Skip to main content
SpringerLink
Log in

A semismooth equation approach to the solution of nonlinear complementarity problems

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.

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.

Similar content being viewed by others

References

  1. D.P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982).

    MATH  Google Scholar 

  2. B. Chen and P.T. Harker, A noninterior continuation method for quadratic and linear programming,SIAM J. on Optimization 3 (1993) 503–515.

    Article  MATH  MathSciNet  Google Scholar 

  3. F.H. Clarke,Optimization and Nonsmooth Analysis (John Wiley & Sons, New York, 1983) (reprinted by SIAM, Philadelphia, PA, 1990).

    MATH  Google Scholar 

  4. R.W. Cottle, J.-S. Pang and R.E. Stone,The Linear Complementarity Problem (Academic Press, Boston, 1992).

    MATH  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. F. Facchinei, Minimization ofSC 1 functions and the Maratos effect,Operations Research Letters 17 (1995), 131–137.

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Facchinei and J. Soares, A new meril function for nonlinear complementarity problems and a related algorithm, Technical Report 15.94, Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Rome, Italy (1994) (to appear in:SIAM J. on Optimization).

    Google Scholar 

  8. F. Facchinei and J. Soares, Testing a new class of algorithms for nonlinear complementarity problems, in: F. Giannessi and A. Maugeri, eds.,Variational Inequalities and Network Equilibrium Problems (Plenum Press, New York, 1995) 69–83.

    Google Scholar 

  9. M.C. Ferris and S. Lucidi, Nonmonotone stabilization methods for nonlinear equations,J. of Optimization Theory and Applications 81 (1994) 53–71.

    Article  MATH  MathSciNet  Google Scholar 

  10. M.C. Ferris and D. Ralph, Projected gradient methods for nonlinear complementarity problems via normal maps, in: D.Z. Du, L. Qi and R.S. Womersley, eds.,Recent Advances in Nonsmooth Optimization (World Scientific Publishers, Singapore) 57–87.

  11. A. Fischer, A special Newton-type optimization method.Optimization 24 (1992) 269–284.

    MATH  MathSciNet  Google Scholar 

  12. A. Fischer, A Newton-type method for positive semidefinite linear complementarity problems,J. of Optimization Theory and Applications 86 (1995) 585–608.

    Article  MATH  Google Scholar 

  13. A. Fischer, An NCP-function and its use for the solution of complementarity problems, in: D.Z. Du, L. Qi and R.S. Womersley, eds.,Recent Advances in Nonsmooth Optimization (World Scientific Publishers, Singapore) 88–105.

  14. A. Fischer and C. Kanzow, On finite termination of an iterative method for linear complementarity problems, Preprint MATH-NM-10-1994 Institute for Numerical Mathematics, Technical University of Dresden, Dresden, Germany (1994) (to appear in:Mathematical Programming).

    Google Scholar 

  15. M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,Mathematical Programming (Series A) 53 (1992) 99–110.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems,Computational Optimization and Applications 5 (1996) 155–173.

    Article  MATH  MathSciNet  Google Scholar 

  17. L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone linesearch technique for Newton's method,SIAM J. on Numerical Analysis 23 (1986) 707–716.

    Article  MATH  MathSciNet  Google Scholar 

  18. S.P. Han, J.-S. Pang and N. Rangaraj, Globally convergent Newton methods for nonsmooth equations,Mathematics of Operations Research 17 (1992) 586–607.

    MATH  MathSciNet  Google Scholar 

  19. P.T. Harker, Accelerating the convergence of the diagonalization and projection algorithm for finitedimensional variational inequalities,Mathematical Programming (Series A) 41 (1988) 25–59.

    MathSciNet  Google Scholar 

  20. P.T. Harker and B. Xiao, Newton's method for the nonlinear complementarity problem: AB-differentiable equation approach,Mathematical Programming (Series A) 48 (1990) 339–357.

    Article  MATH  MathSciNet  Google Scholar 

  21. W. Hock and K. Schittkowski, Test examples for nonlinear programming codes, Lectures Notes in Economics and Mathematical Systems 187 (Springer-Verlag, Berlin, 1981).

    MATH  Google Scholar 

  22. H. Jiang, Unconstrained minimization approaches to nonlinear complementarities, Applied Mathematics Report AMR 94/33, School of Mathematics, University of New South Wales, Sydney, Australia (1994).

    Google Scholar 

  23. H. Jiang and L. Qi, A nonsmooth equations approach to nonlinear complementarities, Applied Mathematics Report AMR 94/31, School of Mathematics, University of New South Wales, Sydney, Australia (1994) (to appear in:SIAM J. on Control and Optimization).

    Google Scholar 

  24. N.H. Josephy, Newton's method for generalized equations, MRC Technical Summary Report 1965, Mathematics Research Center. University of Wisconsin-Madison, WI (1979).

    Google Scholar 

  25. C. Kanzow, Some equation-based methods for the nonlinear complementarity problem,Optimization Methods and Software 3 (1994) 327–340.

    Google Scholar 

  26. C. Kanzow, Nonlinear complementarity as unconstrained optimization,J. of Optimization Theory and Applications 88 (1996) 139–155.

    Article  MATH  MathSciNet  Google Scholar 

  27. C. Kanzow, Global convergence properties of some iterative methods for linear complementarity problems, Preprint 72, Institute of Applied Mathematics, University of Hamburg Hamburg, Germany (1993) revised (1994) (to appear in:SIAM J. on Optimization).

    Google Scholar 

  28. M. Kojima, Computational methods for solving the nonlinear complementarity problem, Keio Engineering Reports 27 (1974) 1–41.

    MathSciNet  Google Scholar 

  29. O.L. Mangasarian, Equivalence of the complementarity problem to a system of nonlinear equations,SIAM J. on Applied Muthematics 31 (1976) 89–92.

    Article  MathSciNet  MATH  Google Scholar 

  30. O.L. Mangasarian and M.V. Solodov, Nonlinear complementarity as unconstrained and constrained minimization,Mathematical Programming (Series B) 62 (1993) 277–297.

    Article  MathSciNet  Google Scholar 

  31. R. Mifflin, Semismooth and semiconvex functions in constrained optimization,SIAM J. on Control and Optimization 15 (1977) 957–972.

    MathSciNet  Google Scholar 

  32. J.J. Moré, Global methods for nonlinear complementarity problems, Preprint MCS-P429-0494 Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL (1994).

    Google Scholar 

  33. J.J. Moré and W.C. Rheinboldt, OnP- andS-functions and related classes ofn-dimensional nonlinear mappings,Linear Algebra and its Applications 6 (1973) 45–68.

    Article  MATH  Google Scholar 

  34. J.-S. Pang, Newton's method for B-differentiable equations.Mathematics of Operations Research 15 (1990) 311–341.

    MATH  MathSciNet  Google Scholar 

  35. J.-S. Pang, A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems,Mathematical Programming (Series A) 51 (1991) 101–132.

    Article  MATH  MathSciNet  Google Scholar 

  36. J.-S. Pang and S.A. Gabriel, NE/SQP: A robust algorithm for the nonlinear complementarity problem,Mathematical Programming (Series A) 60 (1993) 295–337.

    Article  MathSciNet  Google Scholar 

  37. J.-S. Pang and L. Qi, Nonsmooth equations: motivation and algorithms,SIAM J. on Optimization 3 (1993) 443–465.

    Article  MATH  MathSciNet  Google Scholar 

  38. L. Qi, A convergence analysis of some algorithms for solving nonsmooth equations,Mathematics of Operations Research 18 (1993) 227–244.

    MATH  MathSciNet  Google Scholar 

  39. L. Qi and H. Jiang, Karush-Kuhn-Tucker equations and convergence analysis of Newton methods and Quasi-Newton methods for solving these equations, Technical Report AMR 94/5, School of Mathematics, University of New South Wales, Australia (1994) (to appear in:SIAM J. on Control and Optimization).

    Google Scholar 

  40. L. Qi and J. Sun, A nonsmooth version of Newton's method,Mathematical Programming (Series A) 58 (1993) 353–368.

    Article  MathSciNet  Google Scholar 

  41. 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 

  42. S.M. Robinson, Strongly regular generalized equations,Mathematics of Operations Research 5 (1980) 43–62.

    MATH  MathSciNet  Google Scholar 

  43. S.M. Robinson, ImplicitB-differentiability in generalized equations, Technical Summary Report 2854, Mathematics Research Center, University of Wisconsin-Madison. WI (1985).

    Google Scholar 

  44. S.M. Robinson, An implicit function theorem forB-differentiable functions, IIASA Working Paper 88-67, Laxemburg, Austria (1988).

  45. S.M. Robinson, Newton's method for a class of nonsmooth functions, Manuscript, Department of Industrial Engineering, University of Wisconsin-Madison, WI (1988).

    Google Scholar 

  46. S.M. Robinson, Normal maps induced by linear transformations,Mathematics of Operations Research 17 (1992) 691–714.

    MATH  MathSciNet  Google Scholar 

  47. S.M. Robinson, Generalized equations, in: A. Bachem, M. Groetschel and B. Korte, eds.,Mathematical programming: the state of the art (Springer-Verlag, Berlin, 1983) 346–367.

    Google Scholar 

  48. H. Scarf,The Computation of Economic Equilibria (Yale University Press, New Haven, CT, 1973).

    MATH  Google Scholar 

  49. P.K. Subramanian, Gauss-Newton methods for the complementarity problem,J. of Optimization Theory and Applications 77 (1993) 467–482.

    Article  MATH  MathSciNet  Google Scholar 

  50. R.L. Tobin, Variable dimension spatial price equilibrium algorithm,Mathematical Programming (Series A) 40 (1988) 33–51.

    Article  MATH  MathSciNet  Google Scholar 

  51. P.L. Toint, A non-monotone trust-region algorithm for nonlinear optimization subject to convex constraints, Report 92/94, Department of Mathematics, FUNDP, Namur, Belgium (1994).

    Google Scholar 

  52. P. Tseng, Growth behavior of a class of merit functions for the nonlinear complementarity problem, Technical Report, Department of Mathematics, University of Washington, Seattle, WA (1994).

    Google Scholar 

  53. L.T. Watson, Solving the nonlinear complementarity problem by a homotopy method,SIAM J. on Control and Optimization 17 (1979) 36–46.

    Article  MATH  Google Scholar 

  54. B. Xiao and P.T. Harker, A nonsmooth Newton method for variational inequalities, 1: Theory,Mathematical Programming (Series A) 65 (1994) 151–194.

    Article  MathSciNet  Google Scholar 

  55. B. Xiao and P.T. Harker, A nonsmooth Newton method for variational inequalities, II: Numerical results,Mathematical Programming (Series A) 65 (1994) 195–216.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Via Buonarroti 12, 00185, Roma, Italy

    Tecla De Luca & Francisco Facchinei

  2. Institute of Applied Mathematics, University of Hamburg, Bundesstrasse 55, D-20146, Hamburg, Germany

    Christian Kanzow

Authors
  1. Tecla De Luca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francisco Facchinei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Kanzow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Facchinei.

Rights and permissions

Reprints and permissions

About this article

Cite this article

De Luca, T., Facchinei, F. & Kanzow, C. A semismooth equation approach to the solution of nonlinear complementarity problems. Mathematical Programming 75, 407–439 (1996). https://doi.org/10.1007/BF02592192

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02592192

Key words

Search

Navigation