Advertisement

Computational Optimization and Applications

, Volume 37, Issue 2, pp 139–156 | Cite as

The eigenvalue complementarity problem

  • Joaquim J. JúdiceEmail author
  • Hanif D. Sherali
  • Isabel M. Ribeiro
Article

Abstract

In this paper an eigenvalue complementarity problem (EiCP) is studied, which finds its origins in the solution of a contact problem in mechanics. The EiCP is shown to be equivalent to a Nonlinear Complementarity Problem, a Mathematical Programming Problem with Complementarity Constraints and a Global Optimization Problem. A finite Reformulation–Linearization Technique (Rlt)-based tree search algorithm is introduced for processing the EiCP via the lattermost of these formulations. Computational experience is included to highlight the efficacy of the above formulations and corresponding techniques for the solution of the EiCP.

Keywords

Global optimization Complementarity Eigenvalue problems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bard, J., Moore, J.: A branch-and-bound algorithm for the bilevel linear program. SIAM J. Sci. Stat. Comput. 11, 281–292 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brooke, A., Kendrick, D., Meeraus, A., Raman, R.: GAMS a User’s Guide. GAMS Development Corporation, New York (1998) Google Scholar
  3. 3.
    Chung, S.: NP-completeness of the linear complementarity problems. J. Optim. Theory Appl. 60, 393–399 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Costa, A., Martins, J., Figueiredo, I., Júdice, J.: The directional instability problem in systems with frictional contacts. Comput. Methods Appl. Mech. Eng. 193, 357–384 (2004) zbMATHCrossRefGoogle Scholar
  5. 5.
    Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem. Academic, New York (1992) zbMATHGoogle Scholar
  6. 6.
    Dirkse, S., Ferris, M.: The path solver: a nonmonotone stabilization scheme for mixed complementarity problems. Optim. Softw. 5, 123–156 (1995) Google Scholar
  7. 7.
    Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
  8. 8.
    Golub, G., Van Loan, C.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996) zbMATHGoogle Scholar
  9. 9.
    Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Comput. 13(5), 1194–1217 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Júdice, J., Faustino, A.: A sequential LCP algorithm for bilevel linear programming. Ann. Oper. Res. 34, 89–106 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Júdice, J., Ribeiro, I., Faustino, A.: On the solution of NP-hard linear complementarity problems. TOP—Sociedad de Estatística e Investigacion Operativa 10(1), 125–145 (2002) zbMATHGoogle Scholar
  12. 12.
    Júdice, J., Sherali, H., Ribeiro, I., Faustino, A.: A complementarity active-set algorithm for mathematical programming problems with equilibrium constraints. Working paper (2005) Google Scholar
  13. 13.
    Júdice, J., Sherali, H., Ribeiro, I., Faustino, A.: A complementarity-based partitioning and disjunctive cut algorithm for mathematical programming problems with equilibrium constraints. Working paper (2005) Google Scholar
  14. 14.
    Murtagh, B., Saunders, A.: MINOS 5.0 user’s guide. Technical report SOL 83-20R, Department of Operations Research, Stanford University (1987) Google Scholar
  15. 15.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999) zbMATHCrossRefGoogle Scholar
  16. 16.
    Queiroz, M., Júdice, J., Humes Jr., C.: The symmetric eigenvalue complementarity problem. Math. Comput. 73, 1849–1863 (2003) Google Scholar
  17. 17.
    Ribeiro, I.: Global optimization and applications to structural engineering (in Portuguese). Ph.D. thesis, University of Porto, Porto (2005) Google Scholar
  18. 18.
    Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation–linearization technique. J. Glob. Optim. 2, 101–112 (1992) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Joaquim J. Júdice
    • 1
    Email author
  • Hanif D. Sherali
    • 2
  • Isabel M. Ribeiro
    • 3
  1. 1.Departamento de Matemática da Universidade de Coimbra and Instituto de TelecomunicaçõesCoimbraPortugal
  2. 2.Grado Department of Industrial & Systems EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  3. 3.Secção de Matemática do Departamento de Engenharia Civil, Faculdade de EngenhariaUniversidade do PortoPortoPortugal

Personalised recommendations