. (eds.) Nonlinear Optimization and Related Topics, pp. 463–489. Kluwer Academic, Dordrecht (2000) Google Scholar
  41. 41.
    Yamashita, N.: Properties of restricted NCP functions for nonlinear complementarity problems. J. Optim. Theory Appl. 98, 701–717 (1998) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.John von Neumann Faculty of InformaticsÓbuda UniversityBudapestHungary

Personalised recommendations