Computational Optimization and Applications

, Volume 50, Issue 2, pp 351–378 | Cite as

A smoothing Newton-type method for solving the L 2 spectral estimation problem with lower and upper bounds



This paper discusses the L 2 spectral estimation problem with lower and upper bounds. To the best of our knowledge, it is unknown if the existing methods for this problem have superlinear convergence property or not. In this paper we propose a nonsmooth equation reformulation for this problem. Then we present a smoothing Newton-type method for solving the resulting system of nonsmooth equations. Global and local superlinear convergence of the proposed method are proved under some mild conditions. Numerical tests show that this method is promising.


L2 spectral estimation Smoothing Newton-type method Semismoothness Superlinear convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bell, R.J.: Introductory Fourier Transform Spectroscopy. Academic Press, New York (1972) Google Scholar
  2. 2.
    Ben-Tal, A., Borwein, J.M., Teboulle, M.: A dual approach to multidimension L p spectral estimation problems. SIAM J. Control Optim. 26, 985–996 (1988) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Borwein, J.M., Lewis, A.S.: Dual relationships for entropy-like minimization problems. SIAM J. Control Optim. 29, 325–333 (1991) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Borwein, J.M., Lewis, A.S.: Partially finite convex programming I: Quasi relative interiors and duality theory; II: Explicit lattice models. Math. Program. 57, 15–83 (1992) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Borwein, J.M., Wolkowicz, H.: A simple constraint qualification in infinite dimensional programming. Math. Program. 35, 83–96 (1986) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Burg, J.P.: Maximum entropy spectral analysis. Presented at the 37th International Meeting of the Society of Exploratory Geophysics, Oklahoma City, OK, October 1968 Google Scholar
  7. 7.
    Burg, J.P.: Maximum entropy spectral analysis. Ph.D. Dissertation, Stanford University, CA, May 1975 Google Scholar
  8. 8.
    Chamberlain, J.E.: The Principles of Interferometric Spectroscopy. Wiley, New York (1979) Google Scholar
  9. 9.
    Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67, 519–540 (1998) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2000) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) MATHGoogle Scholar
  12. 12.
    Cole, R.E., Goodrich, R.K.: L p-spectral estimation with an L -upper bound. J. Optim. Theory Appl. 76, 321–355 (1993) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dickinson, B.W.: Two-dimensional Markov spectrum estimates need not exist. IEEE Trans. Inf. Theory 22, 120–121 (1980) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dontchev, A.L., Qi, H., Qi, L.: Convergence of Newton’s method for convex best interpolation. Numer. Math. 87, 435–456 (2001) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Dontchev, A.L., Qi, H., Qi, L., Yin, H.: A Newton method for shape-preserving spline interpolation. SIAM J. Optim. 13, 588–602 (2003) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dontchev, A.L., Qi, H., Qi, L.: Quadratic convergence of Newton’s method for convex interpolation and smoothing. Constr. Approx. 19(1), 123–143 (2003) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85, 107–134 (1999) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Goodrich, R.K., Steinhardt, A.: L 2 spectral estimation. SIAM J. Appl. Math. 46, 417–426 (1986) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kay, S.M., Marple, S.L.: Spectrum analysis-a modern perspective. Proc. IEEE 69, 1380–1419 (1981) CrossRefGoogle Scholar
  20. 20.
    Landau, H.J.: Maximum entropy and maximum likelihood in spectral estimation. IEEE Trans. Inf. Theory 44, 1332–1336 (1998) MATHCrossRefGoogle Scholar
  21. 21.
    Lang, S.W., McClellan, J.H.: Spectral estimation for sensor arrays. IEEE Trans. Acoust. Speech Signal Process. 31, 349–358 (1983) MATHCrossRefGoogle Scholar
  22. 22.
    Qi, H.D., Liao, L.Z.: A smoothing Newton method for general nonlinear complementarity problems. Comput. Optim. Appl. 17, 231–253 (2000) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Qi, L., Ling, C., Tong, X., Zhou, G.: A smoothing projected Newton-type algorithm for semi-infinite programming. Comput. Optim. Appl. 42, 1–30 (2009) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Optim. 15, 957–972 (1977) MathSciNetGoogle Scholar
  25. 25.
    Pang, J.-S., Qi, L.: Nonsmooth equation: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Potter, L.C.: Constrained signal reconstruction. Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1990 Google Scholar
  27. 27.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Qi, L., Jiang, H.: Semismooth Karush-Kuhn-Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. Oper. Res. 22, 301–325 (1997) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Qi, L., Sun, D.: Smoothing functions and smoothing Newton method for complementarity and variational inequality problems. J. Optim. Theory Appl. 113, 121–147 (2002) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Qi, L., Sun, D., Zhou, G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Program. 87, 1–35 (2000) MathSciNetMATHGoogle Scholar
  32. 32.
    Qi, L., Shapiro, A., Ling, C.: Differentiability and semismoothness properties of integral functions and their applications. Math. Program. 102, 223–248 (2005) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Stewart, J.: Positive definite functions and generalizations: an historical survey. Rocky Mt. J. Math. 6, 409–434 (1976) MATHCrossRefGoogle Scholar
  34. 34.
    Sun, D.: A further result on an implicit function theorem for locally Lipschitz functions. Oper. Res. Lett. 28, 193–198 (2001) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Yin, H.X., Ling, C., Qi, L.: A semismooth Newton method for L 2 spectral estimation. Math. Program. 107, 539–546 (2006) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of ScienceHangzhou Dianzi UniversityHangzhouChina
  2. 2.Department of Mathematics and StatisticsMinnesota State University MankatoMankatoUSA
  3. 3.Department of Mathematics and StatisticsCurtin University of TechnologyPerthAustralia

Personalised recommendations