Advertisement

Quasi-Newton Methods

Abstract

In Chap. 6, multidimensional optimization methods were considered in which the search for the minimizer is carried out by using a set of conjugate directions. An important feature of some of these methods (e.g., the Fletcher-Reeves and Powell’s methods) is that explicit expressions for the second derivatives of f(x) are not required. Another class of methods that do not require explicit expressions for the second derivatives is the class of quasi-Newton methods. These are sometimes referred to as variable metric methods.

Keywords

Line Search Descent Direction Gradient Evaluation BFGS Method Real Symmetric Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed., Addison-Wesley, Reading, MA, 1984.MATHGoogle Scholar
  2. 2.
    C. G. Broyden, “Quasi-Newton methods and their application to function minimization,” Maths. Comput., vol. 21, pp. 368–381, 1965.CrossRefMathSciNetGoogle Scholar
  3. 3.
    W. C. Davidon, “Variable metric method for minimization,” AEC Res. and Dev. Report ANL-5990, 1959.Google Scholar
  4. 4.
    A. V. Fiacco and G. P. McCormick, Nonlinear Programming, Wiley, New York, 1968.MATHGoogle Scholar
  5. 5.
    B. A. Murtagh and R. W. H. Sargent, “A constrained minimization method with quadratic convergence,” Optimization, ed. R. Fletcher, pp. 215–246, Academic Press, London, 1969.Google Scholar
  6. 6.
    P. Wolfe, “Methods of nonlinear programming,” Nonlinear Programming, ed. J. Abadie, pp. 97–131, Interscience, Wiley, New York, 1967.Google Scholar
  7. 7.
    R. Fletcher and M. J. D. Powell, “A rapidly convergent descent method for minimization,” Computer J., vol. 6, pp. 163–168, 1963.MATHMathSciNetGoogle Scholar
  8. 8.
    T. Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, N.J., 1980.MATHGoogle Scholar
  9. 9.
    P. E. Gill, W. Murray, and W. H. Wright, Numerical Linear Algebra and Optimization, vol. 1, Addison Wesley, Reading, MA, 1991.MATHGoogle Scholar
  10. 10.
    R. Fletcher, “A new approach to variable metric algorithms,” Computer J., vol. 13, pp. 317–322, 1970.MATHCrossRefGoogle Scholar
  11. 11.
    D. Goldfarb, “A family of variable metric methods derived by variational means,” Maths. Comput., vol. 24, pp. 23–26, 1970.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Maths. Comput., vol. 24, pp. 647–656, 1970.CrossRefMathSciNetGoogle Scholar
  13. 13.
    R. Fletcher, Practical Methods of Optimization, vol. 1, Wiley, New York, 1980.MATHGoogle Scholar
  14. 14.
    R. Fletcher, Practical Methods of Optimization, 2nd ed., Wiley, New York, 1987.MATHGoogle Scholar
  15. 15.
    S. Hoshino, “A formulation of variable metric methods,” J. Inst. Maths. Applns. vol. 10, pp. 394–403, 1972.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    H. Y. Huang, “Unified approach to quadratically convergent algorithms for function minimization,” J. Opt. Theo. Applns., vol. 5, pp. 405–423, 1970.MATHCrossRefGoogle Scholar
  17. 17.
    G. P. McCormick and J. D. Pearson, “Variable metric methods and unconstrained optimization,” in Optimization, ed. R. Fletcher, Academic Press, London, 1969.Google Scholar
  18. 18.
    J. D. Pearson, “Variable metric methods of minimization,” Computer J., vol. 12, pp. 171–178, 1969.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    A. Antoniou, Digital Signal Processing: Signals, Systems, and Filters, McGraw-Hill, New York, 2005.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Personalised recommendations