Skip to main content
SpringerLink
Log in

Partial Augmented Lagrangian Method and Mathematical Programs with Complementarity Constraints

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we apply a partial augmented Lagrangian method to mathematical programs with complementarity constraints (MPCC). Specifically, only the complementarity constraints are incorporated into the objective function of the augmented Lagrangian problem while the other constraints of the original MPCC are retained as constraints in the augmented Lagrangian problem. We show that the limit point of a sequence of points that satisfy second-order necessary conditions of the partial augmented Lagrangian problems is a strongly stationary point (hence a B-stationary point) of the original MPCC if the limit point is feasible to MPCC, the linear independence constraint qualification for MPCC and the upper level strict complementarity condition hold at the limit point. Furthermore, this limit point also satisfies a second-order necessary optimality condition of MPCC. Numerical experiments are done to test the computational performances of several methods for MPCC proposed in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.P. Aubin H. Frankowska (1990) Set-Valued Analysis Springer Birkhauser

    Google Scholar 

  2. D.P. Bertsekas (1982) Constrained Optimization and Lagrangian Multiplier Methods Academic Press New York

    Google Scholar 

  3. A.R. Conn N. Gould A. Sartenaer Ph.L. Toint (1996) ArticleTitleConvergence properties of an augmented Lagrangian algorithm for optimization with a combination of general equality and linear constraints SIAM Journal of Optimization 6 674–703 Occurrence Handle10.1137/S1052623493251463

    Article  Google Scholar 

  4. A.R. Conn N. Gould Ph.L. Toint (1991) ArticleTitleA globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds SIAM Journal of Numer. Anal 28 545–572 Occurrence Handle10.1137/0728030

    Article  Google Scholar 

  5. G.B. Dantzig J. Folkman N. Shapiro (1967) ArticleTitleOn continuity of the minimum set of a continuous function Journal of Mathematical Analysis and Applications 17 519–548 Occurrence Handle10.1016/0022-247X(67)90139-4

    Article  Google Scholar 

  6. F. Facchinei H. Jiang L. Qi (1999) ArticleTitleA smoothing method for mathematical programs with equilibrium constraints Mathematical Programming 85 107–134 Occurrence Handle10.1007/s101070050048

    Article  Google Scholar 

  7. B. Fares P. Apkarian D. Noll (2001) ArticleTitleAugmented Lagrangian method for a class of LMI-constrained problems in robust control theory International Journal of Control 74 348–360 Occurrence Handle10.1080/00207170010010605

    Article  Google Scholar 

  8. R. Fletcher (1987) Practical Methods of Optimization Wiley New York

    Google Scholar 

  9. M. Fukushima Z.Q. Luo J.S. Pang (1998) ArticleTitleA globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints Computational Optimization and Applications 10 5–34 Occurrence Handle10.1023/A:1018359900133

    Article  Google Scholar 

  10. Fukushima, M., Pang, J.S. (1999). Convergence of a Smoothing Continuation Method for Mathematical Programs with Complementarity Constraints, Ill-posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Economics and mathematical Systems, Vol. 477, Springer, Berlin, pp. 99–110.

  11. M. Fukushima J.S. Pang (1998) ArticleTitleSome feasibility issues in mathematical programs with equilibrium constraints SIAM Journal of Optimization 8 673–681 Occurrence Handle10.1137/S105262349731577X

    Article  Google Scholar 

  12. M. Fukushima P. Tseng (2001) ArticleTitleAn implementable active-set algorithm for computing a B-stationary point of a mathematical program with linear complementarity constraints SIAM Journal of Optimization 12 724–739 Occurrence Handle10.1137/S1052623499363232

    Article  Google Scholar 

  13. X.M. Hu D. Ralph (2001) ArticleTitleConvergence of a Penalty Method for Mathematical Programming with Complementarity Constraints Journal of Optimization Theory and Applications 123 IssueID2 365–390 Occurrence Handle10.1007/s10957-004-5154-0

    Article  Google Scholar 

  14. Huang, X.X., Yang, X.Q., Zhu, D.L. (2001). A smooth sequential penalization approach to mathematical programs with complementarity constraints, preprint.

  15. H.Y. Jiang D. Ralph (2000) ArticleTitleSmooth SQP methods for mathematical programs with nonlinear complementarity constraints SIAM Journal of Optimization 10 779–808 Occurrence Handle10.1137/S1052623497332329

    Article  Google Scholar 

  16. H.Y. Jiang D. Ralph (1999) ArticleTitleQPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints Computational Optimization and Applications 13 25–59 Occurrence Handle10.1023/A:1008696504163

    Article  Google Scholar 

  17. Lin, G.H. and Fukushima, M. (to appear), A modified relaxation scheme for mathematical programs with complementarity constraints, Annals of Operations Research

  18. Z.Q. Luo J.S. Pang D. Ralph (1996) Mathematical Programs with Equilibrium Constraints Cambridge University Press Cambridge

    Google Scholar 

  19. Q. Meng H. Yang M.G.H. Bell (2001) ArticleTitleAn equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem Transportation Research Part B 35 83–105 Occurrence Handle10.1016/S0191-2615(00)00016-3

    Article  Google Scholar 

  20. J.V. Outrata (1994) ArticleTitleOn optimization problems with variational inequality constraints SIAM Journal of Optimization 4 340–357 Occurrence Handle10.1137/0804019

    Article  Google Scholar 

  21. J.V. Outrata J. Zowe (1995) ArticleTitleA numerical approach to optimization problems with variational inequality constraints Mathematical Programming 68 105–130

    Google Scholar 

  22. J.S. Pang (1997) ArticleTitleError bounds in mathematical programming Mathematical Programming 79 299–332 Occurrence Handle10.1016/S0025-5610(97)00042-7

    Article  Google Scholar 

  23. J.S. Pang M. Fukushima (1999) ArticleTitleComplementarity constraint qualifications and simplified B stationarity conditions for mathematical programs with equilibrium constraints. Computational Optimization–a tribute to Olvi Mangasarian, Part II. Computational Optimization and Applications 13 111–136 Occurrence Handle10.1023/A:1008656806889

    Article  Google Scholar 

  24. S. Scholtes (2001) ArticleTitleConvergence properties of a regularization scheme for mathematical programs with complementarity constraints SIAM Journal of Optimization 11 918–936 Occurrence Handle10.1137/S1052623499361233

    Article  Google Scholar 

  25. H. Scheel S. Scholtes (1999) ArticleTitleExact penalization of mathematical programs with equilibrium constraints SIAM Journal of Control and Optimization 37 617–652

    Google Scholar 

  26. S. Scholtes M. Stohr (2000) ArticleTitleMathematical programs with complementarity constraints: stationarity, optimality and sensitivity Mathematics of Operations Research 25 1–22 Occurrence Handle10.1287/moor.25.1.1.15213

    Article  Google Scholar 

  27. S. Scholtes M. Stohr (2001) ArticleTitleHow stringent is the linear independence assumption for mathematical programs with complementarity constraints Mathematics of Operations Research 26 851–863 Occurrence Handle10.1287/moor.26.4.851.10007

    Article  Google Scholar 

  28. R.T. Rockafellar (1974) ArticleTitleAugmented Lagrangian multiplier functions and duality in nonconvex programming SIAM Journal on Control and Optimization 12 268–285 Occurrence Handle10.1137/0312021

    Article  Google Scholar 

  29. R.T. Rockafellar (1993) ArticleTitleLagrangian multipliers and optimality SIAM Review 35 183–238 Occurrence Handle10.1137/1035044

    Article  Google Scholar 

  30. R.T. Rockafellar R.J.-B. Wets (1998) Variational Analysis Springer-Verlag Berlin

    Google Scholar 

  31. X.Q. Yang (1994) ArticleTitleAn exterior method for computing points that satisfy second-order necessary conditions for a C1,1 optimization problem Journal of Mathematical Analysis and Applications 187 118–133 Occurrence Handle10.1006/jmaa.1994.1348

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Chongqing Normal University, Chongqing, 400047, China

    X. X. Huang

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    X. Q. Yang

  3. Department of Mathematics and Statistics, Curtin University of Technology, Western Australia, Australia

    K. L. Teo

Authors
  1. X. X. Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. X. Q. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. L. Teo
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was partially supported by the Research Grants Council (BQ654) of Hong Kong and the Postdoctoral Fellowship of The Hong Kong Polytechnic University. Dedicated to Alex Rubinov on the occassion of his 65th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, X.X., Yang, X.Q. & Teo, K.L. Partial Augmented Lagrangian Method and Mathematical Programs with Complementarity Constraints. J Glob Optim 35, 235–254 (2006). https://doi.org/10.1007/s10898-005-3837-1

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-005-3837-1

Keywords

Search

Navigation