Skip to main content
SpringerLink
Log in

The extended linear complementarity problem

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we define the Extended Linear Complementarity Problem (ELCP), an extension of the well-known Linear Complementarity Problem (LCP). We show that the ELCP can be viewed as a kind of unifying framework for the LCP and its various generalizations. We study the general solution set of an ELCP and we develop an algorithm to find all its solutions. We also show that the general ELCP is an NP-hard problem.

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. F.A. Al-Khayyal, “An implicit enumeration procedure for the general linear complementarity problem,”Mathematical Programming Study 31 (1987) 1–20.

    Google Scholar 

  2. F. Baccelli, G. Cohen, G.J. Olsder and J.P. Quadrat,Synchronization and Linearity (Wiley, New York, 1992).

    Google Scholar 

  3. R. Carraghan and P.M. Pardalos, “An exact algorithm for the maximum clique problem,”Operations Research Letters 9 (6) (1990) 375–382.

    Google Scholar 

  4. S. Chung, “NP-completeness of the linear complementarity problem,”Journal of Optimization Theory and Applications 60 (3) (1989) 393–399.

    Google Scholar 

  5. R.W. Cottle and G.B. Dantzig, “A generalization of the linear complementarity problem,”Journal of Combinatorial Theory 8 (1) (1970) 79–90.

    Google Scholar 

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

    Google Scholar 

  7. R.A. Cuninghame-Green,Minimax Algebra, Lecture Notes in Economics and Mathematical Systems, Vol. 166 (Springer, Berlin, 1979).

    Google Scholar 

  8. B. De Moor, “Mathematical concepts and techniques for modelling of static and dynamic systems,” Ph.D. Thesis, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium (1988).

    Google Scholar 

  9. B. De Moor, L. Vandenberghe and J. Vandewalle, “The generalized linear complementarity problem and an algorithm to find all its solutions,”Mathematical Programming 57 (1992) 415–426.

    Google Scholar 

  10. B. De Schutter and B. De Moor, “Minimal realization in the max algebra is an extended linear complementarity problem,”Systems & Control Letters 25 (2) (1995) 103–111.

    Google Scholar 

  11. B. De Schutter and B. De Moor, “A method to find all solutions of a system of multivariate polynomial equalities and inequalities in the max algebra,”Discrete Event Dynamic Systems: Theory and Applications, to appear.

  12. B. De Schutter and B. De Moor, “The characteristic equation and minimal state space realization of SISO systems in the max algebra,” in: G. Cohen and J.-P. Quadrat, eds.,Proceedings of the 11th International Conference on Analysis and Optimization of Systems, Sophia-Antipolis, France, Lecture Notes in Control and Information Sciences, Vol. 199 (Springer, New York, 1994) pp. 273–282.

    Google Scholar 

  13. A.A. Ebiefung and M.K. Kostreva, “Global solvability of generalized linear complementarity problems and a related class of polynomial complementarity problems,” in: C.A. Floudas and P.M. Pardalos, eds.,Recent Advances in Global Optimization, Princeton Series in Computer Science (Princeton University Press, Princeton, NJ, 1992) pp. 102–124.

    Google Scholar 

  14. M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, CA, 1979).

    Google Scholar 

  15. P.E. Gill and W. Murray,Numerical Methods for Constrained Optimization (Academic Press, London, 1974).

    Google Scholar 

  16. M.S. Gowda and R. Sznajder, “The generalized order linear complementarity problem,”SIAM Journal on Matrix Analysis and Applications 15 (3) (1994) 779–795.

    Google Scholar 

  17. M.S. Gowda, “On the extended linear complementarity problem,”Mathematical Programming, to appear.

  18. J.J. Júdice and L.N. Vicente, “On the solution and complexity of a generalized linear complementarity problem,”Journal of Global Optimization 4 (4) (1994) 415–424.

    Google Scholar 

  19. J.A. Kaliski and Y. Ye, “An extension of the potential reduction algorithm for linear complementarity problems with some priority goals,”Linear Algebra and its Applications 193 (1993) 35–50.

    Google Scholar 

  20. O.L. Mangasarian and J.S. Pang, “The extended linear complementarity problem,”SIAM Journal on Matrix Analysis and Applications 16 (12) (1995) 359–368.

    Google Scholar 

  21. O.L. Mangasarian and M.V. Solodov, “Nonlinear complementarity as unconstrained and constrained minimization,”Mathematical Programming 62 (2) (1993) 277–297.

    Google Scholar 

  22. T.H. Mattheiss and D.S. Rubin, “A survey and comparison of methods for finding all vertices of convex polyhedral sets,”Mathematics of Operations Research 5 (2) (1980) 167–185.

    Google Scholar 

  23. K. McShane, “Superlinearly convergent\(O\left( {\sqrt n L} \right)\)-iteration interior-point algorithms for linear programming and the monotone linear complementarity problem,”SIAM Journal on Optimization 4 (2) (1994) 247–261.

    Google Scholar 

  24. S. Mehrotra and R.A. Stubbs, “Predictor-corrector methods for a class of linear complementarity problems,”SIAM Journal on Optimization 4 (2) (1994) 441–453.

    Google Scholar 

  25. T.S. Motzkin, H. Raiffa, G.L. Thompson and R.M. Thrall, “The double description method,” in: H.W. Kuhn and A.W. Tucker, eds.,Contributions to the Theory of Games, Annals of Mathematics Studies, Vol. 28 (Princeton University Press, Princeton, NJ, 1953) pp. 51–73.

    Google Scholar 

  26. P.M. Pardalos and J.B. Rosen,Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science, Vol. 268 (Springer, Berlin, 1987).

    Google Scholar 

  27. P.M. Pardalos and J. Xue, “The maximum clique problem,”Journal of Global Optimization 4 (1994) 301–328.

    Google Scholar 

  28. A. Schrijver,Theory of Linear and Integer Programming (Wiley, Chichester, 1986).

    Google Scholar 

  29. M. Sun, “Monotonicity of Mangasarian's iterative algorithm for generalized linear complementarity problems,”Journal of Mathematical Analysis and Applications 144 (1989) 474–485.

    Google Scholar 

  30. R. Sznajder and M.S. Gowda, “Generalizations ofP 0- andP-properties; extended vertical and horizontal LCPs,”Linear Algebra and its Applications 223–224 (1995) 695–716.

    Google Scholar 

  31. R.H. Tütüncü and M.J. Todd, “Reducing horizontal linear complementarity problems,”Linear Algebra and its Application 223–224 (1995) 717–730.

    Google Scholar 

  32. L. Vandenberghe, B. De Moor and J. Vandewalle, “The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits,”IEEE Transactions on Circuits and Systems 36 (11) (1989) 1382–1391.

    Google Scholar 

  33. Y. Ye, “A fully polynomial-time approximation algorithm for computing a stationary point of the general linear complementarity problem,”Mathematics of Operations Research 18 (2) (1993) 334–345.

    Google Scholar 

  34. Y. Zhang, “On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem,”SIAM Journal on Optimization 4 (1) (1994) 208–227.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ESAT — Katholieke Universiteit Leuven, Kardinaal Mercierlaan 94, B-3001, Leuven, Belgium

    Bart De Schutter & Bart De Moor

Authors
  1. Bart De Schutter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bart De Moor
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This paper presents research results of the Belgian programme on interuniversity attraction poles (IUAP-50) initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The scientific responsibility is assumed by its authors.

Supported by the N.F.W.O. (Belgian National Fund for Scientific Research).

Rights and permissions

Reprints and permissions

About this article

Cite this article

De Schutter, B., De Moor, B. The extended linear complementarity problem. Mathematical Programming 71, 289–325 (1995). https://doi.org/10.1007/BF01590958

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01590958

Keywords

Search

Navigation