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Nonnegative inverse eigenvalue problems with partial eigendata

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Abstract

In this paper we consider the inverse problem of constructing an n × n real nonnegative matrix A from the prescribed partial eigendata. We first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem. To find a nonnegative solution, we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth Newton-type method for solving its equivalent nonsmooth equation. Under some mild assumptions, the global and quadratic convergence of our method is established. We also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries. Numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings.

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References

  1. Bapat, R.B., Raghavan, T.E.S.: Nonnegative matrices and applications. In: Encyclopedia of Mathematics and its Applications, vol. 64. Cambridge University Press, Cambridge (1997)

  2. Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. In: Classics in Applied Mathematics, vol. 9. SIAM, Philadelphia (1994)

  3. Bonnans J.F., Shapiro A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    MATH  Google Scholar 

  4. Chen B., Harker P.T.: Smooth approximations to nonlinear complementarity problems. SIAM J. Optim. 7, 403–420 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen X., Liu D.L.: Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure. J. Comput. Appl. Math. 235, 3990–4002 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chu M.T., Diele F., Sgura I.: Gradient flow method for matrix completion with prescribed eigenvalues. Linear Algebra Appl. 379, 85–112 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chu M.T., Driessel K.R.: Constructing symmetric nonnegative matrices with prescribed eigenvalues by differential equations. SIAM J. Math. Anal. 22, 1372–1387 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chu M.T., Golub G.H.: Structured inverse eigenvalue problems. Acta Numer. 11, 1–71 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  10. Chu M.T., Guo Q.: A numerical method for the inverse stochastic spectrum problem. SIAM J. Matrix Anal. Appl. 19, 1027–1039 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. Datta B.N.: Numerical Methods for Linear Control Systems: Design and Analysis. Elsevier, Amsterdam (2003)

    Google Scholar 

  13. De Luca T., Facchinei F., Kanzow C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eaves B.C.: On the basic theorem for complemenarity. Math. Program. 1, 68–75 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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 

  16. Facchinei F., Soares J.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225–247 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fischer A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fischer A.: On the local superlinear convergence of a Newton-type method for LCP under weak conditions. Optim. Methods Softw. 6, 83–107 (1995)

    Article  Google Scholar 

  19. Fischer A.: Solution of monotone complementarity problems with Lipschitzian functions. Math. Program. 76, 513–532 (1997)

    MATH  Google Scholar 

  20. Freund R.W.: A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Comput. 14, 470–482 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Freund R.W., Nachtigal N.M.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60, 315–339 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Friswell M.I., Mottershead J.E.: Finite Element Model Updating in Structural Dynamics. Kluwer, Dordrecht (1995)

    MATH  Google Scholar 

  23. Geiger C., Kanzow C.: On the resolution of monotone complementarity problems. Comput. Optim. Appl. 5, 155–173 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gladwell G.M.L.: Inverse Problems in Vibration. Kluwer, Dordrecht (2004)

    Google Scholar 

  25. Jiang H.Y., Qi L.Q.: A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J. Control Optim. 35, 178–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kanzow C.: Some noninterior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17, 851–868 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Meng F.W., Sun D.F., Zhao G.Y.: Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization. Math. Program. 104, 561–581 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957–972 (1977)

    Article  MathSciNet  Google Scholar 

  29. Minc H.: Nonnegative Matrices. Wiley, New York (1988)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  31. Orsi R.: Numerical methods for solving inverse eigenvalue problems for nonnegative matrices. SIAM J. Matrix Anal. Appl. 28, 190–212 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pang J.S., Sun D.F., Sun J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28, 39–63 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Qi L.Q., Jiang H.Y.: Semismooth Karush-Kuhn-Tucker equations and convergence analysis of Newton methods and quasi-Newton methods for solving these equations. Math. Oper. Res. 22, 301–325 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Robinson S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. Robinson S.M.: Local structure of feasible sets in nonlinear programming. Part II: nondegeneracy. Math. Program. Study 22, 217–230 (1984)

    MATH  Google Scholar 

  37. Rockafellar R.T.: Convex Analysis. Princeton, New Jersey (1970)

    MATH  Google Scholar 

  38. Rockafellar R.T., Wets R.J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  39. Saad Y., Schultz M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  40. Senata E.: Non-negative Matrices and Markov Chains, 2nd rev. edn. Springer, New York (2006)

    Google Scholar 

  41. Shapiro A.: Sensitivity analysis of generalized equations. J. Math. Sci. 115, 2554–2565 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sonneveld P.: CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10, 36–52 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sun D.F.: The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31, 761–776 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sun D.F., Sun J.: Semismooth matrix valued functions Math. Oper. Res. 27, 150–169 (2002)

    MATH  Google Scholar 

  45. Sun J.G.: Backward perturbation analysis of certain characteristic subspaces. Numer. Math. 65, 357–382 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  46. van der Vorst H.A.: Bi-CGSTAB: a fast and smoothly converging variant of the BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)

    Article  MATH  Google Scholar 

  47. Xu S.F.: An Introduction to Inverse Eigenvalue Problems. Peking University Press and Vieweg Publishing, Beijing (1998)

    MATH  Google Scholar 

  48. Yamashita N., Fukushima M.: Modified Newton methods for solving semismooth reformulations of monotone complementarity problems. Math. Program. 76, 469–491 (1997)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Zheng-Jian Bai & Zhi Zhao

  2. Dipartimento di Fisica e Matematica, Università dell’Insubria-Sede di Como, Via Valleggio 11, 22100, Como, Italy

    Zheng-Jian Bai & Stefano Serra-Capizzano

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  1. Zheng-Jian Bai
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  2. Stefano Serra-Capizzano
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  3. Zhi Zhao
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Correspondence to Zheng-Jian Bai.

Additional information

The research of Z.-J. Bai was partially supported by the Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars (No. 2010J06002), NCET, and Internationalization Grant of U. Insubria 2008, 2009.

The work of S. Serra-Capizzano was partially supported by MIUR (No. 20083KLJEZ).

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Bai, ZJ., Serra-Capizzano, S. & Zhao, Z. Nonnegative inverse eigenvalue problems with partial eigendata. Numer. Math. 120, 387–431 (2012). https://doi.org/10.1007/s00211-011-0415-y

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  • DOI: https://doi.org/10.1007/s00211-011-0415-y

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