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The eigenvalue complementarity problem

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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.

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References

  1. Bard, J., Moore, J.: A branch-and-bound algorithm for the bilevel linear program. SIAM J. Sci. Stat. Comput. 11, 281–292 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brooke, A., Kendrick, D., Meeraus, A., Raman, R.: GAMS a User’s Guide. GAMS Development Corporation, New York (1998)

    Google Scholar 

  3. Chung, S.: NP-completeness of the linear complementarity problems. J. Optim. Theory Appl. 60, 393–399 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  5. Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem. Academic, New York (1992)

    MATH  Google Scholar 

  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. Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

  8. Golub, G., Van Loan, C.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Júdice, J., Faustino, A.: A sequential LCP algorithm for bilevel linear programming. Ann. Oper. Res. 34, 89–106 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  12. Júdice, J., Sherali, H., Ribeiro, I., Faustino, A.: A complementarity active-set algorithm for mathematical programming problems with equilibrium constraints. Working paper (2005)

  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)

  14. Murtagh, B., Saunders, A.: MINOS 5.0 user’s guide. Technical report SOL 83-20R, Department of Operations Research, Stanford University (1987)

  15. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  16. Queiroz, M., Júdice, J., Humes Jr., C.: The symmetric eigenvalue complementarity problem. Math. Comput. 73, 1849–1863 (2003)

    Google Scholar 

  17. Ribeiro, I.: Global optimization and applications to structural engineering (in Portuguese). Ph.D. thesis, University of Porto, Porto (2005)

  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)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática da Universidade de Coimbra and Instituto de Telecomunicações, Coimbra, Portugal

    Joaquim J. Júdice

  2. Grado Department of Industrial & Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA

    Hanif D. Sherali

  3. Secção de Matemática do Departamento de Engenharia Civil, Faculdade de Engenharia, Universidade do Porto, Porto, Portugal

    Isabel M. Ribeiro

Authors
  1. Joaquim J. Júdice
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  2. Hanif D. Sherali
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  3. Isabel M. Ribeiro
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Corresponding author

Correspondence to Joaquim J. Júdice.

Additional information

J.J. Júdice was supported by Instituto de Telecomunicações and by FCT under grant POCTI/35059/MAT/2000. H.D. Sherali was supported by the National Science Foundation under grant DMI-0094462.

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Júdice, J.J., Sherali, H.D. & Ribeiro, I. The eigenvalue complementarity problem. Comput Optim Appl 37, 139–156 (2007). https://doi.org/10.1007/s10589-007-9017-0

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  • DOI: https://doi.org/10.1007/s10589-007-9017-0

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