Skip to main content
SpringerLink
Log in

QPCOMP: A quadratic programming based solver for mixed complementarity problems

  • Published:
Mathematical Programming Submit manuscript

Abstract

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel [14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate 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. S.C. Billups, Algorithms for complementarity problems and generalized equations, Ph.D. thesis, University of Wisconsin-Madison (Madison, WI, 1995).

    Google Scholar 

  2. A. Brooke, D. Kendrick and A. Meeraus,GAMS: A User’s Guide (Scientific Press, South San Francisco, CA, 1988).

    Google Scholar 

  3. C. Chen and O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems,Computational Optimization and Applications 5 (1996) 97–138.

    Article  MATH  MathSciNet  Google Scholar 

  4. S.P. Dirkse and M.C. Ferris. MCPLIB: A collection of nonlinear mixed complementarity problems.Optimization Methods and Software 5 (1995) 319–345.

    Google Scholar 

  5. S.P. Dirkse and M.C. Ferris. The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems.Optimization Methods and Software 5 (1995) 123–156.

    Google Scholar 

  6. S.P. Dirkse, M.C. Ferris, P.V. Preckel and T. Rutherford, The GAMS callable program library for variational and complementarity solvers, Mathematical Programming Technical Report 94-07, Computer Sciences Department, University of Wisconsin (Madison, WI, 1994) available via ftp://ftp.cs.wisc.edu/math-prog/tech-reports/.

    Google Scholar 

  7. M.C. Ferris and J.S. Pang, Engineering and economic applications of complementarity problems. Discussion Papers in Economics 95-4, Department of Economics, University of Colorado (Boulder, CO, 1995) available via ftp://ftp.cs.wisc.edu/math-prog/tech-reports/.

    Google Scholar 

  8. S.A. Gabriel, Algorithms for the Nonlinear Complementarity Problem: The NE/SQP Method and Extensions. Ph.D. thesis, The Johns Hopkins University (Baltimore, MD, 1992).

    Google Scholar 

  9. P.T. Harker and J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,Mathematical Programming 48 (1990) 161–220.

    Article  MATH  MathSciNet  Google Scholar 

  10. P.T. Harker and B. Xiao, Newton’s method for the nonlinear complementarity problem: A B-differentiable equation approach,Mathematical Programming 48 (1990) 339–358.

    Article  MATH  MathSciNet  Google Scholar 

  11. O.L. Mangasarian,Nonlinear Programming (McGraw-Hill, New York, 1969); SIAM Classics in Applied Mathematics 10 (SIAM, Philadelphia, PA, 1994).

    MATH  Google Scholar 

  12. B.A. Murtagh and M.A. Saunders, MINOS 5.0 user’s guide. Technical Report SOL 83.20, Stanford University (Stanford, CA, 1983).

    Google Scholar 

  13. J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, CA, 1970).

    MATH  Google Scholar 

  14. J.S. Pang and S.A. Gabriel, NE/SQP: A robust algorithm for the nonlinear complementarity problem,Mathematical Programming 60 (1993) 295–338.

    Article  MathSciNet  Google Scholar 

  15. D. Ralph, Global convergence of damped Newton’s method for nonsmooth equations, via the path search,Mathematics of Operations Research 19 (1994) 352–389.

    Article  MATH  MathSciNet  Google Scholar 

  16. R.T. Rockafellat, Monotone operators and augmented Lagrangian methods in nonlinear programming, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 3 (Academic, London, 1978) pp. 1–26.

    Google Scholar 

  17. T.F. Rutherford, MILES: A mixed inequality and nonlinear equation solver, Working Paper, Department of Economics, University of Colorado, Boulder, CO.

  18. A.N. Tikhonov and V.Y. Arsenin,Solutions of Ill-Posed Problems (Wiley, New York, 1977).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Colorado, 80217, Denver, CO, USA

    Stephen C. Billups

  2. Computer Sciences Department, University of Wisconsin, 53706, Madison, WI, USA

    Michael C. Ferris

Authors
  1. Stephen C. Billups
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael C. Ferris
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This material is based on research supported by National Science Foundation Grant CCR-9157632, Department of Energy Grant DE-FG03-94ER61915, and the Air Force Office of Scientific Research Grant F49620-94-1-0036.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Billups, S.C., Ferris, M.C. QPCOMP: A quadratic programming based solver for mixed complementarity problems. Mathematical Programming 76, 533–562 (1997). https://doi.org/10.1007/BF02614397

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02614397

Keywords

Search

Navigation