Advertisement

Computational Optimization and Applications

, Volume 55, Issue 3, pp 703–731 | Cite as

A new method for solving Pareto eigenvalue complementarity problems

  • Samir AdlyEmail author
  • Hadia Rammal
Article

Abstract

In this paper, we introduce a new method, called the Lattice Projection Method (LPM), for solving eigenvalue complementarity problems. The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues and eigenvectors of the complementarity problems. The LPM is compared to SNMmin and SNMFB, two methods widely discussed in the literature for solving nonlinear complementarity problems, by using the performance profiles as a comparing tool (Dolan, Moré in Math. Program. 91:201–213, 2002). The performance measures, used to analyze the three solvers on a set of matrices mostly taken from the Matrix Market (Boisvert et al. in The quality of numerical software: assessment and enhancement, pp. 125–137, 1997), are computing time, number of iterations, number of failures and maximum number of solutions found by each solver. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving eigenvalue complementarity problems. Finally, Pareto bi-eigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method.

Keywords

Complementarity problems One-constrained eigenvalue problems Complementarity functions Semismooth Newton Method Lattice Projection Method Bi-eigenvalue complementarity problems 

Notes

Acknowledgements

The first author wishes to thank Professor Jonathan Borwein for some stimulating discussions about the new method LPM and particularly Lemma 3.

The authors wish to thank also Professor Alberto Seeger for his careful reading of the first version of the manuscript.

References

  1. 1.
    Adly, S., Seeger, A.: A nonsmooth algorithm for cone-constrained eigenvalue problems. Comput. Optim. Appl. 49, 299–318 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amri, A., Seeger, A.: Spectral analysis of coupled linear complementarity problems. Linear Algebra Appl. 432, 2507–2523 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Boisvert, R.F., Pozo, R., Remington, K., Barrett, R.F., Dongarra, J.J.: Matrix market: a web resource for test matrix collections. In: The Quality of Numerical Software: Assessment and Enhancement, pp. 125–137. Chapman and Hall, London (1997) Google Scholar
  4. 4.
    Chu, M.T., Watterson, J.L.: On a multivariate eigenvalue problem: I. Algebric theory and a power method. SIAM J. Sci. Comput. 14, 1089–1106 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983). Reprinted by, SIAM, Philadelphia, PA, 1990 zbMATHGoogle Scholar
  6. 6.
  7. 7.
    De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996) zbMATHCrossRefGoogle Scholar
  8. 8.
    Dirkse, S.P., Ferris, M.C.: A pathsearch damped Newton method for computing general equilibria. Ann. Oper. Res. 68, 211–232 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dirkse, S.P., Ferris, M.C.: The path solver: a non-monotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5, 123–156 (1995) CrossRefGoogle Scholar
  10. 10.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. J. Soc. Ind. Appl. Math. 39, 669–713 (1997) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hanafi, M., Ten Berge, J.M.F.: Global optimality of the successive Maxbet algorithm. Psychometrika 68, 97–103 (2003) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Horst, P.: Relations among m sets of measures. Psychometrika 26, 129–149 (1961) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hotelling, H.: The most predictable criterion. J. Educ. Psychol. 26, 139–142 (1935) CrossRefGoogle Scholar
  17. 17.
    Hotelling, H.: Relations between two sets of variates. Biometrika 28, 321–377 (1936) zbMATHGoogle Scholar
  18. 18.
    Judice, J.J., Sherali, H.D., Ribeiro, I.M.: The eigenvalue complementarity problem. Comput. Appl. 37(2), 139–156 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Júdice, J.J., Raydan, M., Rosa, S.S., Santos, S.A.: On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm. Numer. Algorithms 47, 391–407 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Júdice, J.J., Sherali, H.D., Ribeiro, I.M., Rosa, S.S.: On the asymmetric eigenvalue complementarity problem. Optim. Methods Softw. 24(4–5), 549–568 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kanzow, C., Kleinmichel, H.: A new class of semismooth Newton type methods for nonlinear complementarity problems. Comput. Optim. Appl. 11, 227–251 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Kanzow, C., Yamashita, N., Fukushima, M.: New NCP-functions and their properties. J. Optim. Theory Appl. 94, 115–135 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Kettenring, J.R.: Canonical analysis of several sets of variables. Biometrika 58, 433–451 (1971) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Martins, J.A.C., Barbarin, S., Raous, M., Pinto da Costa, A.: Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction. Comput. Methods Appl. Mech. Eng. 177, 289–328 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Martins, J.A.C., Pinto da Costa, A.: Stability of finite-dimensional nonlinear elastic systems with unilateral contact and friction. Int. J. Solids Struct. 37, 2519–2564 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Martins, J.A.C., Pinto da Costa, A.: Bifurcations and instabilities in frictional contact problems: theoretical relations, computational methods and numerical results. In: European Congress on Computational Methods in Applied Sciences and Engineering: ECCOMAS (2004) Google Scholar
  27. 27.
    Martins, J.A.C., Pinto da Costa, A.: Computation of bifurcations and instabilities in some frictional contact problems. In: European Conference on Computational Mechanics: ECCM (2001) Google Scholar
  28. 28.
    Martins, J.A.C., Pinto da Costa, A., Figueiredo, I.N., Júdice, J.J.: The directional instability problem in systems with frictional contacts. Comput. Methods Appl. Mech. Eng. 193, 357–384 (2004) zbMATHCrossRefGoogle Scholar
  29. 29.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957–972 (1977) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Pang, J.S., Facchinei, F.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Operations Research, vol. 2. Springer, New York (2003) Google Scholar
  31. 31.
    Pang, J.S.: Newton’s method for B-differentiable equations. Math. Oper. Res. 15, 311–465 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Pinto da Costa, A., Seeger, A.: Cone-constrained eigenvalue problems: theory and algorithms. Comput. Optim. Appl. 45(1), 25–57 (2008) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Pinto da Costa, A., Seeger, A.: Numerical resolution of cone-constrained eigenvalues problems. J. Comput. Appl. Math. 28, 37–61 (2009) MathSciNetzbMATHGoogle Scholar
  34. 34.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Qi, L.: Regular pseudo-smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems. Technical report AMR 97/14, The University of New South Wales, Sydney, Australia (1997) Google Scholar
  36. 36.
    Queiroz, M., Júdice, J.J., Humes, C.: The symmetric eigenvalue complementarity problem. Math. Comput. 73(248), 1849–1863 (2003) CrossRefGoogle Scholar
  37. 37.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Seeger, A.: Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions. Linear Algebra Appl. 292, 1–14 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Seeger, A., Torki, M.: On eigenvalues induced by a cone constraint. Linear Algebra Appl. 372, 181–206 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Seeger, A., Vicente-Pérez, J.: On cardinality of Pareto spectra. Electron. J. Linear Algebra 22, 758–766 (2011) MathSciNetzbMATHGoogle Scholar
  41. 41.
    Tanaka, Y.: Some generalized methods of optimal scaling and their asymptotic theories: the case of multiple responses-multiple factors. Ann. Inst. Stat. Math. 30, 329–348 (1978) zbMATHCrossRefGoogle Scholar
  42. 42.
    Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. J. Soc. Ind. Appl. Math. 43, 235–286 (2001) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Ulbrich, M.: In: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. MOS-SIAM Series on Optimization (2011) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.XLIM UMR-CNRS 7252Université de LimogesLimogesFrance

Personalised recommendations