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Numerical Algorithms

, Volume 32, Issue 1, pp 57–66 | Cite as

Hybrid Methods for Approximating Hankel Matrix

  • Suliman Al-Homidan
Article

Abstract

Hybrid methods for minimizing least distance functions with Hankel positive semi-definite matrix constraints are considered. Our approach is based on (i) a projection algorithm which converges globally but slowly; and (ii) the Newton method which is faster. Hybrid methods that attempt to combine the best features of both methods are then considered. Comparative numerical results are reported.

alternating projections Hankel matrix least distance functions non-smooth optimization positive semi-definite matrix Newton method 

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References

  1. [1]
    N.I. Akhiezer and M.G. Krein, Some Questions in the Theory of Moments, Translations of Mathematical Monographs, Vol. 2 (Amer. Math. Soc., Providence, RI, 1962).Google Scholar
  2. [2]
    S. Al-Homidan, Hybrid methods for optimization problems with positive semi-definite matrix constraints, Ph.D. thesis, Department of Mathematics and Computer Science, University of Dundee, Dundee, Scotland (1993).Google Scholar
  3. [3]
    S. Al-Homidan and R. Fletcher, Hybrid methods for finding the nearest Euclidean distance matrix, in: Recent Advances in Nonsmooth Optimization, eds. D. Du, L. Qi and R. Womersley (World Scientific, Singapore, 1995) pp. 1–17.Google Scholar
  4. [4]
    J.P. Boyle and R.L. Dykstra, A method for finding projections onto the intersection of convex sets in Hilbert space, in: Advances in Order Restricted Statistical Inference, eds. R. Dykstra, T. Robertson and F.T. Wright, Lecature Notes in Statistics, Vol. 37 (Springer, Berlin, 1986) pp. 28–47.Google Scholar
  5. [5]
    R.L. Dykstra, An algorithm for restricted least squares regression, J. Amer. Statist. Assoc. 78 (1983) 839–842.Google Scholar
  6. [6]
    E. Fischer, Ñber das Caratheodorysche Problem, Potenzreihen mit positivem reellen Teil betreffend, Rend Palermo 32 (1911).Google Scholar
  7. [7]
    R. Fletcher, Practical Methods of Optimization (Wiley, Chichester, 1987).Google Scholar
  8. [8]
    G.H. Golub and V. Pereyra, The differentiation of pseudoinverses and nonlinear problems whose variables separate, SIAM J. Numer. Anal. 10(2) (1973) 413–432.Google Scholar
  9. [9]
    S.P. Han, A successive projection method, Math. Programming 40 (1988) 1–14.Google Scholar
  10. [10]
    N. Higham, Computing a nearest symmetric positive semi-definite matrix, Linear Algebra Appl. 103 (1988) 103–118.Google Scholar
  11. [11]
    I.S. Iohvidov, Toeplitz and Hankel Matrices and Forms: Algebraic Theory (Birkhäuser, Boston, 1987).Google Scholar
  12. [12]
    C.S. MacInnes, The solution to a structured matrix approximation problem using Grassman coordinates, SIAM J. Matrix Anal. Appl. 21 (1999) 446–453.Google Scholar
  13. [13]
    S. Oh and R.J. Marks II, Alternating projection onto fuzzy convex sets, Proc. IEEE 81 (1993) 148–155.Google Scholar
  14. [14]
    J. Ponstein, A matrix minimization problem involving ranks, Linear Algebra Appl. 92 (1987) 81–106.Google Scholar
  15. [15]
    A.K. Shaw and S. Kumaresan, Some structured matrix approximation problems, in: Proc. of Internat. Conf. on Acoustics, Speech, and Signal Proccessing, IEEE ICASSP, Vol. 4, 1988, pp. 2324–2327.Google Scholar
  16. [16]
    A.K. Shaw, S. Pokala and S. Kumaresan, Toeplitz and Hankel matrix approximation using structured approach, Proc. IEEE 86 (1998) 2349–2352.Google Scholar
  17. [17]
    M.J. Todd, Semidefinite optimization, Acta Numerica 10 (2001) 515–560.Google Scholar
  18. [18]
    L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38 (1996) 49–95.Google Scholar
  19. [19]
    H. Wolkwicz, R. Saigal and L. Vandenberghe, Handbook of Semidefinite Programming, Theory, Algorithms, and Applications (Kluwer Academic, Dordrecht, 2000).Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Suliman Al-Homidan
    • 1
  1. 1.Department of Mathematical SciencesKing Fahd University of Petroleum & MineralsDhahranSaudi Arabia

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