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On the uniform nonsingularity of matrices of search directions and the rate of convergence in minimization algorithms

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Abstract

A convergent minimization algorithm made up of repetitive line searches is considered in ℝn. It is shown that the uniform nonsingularity of the matrices consisting ofn successive normalized search directions guarantees a speed of convergence which is at leastn-step Q-linear. Consequences are given for multistep methods, including Powell's 1964 procedure for function minimization without calculating derivatives as well as Zangwill's modifications of this procedure.

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References

  1. Ostrowski, A.,Contribution to the Theory of Steepest Descent, Archive for Rational Mechanics, Vol. 26, pp. 257–280, 1967.

    Google Scholar 

  2. Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.

    Google Scholar 

  3. Tornheim, L.,Convergence of Multipoint Iterative Methods, Journal of the ACM, Vol. 11, pp. 210–220, 1964.

    Google Scholar 

  4. Schuller, G.,On the Order of Convergence of Certain Quasi-Newton Methods, Numerische Mathematik, Vol. 23, pp. 181–192, 1974.

    Google Scholar 

  5. Powell, M. J. D.,An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives, Computer Journal, Vol. 7, pp. 155–162, 1964.

    Google Scholar 

  6. Zangwill, W. I.,Minimizing a Function without Calculating Derivatives, Computer Journal, Vol. 10, pp. 293–296, 1967.

    Google Scholar 

  7. Polak, E. L.,Computational Methods in Optimization, Academic Press, New York, New York, 1971.

    Google Scholar 

  8. Toint, P. L., andCallier, F. M.,On the Accelerating Property of an Algorithm for Function Minimization without Calculating Derivatives, Journal of Optimization Theory and Applications, Vol. 23, No. 4, 1977.

  9. Vainberg, M. M.,Variational Methods for the Study of Nonlinear Operators, Holden-Day, San Francisco, California, 1964.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Facultés Universitaires de Namur, Namur, Belgium

    P. L. Toint (Assistant) & F. M. Callier (Associate Professor)

  2. Belgian National Fund for Scientific Research, Brussels, Belgium

    F. M. Callier (Associate Professor)

Authors
  1. P. L. Toint
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  2. F. M. Callier
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Additional information

Communicated by P. Varaiya

The authors wish to thank the Namur Department of Mathematics, especially its optimization group, for many discussions and encouragement. They also thank the reviewers for many helpful suggestions.

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Toint, P.L., Callier, F.M. On the uniform nonsingularity of matrices of search directions and the rate of convergence in minimization algorithms. J Optim Theory Appl 23, 511–529 (1977). https://doi.org/10.1007/BF00933294

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  • DOI: https://doi.org/10.1007/BF00933294

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