Skip to main content
SpringerLink
Log in

Reconstructing a matrix from a partial sampling of Pareto eigenvalues

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

Abstract

Let Λ={λ 1,…,λ p } be a given set of distinct real numbers. This work deals with the problem of constructing a real matrix A of order n such that each element of Λ is a Pareto eigenvalue of A, that is to say, for all k∈{1,…,p} the complementarity system

$$x\geq \mathbf{0}_n,\quad Ax-\lambda_k x\geq \mathbf{0}_n,\quad \langle x, Ax-\lambda_k x\rangle = 0$$

admits a nonzero solution x∈ℝn.

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. Abaffy, J., Galántai, A., Spedicato, E.: The local convergence of ABS methods for nonlinear algebraic equations. Numer. Math. 51, 429–439 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adly, S., Seeger, A.: A nonsmooth algorithm for cone-constrained eigenvalue problems. Comput. Optim. Appl. (2010), in press, online Oct. 2009

  3. Bai, Z.-J.: Inexact Newton methods for inverse eigenvalue problems. Appl. Math. Comput. 172, 682–689 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-J., Chan, R.H., Morini, B.: On the convergence rate of a Newton-like method for inverse eigenvalue and inverse singular value problems. Int. J. Appl. Math. 13, 59–69 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Ben-Israel, A.: A Newton-Raphson method for the solution of systems of equations. J. Math. Anal. Appl. 15, 243–252 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borobia, A.: On the nonnegative eigenvalue problem. Linear Algebra Appl. 223/224, 131–140 (1995)

    Article  MathSciNet  Google Scholar 

  7. Chan, R.H., Chung, H.L., Xu, S.-F.: The inexact Newton-like method for inverse eigenvalue problem. BIT 43, 7–20 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, X.-D., Qi, Y.-C.: The Pareto eigenvalue problem of some matrices. J. Shanghai Second Polytech. Univ. 27, 54–64 (2010) (Chinese)

    MATH  Google Scholar 

  9. Chen, X., Nashed, Z., Qi, L.: Convergence of Newton’s method for singular smooth and nonsmooth equations using adaptive outer inverses. SIAM J. Optim. 7, 445–462 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Egleston, P.D., Lenker, T.D., Narayan, S.K.: The nonnegative inverse eigenvalue problem. Linear Algebra Appl. 379, 475–490 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  12. Figueiredo, I.N., Júdice, J.J., Martins, J.A.C., Pinto da Costa, A.: A complementarity eigenproblem in the stability of finite dimensional elastic systems with frictional contact. In: Ferris, M., Mangasarian, O., Pang, J.S. (eds.) Complementarity: Applications, Algorithms and Extensions. Applied Optimization Series, vol. 50, pp. 67–83. Kluwer Acad., Dordrecht (1999)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Friedland, S., Nocedal, J., Overton, M.L.: The formulation and analysis of numerical methods for inverse eigenvalue problems. SIAM J. Numer. Anal. 24, 634–667 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. He, J.-S., Li, C., Wang, J.-H.: Newton’s method for underdetermined systems of equations under the γ-condition. Numer. Funct. Anal. Optim. 28, 663–679 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, Z.J.: A new method for solving nonlinear underdetermined systems. Math. Appl. Comput. 13, 33–48 (1994)

    MATH  Google Scholar 

  17. Humes, C., Júdice, J.J., Queiroz, M.: The symmetric eigenvalue complementarity problem. Math. Comput. 73, 1849–1863 (2004)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Kellogg, R.B.: Matrices similar to a positive or essentially positive matrix. Linear Algebra Appl. 4, 191–204 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer Academic, Dordrecht (2002)

    MATH  Google Scholar 

  23. Martínez, J.M.: Quasi-Newton methods for solving underdetermined nonlinear simultaneous equations. J. Comput. Appl. Math. 34, 171–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nocedal, J., Overton, M.L.: Numerical methods for solving inverse eigenvalue problems. In: Numerical Methods, Caracas, 1982. Lect. Notes in Math., vol. 1005, pp. 212–226. Springer, Berlin (1983)

    Chapter  Google Scholar 

  25. Pang, J.S.: Newton’s method for B-differentiable equations. Math. Oper. Res. 15, 311–341 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pinto da Costa, A., Seeger, A.: Numerical resolution of cone-constrained eigenvalue problems. Comput. Appl. Math. 28, 37–61 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Salzmann, F.L.: A note on eigenvalues of nonnegative matrices. Linear Algebra Appl. 5, 329–338 (1972)

    MathSciNet  MATH  Google Scholar 

  30. Scholtyssek, V.: Solving inverse eigenvalue problems by a projected Newton method. Numer. Funct. Anal. Optim. 17, 925–944 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Seeger, A., Vicente-Pérez, J.: Inverse eigenvalue problems for linear complementarity systems. April (2010), submitted

  34. Suleimanova, H.R.: Stochastic matrices with real characteristic numbers. Dokl. Akad. Nauk SSSR 66, 343–345 (1949)

    MathSciNet  MATH  Google Scholar 

  35. Suleimanova, H.R.: The question of a necessary and sufficient condition for the existence of a stochastic matrix with prescribed characteristic numbers. Trudy Vsesojuz. Zaocn. Energet. Inst. Vyp 28, 33–49 (1965)

    MathSciNet  Google Scholar 

  36. Tanabe, K.: Continuous Newton-Raphson method for solving an underdetermined system of nonlinear equations. Nonlinear Anal. 3, 495–503 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tanabe, K.: Global analysis of continuous analogues of the Levenberg-Marquardt and Newton-Raphson methods for solving nonlinear equations. Ann. Inst. Stat. Math. 37, 189–203 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  38. Walker, H.F.: Newton-like methods for underdetermined systems. In: Computational Solution of Nonlinear Systems of Equations, Fort Collins, 1988. Lectures in Appl. Math., vol. 26, pp. 679–699. Am. Math. Soc., Providence (1990)

    Google Scholar 

  39. Walker, H.F., Watson, L.T.: Least-change secant update methods for underdetermined systems. SIAM J. Numer. Anal. 27, 1227–1262 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile

    Pedro Gajardo

  2. Department of Mathematics, University of Avignon, 33 rue Louis Pasteur, 84000, Avignon, France

    Alberto Seeger

Authors
  1. Pedro Gajardo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alberto Seeger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Seeger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gajardo, P., Seeger, A. Reconstructing a matrix from a partial sampling of Pareto eigenvalues. Comput Optim Appl 51, 1119–1135 (2012). https://doi.org/10.1007/s10589-010-9391-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-010-9391-x

Keywords

Search

Navigation