Skip to main content
SpringerLink
Log in

On an enumerative algorithm for solving eigenvalue complementarity problems

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In this paper, we discuss the solution of linear and quadratic eigenvalue complementarity problems (EiCPs) using an enumerative algorithm of the type introduced by Júdice et al. (Optim. Methods Softw. 24:549–586, 2009). Procedures for computing the interval that contains all the eigenvalues of the linear EiCP are first presented. A nonlinear programming (NLP) model for the quadratic EiCP is formulated next, and a necessary and sufficient condition for a stationary point of the NLP to be a solution of the quadratic EiCP is established. An extension of the enumerative algorithm for the quadratic EiCP is also developed, which solves this problem by computing a global minimum for the NLP formulation. Some computational experience is presented to highlight the efficiency and efficacy of the proposed enumerative algorithm for solving linear and quadratic EiCPs.

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. Júdice, J., Sherali, H.D., Ribeiro, I., Rosa, S.: On the asymmetric eigenvalue complementarity problem. Optim. Methods Softw. 24, 549–586 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Pinto da Costa, A., Martins, J., Figueiredo, I., Júdice, J.: The directional instability problem in systems with frictional contacts. Comput. Methods Appl. Mech. Eng. 193(35), 357–384 (2004)

    Article  MATH  Google Scholar 

  3. Seeger, A.: Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions. Linear Algebra Appl. 292(13), 1–14 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Seeger, A., Torki, M.: On eigenvalues induced by a cone constraint. Linear Algebra Appl. 372, 181–206 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Zhou Yihuiand Gowda, M.S.: On the finiteness of the cone spectrum of certain linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 431(57), 772–782 (2009)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  7. Brás, C., Fukushima, M., Júdice, J., Rosa, S.: Variational inequality formulation of the asymmetric eigenvalue complementarity problem and its solution by means of gap functions. Pac. J. Optim. 8, 197–215 (2012)

    MATH  MathSciNet  Google Scholar 

  8. Pinto da Costa, A., Seeger, A.: Cone-constrained eigenvalue problems: theory and algorithms. Comput. Optim. Appl. 45, 25–57 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Adly, S., Seeger, A.: A nonsmooth algorithm for cone-constrained eigenvalue problems. Comput. Optim. Appl. 49, 299–318 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Niu, Y.S., Pham, T., Le Thi, H.A., Júdice, J.: Efficient dc programming approaches for the asymmetric eigenvalue complementarity problem. Optim. Methods Softw. (to appear)

  11. Dirkse, S., Ferris, M.: The path solver: a non-monotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5, 123–156 (1995)

    Article  Google Scholar 

  12. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  13. Vanderbei, R.: LOQO. User’s Manual, Technical Report, Princeton University (2003)

  14. Júdice, J., Raydan, M., Rosa, S., Santos, S.: On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm. Numer. Algorithms 44, 391–407 (2008)

    Article  Google Scholar 

  15. Le Thi, H., Moeini, M., Pham Dinh, T., Júdice, J.: A DC programming approach for solving the symmetric eigenvalue complementarity problem. Comput. Optim. Appl. 51, 1097–1117 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  17. Seeger, A., Torki, M.: Local minima of quadratic forms on convex cones. J. Glob. Optim. 44, 1–28 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Júdice, J., Sherali, H.D., Ribeiro, I.: The eigenvalue complementarity problem. Comput. Optim. Appl. 37, 139–156 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Seeger, A.: Quadratic eigenvalue problems under conic constraints. SIAM J. Matrix Analysis Appl. 32(3), 700–721 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sahinidis, N., Tawarmalani, M.: BARON 7.2.5: Global Optimization of Mixed-Integer Nonlinear Programs. User’s Manual (2005)

  21. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. John Hopkins University, Baltimore (1996)

    MATH  Google Scholar 

  22. Martos, B.: Nonlinear Programming Theory and Methods. North-Holland, Amsterdam (1975)

    MATH  Google Scholar 

  23. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006)

    Book  Google Scholar 

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

    MATH  Google Scholar 

  25. Sherali, H.D., Tuncbilek, C.: 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 

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

    Google Scholar 

  27. Murtagh, B., Saunders, A.: MINOS 5.0 User’s Guide. Technical Report SOL 83-20, Department of Operations Research, Stanford University, (1983)

  28. Seeger, A., Vicente-Pérez, J.: On cardinality of Pareto spectra. Electron. J. Linear Algebra 22, 758–766 (2011)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is supported in part by the National Science Foundation, under Grant Number CMMI - 0969169. The authors also thank two anonymous referees for their constructive and insightful comments.

Author information

Authors and Affiliations

  1. Instituto Politécnico de Tomar, Tomar, Portugal

    Luís M. Fernandes

  2. Instituto de Telecomunicações, Coimbra, Portugal

    Luís M. Fernandes & Joaquim J. Júdice

  3. Grado Department of Industrial & Systems Engineering, Virginia Tech, Blacksburg, VA, USA

    Hanif D. Sherali

  4. Departamento de Matemática, Universidade do Minho, Braga, Portugal

    Maria A. Forjaz

Authors
  1. Luís M. Fernandes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joaquim J. Júdice
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hanif D. Sherali
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Maria A. Forjaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luís M. Fernandes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fernandes, L.M., Júdice, J.J., Sherali, H.D. et al. On an enumerative algorithm for solving eigenvalue complementarity problems. Comput Optim Appl 59, 113–134 (2014). https://doi.org/10.1007/s10589-012-9529-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-012-9529-0

Keywords

Search

Navigation