Skip to main content
Log in

A method for solving the general parametric linear complementarity problem

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

  • Chandrasekaran, R., Kabadi, S. N., & Sridhar, R. (1998). Integer solution for linear complementarity problem. Mathematics of Operations Research, 23, 390–402.

    Article  Google Scholar 

  • Cottle, R. W. (1972). Monotone solutions of the parametric linear complementarity problem. Mathematical Programming, 3, 210–224.

    Article  Google Scholar 

  • Cottle, R. W., Pang, J. S., & Stone, R. E. (1992). The linear complementarity problem. San Diego: Academic Press.

    Google Scholar 

  • Danao, R. A. (1997). On the parametric linear complementarity problem. Journal of Optimization Theory and Applications, 95, 445–454.

    Article  Google Scholar 

  • Dua, V., & Pistikopoulos, E. N. (2000). An algorithm for the solution of multiparametric mixed integer linear programming problems. Annals of Operation Research, 99, 123–139.

    Article  Google Scholar 

  • Eaves, B. C. (1976). A finite algorithm for the linear exchange model. Journal of Mathematical Economics, 3, 197–203.

    Article  Google Scholar 

  • Ferris, M. C., & Pang, J. S. (1997). Engineering and economic applications of complementarity problems. SIAM Review, 39, 669–713.

    Article  Google Scholar 

  • Gailly, B., Installe, M., & Smeers, Y. (2001). A new resolution method for the parametric linear complementarity problem. European Journal of Operational Research, 128, 639–646.

    Article  Google Scholar 

  • Jones, C. N., & Morrari, M. (2006). Multiparametric linear complementarity problems. In 45th IEEE conference on decision and control (pp. 5687–5692).

  • Kaneko, I. (1977). Isotone solutions of parametric linear complementarity problems. Mathematical Programming, 12, 48–59.

    Article  Google Scholar 

  • Li, Z., & Ierapetritou, M. G. (2007). Process scheduling under uncertainty using multiparametric programming. AIChE Journal, 53, 3183–3203.

    Article  Google Scholar 

  • Maier, G. (1972). Problem—on parametric linear complementarity problems. SIAM Review, 14, 364–365.

    Article  Google Scholar 

  • Martin, R. K. (1999). Large scale linear and integer optimization: a unified approach. Norwell: Kluwer Academic.

    Google Scholar 

  • Megiddo, N. (1977). On monotonicity in parametric linear complementarity problems. Mathematical Programming, 12, 60–66.

    Article  Google Scholar 

  • Murty, K. G. (1971). On the parametric complementarity problem. In Engineering summer conference notes. University of Michigan.

  • Pardalos, P. M., & Nagurney, A. (1990). The integer linear complementarity problem. International Journal of Computer Mathematics, 31, 205–214.

    Article  Google Scholar 

  • Pardalos, P. M., & Rosen, J. B. (1988). Global optimization approach to the linear complementarity problem. SIAM Journal on Scientific and Statistical Computing, 9, 341–353.

    Article  Google Scholar 

  • Rosen, J. B. (1990). Minimum norm solution to the linear complementarity problem. In Functional analysis, optimization, and mathematical economics. London: Oxford University Press.

    Google Scholar 

  • Tammer, K. (1998). Parametric linear complementarity problems. In A. V. Fiacco (Ed.), Mathematical programming with data perturbations (pp. 399–415). Boca Raton: CRC Press.

    Google Scholar 

  • Tondel, P., Johansen, T., & Bemporad, A. (2003). An algorithm for multi-parametric quadratic programming and explicit MPC solutions. Automatica, 39, 489–497.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marianthi G. Ierapetritou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Z., Ierapetritou, M.G. A method for solving the general parametric linear complementarity problem. Ann Oper Res 181, 485–501 (2010). https://doi.org/10.1007/s10479-010-0770-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-010-0770-6

Keywords

Navigation