Skip to main content
SpringerLink
Log in

A globally convergent approximately active search algorithm for solving mathematical programs with linear complementarity constraints

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary.

In this paper, we propose a new algorithm for solving mathematical programs with linear complementarity constraints. The algorithm uses a method of approximately active search and introduces the idea of acceptable descent face. The main advantage of the new algorithm is that it is globally convergent without requiring strong assumptions such as nondegeneracy or linear independence condition. Numerical results are presented to show the effectiveness of the algorithm.

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. Harker, P.T., Pang, J.S.: On the existence of optimal solution to mathematical program with equilibrium constraints. Oper. Res. Lett. 7, 61–64 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dempe, S.: On generalized differentiability of optimal solutions and its application to an algorithm for solving bilevel optimization problems. In: D.Z. Du, L.Q. Qi and R.S. Womersley eds., Recent Advancesin Nonsmooth Optimization, World Scientific Publishing Co., Singapore, pp. 36–56, 1995

  3. Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85, 107–134 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Falk, J.E., Liu, J.: On bilevel programming, Part I: General cases. Math. Program. 70, 47–72 (1995)

    Article  MathSciNet  Google Scholar 

  5. Fukushima, M., Luo, Z.Q., Pang, J.S.: A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints. Comput. Optim. Appl. 10, 5–34 (1998)

    Article  MathSciNet  Google Scholar 

  6. Luo, Z.Q., Pang, J.S., Ralph, D.: Piece-wise sequential quadratic programming for mathematical Programs with nonlinear complementarity constraints. In: M.C. Ferris and J.S. Pang, eds., Complementarity and Variational Problems: State of the Art, SIAM Publications, 1997

  7. Outrata, J., Zowe, J.: A numerical approach to optimization problems with variational inequality constraints. Math. Program. 68, 105–130 (1995)

    Article  MathSciNet  Google Scholar 

  8. Pang, J.S., Fukushima, M.: Complementarity constraint qualifications and simplified B-stationary conditions for mathematical programs with equilibrium constraints. Comput. Optim. Appl. 13, 111–136 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ralph, D.: Sequential quadratic programming for mathematical programs with linear complementarity constraints. In: R.L. May and A.K. Easton eds., CTAC95 Computational Techniques and Applications, World Scientific, 1996

  10. Scholtes, S., Stöhr, M.: Exact penalization of mathematical programs with equilibrium constraints. SIAM J. Cont. Optim. 37, 617–652 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Vicente, L., Savard, G., Judice, J.: Descent approaches for quadratic bilevel programming. J. Optim. Theory Appl. 81, 379–399 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Zhang, J.Z., Liu, G.S.: A new extreme point algorithm and its application in PSQP algorithms for solving mathematical programs with linear Complementarity constraints. J. Global Optim. 19, 345–361 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Marcotte, P., Zhu, D.L.: Exact and inexact penalty methods for the generalized bilevel programming problems. Math. Program. 74, 141–157 (1996)

    Article  MathSciNet  Google Scholar 

  14. Ye, J., Zhu, D., Zhu, Q.: Generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Anandalingam, G., Friesz, T. (eds.): Hierarchical optimization. Ann. Oper. Res. 34, (1992)

  16. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, 1996

  17. Migdalas, A., Pardalos, P.M. (eds.): Hierarchical and bilevel programming. J. Global Optim. 8, (1996)

  18. Vicente, L.N., Calamai, P.H.: Bilevel and multilevel programming: A bibliography review. J. Global Optim. 3, 291–306 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Scholtes, S.: Active set method for inverse linear complementarity problems. Technique Report, Department of Engineering and Judge Institut of Management Studies, University of Cambridge, Cambridge CB2 1AG, England

  20. Fukushima, M., Tseng, P.: An implementable active-set algorithm for computing a B-stationary point of a mathematical programs with linear complementarity constraints. SIAM J. Optim. 12(3), 724–739 (2002)

    Article  MATH  Google Scholar 

  21. Jiang, H.Y., Ralph, D.: QPECgen, a MATLAB Generator for Mathematical Programs with Quadratic Objectives and Affine Variational Inequality Constraints. Comput. Optim. Appl. 13, 25–59 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York, 1983

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China

    J. Z. Zhang

  2. School of Business, Renmin University of China, Beijing, 100872, People’s Republic of China

    G. S. Liu

  3. Institute of Systems Science, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

    S. Y. Wang

Authors
  1. J. Z. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. S. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Y. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Mathematics Subject Classification (2000): 90C30, 90C33, 65K05

This research is partially supported by City University of Hong Kong under its Strategic Research Grant #7001339 and the National Natural Science Foundation of China grant # 10171108 and # 70271014

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, J., Liu, G. & Wang, S. A globally convergent approximately active search algorithm for solving mathematical programs with linear complementarity constraints. Numer. Math. 98, 539–558 (2004). https://doi.org/10.1007/s00211-004-0542-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-004-0542-9

Keywords

Search

Navigation