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Solving inverse cone-constrained eigenvalue problems

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Abstract

We compare various algorithms for constructing a matrix of order \(n\) whose Pareto spectrum contains a prescribed set \(\Lambda =\{\lambda _1,\ldots , \lambda _p\}\) of reals. In order to avoid overdetermination one assumes that \(p\) does not exceed \(n^2.\) The inverse Pareto eigenvalue problem under consideration is formulated as an underdetermined system of nonlinear equations. We also address the issue of computing Lorentz spectra and solving inverse Lorentz eigenvalue problems.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. 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 

  4. Chu, M.T., Golub, G.H.: Inverse Eigenvalue Problems: Theory, Algorithms, and Applications. Oxford University Press, New York (2005)

    Book  MATH  Google Scholar 

  5. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  6. Gowda, M.S., Sznajder, R., Tao, J.: Some \(P\)-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393, 203–232 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  8. Forsythe, G.E., Golub, G.H.: On the stationary values of a second-degree polynomial on the unit sphere. J. Soc. Indust. Appl. Math. 13, 1050–1068 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fiedler, M.: Positivity with respect to the round cone. Mat. Casopis Sloven. Akad. Vied 24, 155–159 (1974)

    Google Scholar 

  10. Gajardo, P., Seeger, A.: Reconstructing a matrix from a partial sampling of Pareto eigenvalues. Comput. Optim. Appl. 51, 1119–1135 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gander, W.: Least squares with a quadratic constraint. Numer. Math. 36, 291–307 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gander, W., Golub, G.H., von Matt, U.A.: A constrained eigenvalue problem. Linear Algebra Appl. 114(115), 815–839 (1989)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. 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. Algor. 47, 391–407 (2008)

    Article  MATH  Google Scholar 

  17. 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 

  18. 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 

  19. Loewy, R., Schneider, H.: Positive operators on the \(n\)-dimensional ice cream cone. J. Math. Anal. Appl. 49, 375–392 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Quittner, P.: Spectral analysis of variational inequalities. Comment. Math. Univ. Carolin. 27, 605–629 (1986)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. 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 

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

  30. 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 

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Acknowledgments

The first author has been supported by Projet Fondecyt Nr. 1080173 (Chile) and “Programa de Financiamiento Basal” from the Center of Mathematical Modeling, Universidad de Chile. He thanks also the University of Avignon for the hospitality and working facilities offered during a visit at this institution.

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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

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  1. Pedro Gajardo
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  2. Alberto Seeger
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Correspondence to Alberto Seeger.

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Gajardo, P., Seeger, A. Solving inverse cone-constrained eigenvalue problems. Numer. Math. 123, 309–331 (2013). https://doi.org/10.1007/s00211-012-0487-3

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  • DOI: https://doi.org/10.1007/s00211-012-0487-3

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