Skip to main content
SpringerLink
Log in

A family of variable metric updates

  • Published:
Mathematical Programming Submit manuscript

Abstract

We develop a one parameter family of variable metric updates by considering a fundamental decomposition of the Hessian that underlies Variable Metric Algorithms. The relationship with other Variable Metric Updates is discussed. Considerations based on the condition of the Hessian inverse approximation indicate particular choices of the parameter and these are discussed in the second half of this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W.C. Davidon, “Variable metric method for minimization”, AEC Research and Development Rept., Argonne National Laboratory, ANL-5990 (Rev.) (1959).

  2. R. Fletcher and M.J.D. Powell, “A rapidly convergent descent method for minimization”.The Computer Journal 7 (1963) 149–154.

    Google Scholar 

  3. C.G. Broyden, “The convergence of a class of double rank minimization algorithms”,Journal of the Institute of Mathematics and its Applications 6 (1970) 76–90.

    Google Scholar 

  4. H.U. Huang, “Unified approach to quadratically convergent algorithms for function minimization,”Journal of Optimization Theory and Applications 5 (1970) 405–423.

    Google Scholar 

  5. L.C.W. Dixon, “Quasi-Newton techniques generate identical points, II: The proofs of four new theorems”,Mathematical Programming 3 (1972) 345–358.

    Google Scholar 

  6. J.L. Nazareth, “Unified approach to unconstrained optimization via basic matrix factorizations”,Journal of Linear Algebra and Applications, to appear.

  7. P.E. Gill and W. Murray, “Quasi-Newton methods for unconstrained optimization”,Journal of the Institute of Mathematics and its Applications 9 (1971) 91–108.

    Google Scholar 

  8. Y. Bard, “On a numerical instability of Davidon-like methods”,Mathematics of Computation 22 (1968) 665–666.

    Google Scholar 

  9. D.F. Shanno and P.C. Kettler, “Optimal conditioning of quasi-Newton Methods”,Mathematics of Computation 24 (1970) 657–664.

    Google Scholar 

  10. D.F. Shanno, “Conditioning of quasi-Newton methods for function minimization”,Mathematics of Computation 24 (1970) 647–656.

    Google Scholar 

  11. E. Spedicato, “Stability of Huang's update for the conjugate gradient method”,Journal of Optimization Theory and Applications 11 (1973) 469–479.

    Google Scholar 

  12. M.C. Biggs, “Minimization algorithms making use of nonquadratic properties of the objective functions”,Journal of the Institute of Mathematics and its Applications 8 (1971) 315–327.

    Google Scholar 

  13. S.S. Oren, “On the selection of parameters in self scaling variable metric algorithms”,Mathematical Programming 7 (1974) 351–367.

    Google Scholar 

  14. W.C. Davidon, “Optimally conditioned optimization algorithms without line searches”,Mathematical Programming 9 (1975) 1–30.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Argonne National Laboratory, Arponne, Ill., USA

    Larry Nazareth

Authors
  1. Larry Nazareth
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work performed under the auspices of the U.S. Energy Research and Development Administration.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nazareth, L. A family of variable metric updates. Mathematical Programming 12, 157–172 (1977). https://doi.org/10.1007/BF01593786

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01593786

Keywords

Search

Navigation