Journal of Optimization Theory and Applications

, Volume 123, Issue 2, pp 365–390 | Cite as

Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints

  • X. M. Hu
  • D. Ralph


We adapt the convergence analysis of the smoothing (Ref. 1) and regularization (Ref. 2) methods to a penalty framework for mathematical programs with complementarity constraints (MPCC); we show that the penalty framework shares convergence properties similar to those of these methods. Moreover, we give sufficient conditions for a sequence generated by the penalty framework to be attracted to a B-stationary point of the MPCC.

Complementarity constraints penalty methods B-stationarity linear independence constraint qualification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fukushima, M., and Pang, J. S., Convergence of a Smoothing Continuation Method for Mathematical Programs with Complementarity Constraints, Ill-Posed Variational Problems and Regularization Techniques, Edited by M. Th´era and R. Tichatschke, Springer, New York, NY, pp. 99–110, 2000.Google Scholar
  2. 2.
    Scholtes, S., Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints, SIAM Journal on Optimization, Vol. 11, pp. 918–936, 2001.Google Scholar
  3. 3.
    Luo, Z. Q., Pang, J. S., and Ralph, D., Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, England, 1996.Google Scholar
  4. 4.
    Outrata, J., Kocvara, M., and Zowe, J., Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, Holland, Vol. 28, 1998.Google Scholar
  5. 5.
    Chen, Y., and Florian, M., The Nonlinear Bilevel Programming Problem: Formulations, Regularity, and Optimality Conditions, Optimization, Vol. 32, pp. 193–209, 1995.Google Scholar
  6. 6.
    Ye, J. J., Zhu, D. L., and Zhu, Q. J., Exact Penalization and Necessary Optimality Conditions for Generalized Bilevel Programming Problems, SIAM Journal on Optimization, Vol. 7, pp. 481–507, 1997.Google Scholar
  7. 7.
    Robinson, S. M., Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems, SIAM Journal on Numerical Analysis, Vol. 13, pp. 497–513, 1976.Google Scholar
  8. 8.
    Scheel, H., and Scholtes, S., Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity, Mathematics of Operations Research, Vol. 25, pp.1–22, 2000.Google Scholar
  9. 9.
    Ferris, M. C., and Tin-Loi, F., Nonlinear Programming Approach for a Class of Inverse Problems in Elastoplasticity, Structural Engineering and Mechanics, Vol. 6, pp. 857–870, 1998.Google Scholar
  10. 10.
    Ferris, M. C., and Tin-Loi, F., On the Solution of a Minimum Weight Elastoplastic Problem Involving Displacement and Complementarity Constraints, Computer Methods in Applied Mechanics and Engineering, Vol. 174, pp. 107–120, 1999.Google Scholar
  11. 11.
    Tin-Loi, F., On the Numerical Solution of a Class of Unilateral Contact Structural Optimization Problems, Structural Optimization, Vol. 17, pp. 155–161, 1999.Google Scholar
  12. 12.
    Tin-Loi, F., and Que, N. S., On a Class of Parameter Identification Problems, Computational Mechanics for the Next Millennium, Edited by C. M. Wang, K. H. Lee and K. K. Ang, Proceedings of APCOM 99, Fourth Asia-Pacific Conference on Computational Mechanics, Elsevier Science, Oxford, England, Vol. 1, pp. 311–316, 1999.Google Scholar
  13. 13.
    Tin-Loi, F., and Que, N. S., Nonlinear Programming Approaches for an Inverse Problem in Quasibrittle Fracture, International Journal of Mechanical Sciences, Vol. 44, pp. 843–858, 2002.Google Scholar
  14. 14.
    Huang, X. X., Yang, X. Q., and Zhu, D. L., A Sequential Smooth Penalization Approach to Mathematical Programs with Complementarity Constraints, Manuscript, Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong, China, 2001.Google Scholar
  15. 15.
    Fischer, A., A Special Newton-Type Optimization Method, Optimization, Vol. 24, pp. 269–284, 1992.Google Scholar
  16. 16.
    Kanzow, C., Some Noninterior Continuation Methods for Linear Complementarity Problems, SIAM Journal on Matrix Analysis and Applications, Vol. 17, pp. 851–868, 1996.Google Scholar
  17. 17.
    Facchinei, F., Jiang, H., and Qi, L., A Smoothing Method for Mathematical Programs with Equilibrium Constraints, Mathematical Programming, Vol. 85, pp. 107–134, 1999.Google Scholar
  18. 18.
    Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, 1982.Google Scholar
  19. 19.
    Mangasarian, O. L., Nonlinear Programming, McGraw-Hill, New York, NY, 1969.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • X. M. Hu
    • 1
  • D. Ralph
    • 2
  1. 1.CSIRO Manufacturing and Infrastructure TechnologyAustralia
  2. 2.University of CambridgeEngland

Personalised recommendations