Advertisement

Mathematical Programming

, Volume 99, Issue 2, pp 377–397 | Cite as

On the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm

  • Nobuo Yamashita
  • Hiroshige Dan
  • Masao Fukushima
Article

Abstract.

In this paper we focus on the problem of identifying the index sets P(x):={i|x i >0}, N(x):={i|F i (x)>0} and C(x):={i|x i =F i (x)=0} for a solution x of the monotone nonlinear complementarity problem NCP(F). The correct identification of these sets is important from both theoretical and practical points of view. Such an identification enables us to remove complementarity conditions from the NCP and locally reduce the NCP to a system which can be dealt with more easily. We present a new technique that utilizes a sequence generated by the proximal point algorithm (PPA). Using the superlinear convergence property of PPA, we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution.

Keywords

Nonlinear complementarity problem proximal point algorithm degeneracy 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Billups, S.C.: Improving the robustness of descent-based methods for semismooth equations using proximal perturbations. Math. Program. 87, 269–284 (2000)Google Scholar
  2. 2.
    Bongartz, I., Conn, A.R., Gould, N., Toint, Ph.L.: CUTE: Constrained and unconstrained testing environment. ACM Trans. Math. Softw. 21, 123–160 (1995)CrossRefzbMATHGoogle Scholar
  3. 3.
    El-Bakry, A.S., Tapia, R.A., Zhang, Y.: A study of indicators for identifying zero variables in interior-point methods. SIAM Rev. 36, 45–72 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    El-Bakry, A.S., Tapia, R.A., Zhang, Y.: On the convergence rate of Newton interior-point methods in the absence of strict complementarity. Comput. Optim. Appl. 6, 157–167 (1996)MathSciNetzbMATHGoogle Scholar
  5. 5.
    De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)CrossRefGoogle Scholar
  6. 6.
    Facchinei, F., Fischer, A., Kanzow, C.: On the accurate identification of active constraints. SIAM J. Optim. 9, 14–32 (1998)CrossRefzbMATHGoogle Scholar
  7. 7.
    Facchinei, F., Fischer, A., Kanzow, C.: On the identification of zero variables in an interior-point framework. SIAM J. Optim. 10, 1058–1078 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Fischer, A.: A special Newton-type optimization method. Optimization 24, 153–176 (1992)Google Scholar
  9. 9.
    Fischer, A.: An NCP-function and its use for the solution of complementarity problems. In: Du, D.Z., Qi, L., Womersley, R.S. (eds.), Recent Advances in Nonsmooth Optimization, World Scientific, Singapore, 1995, pp. 88–105Google Scholar
  10. 10.
    Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes. Lect. Notes in Econ. Math. Syst. 187, Springer-Verlag, Berlin, 1981Google Scholar
  11. 11.
    Jiang, H.: Global convergence analysis of the generalized Newton and Gauss-Newton methods of the Fischer-Burmeister equation for the complementarity problem. Math. Oper. Res. 24, 529–543 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kojima, M., Shindo, S.: Extensions of Newton and quasi-Newton methods to systems of PC 1 equations. J. Oper. Res. Soc. Japan 29, 352–374 (1986)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Luo, Z.-Q., Mangasarian, O.L., Ren, J., Solodov, M.V.: New error bounds for the linear complementarity problem. Math. Oper. Res. 19, 880–892 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Pang, J.-S.: A posteriori error bounds for the linearly–constrained variational inequality problem. Math. Oper. Res. 12, 474–484 (1987)MathSciNetGoogle Scholar
  15. 15.
    Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. Study 14, 206–214 (1981)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optimization 14, 877–898 (1976)zbMATHGoogle Scholar
  17. 17.
    Stoer, J., Wechs, M., Mizuno, S.: High order infeasible-interior-point methods for sufficient linear complementarity problems. Math. Oper. Res. 23, 832–862 (1998)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Yamashita, N., Fukushima M.: Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems. Math. Program. 76, 469–491 (1997)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Yamashita, N., Fukushima M.: The proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem. SIAM J. Optim. 11, 364–379 (2001)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Nobuo Yamashita
    • 1
  • Hiroshige Dan
    • 1
    • 2
  • Masao Fukushima
    • 1
  1. 1.Department of Applied Mathematics and PhysicsGraduate School of Informatics, Kyoto UniversityKyotoJapan
  2. 2.Mathematical Systems Inc.TokyoJapan

Personalised recommendations