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On the accelerating property of an algorithm for function minimization without calculating derivatives

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Abstract

This paper studies the speed of convergence of a general algorithm for function minimization without calculating derivatives. This algorithm contains Powell's 1964 algorithm as well as Zangwill's second modification of this procedure. The main results are Theorems 3.1 and 4.1 which show that, if the algorithm behaves well, then asymptotically almost conjugate directions are built; therefore, the algorithm has an every-iteration superlinear speed of convergence. The paper hinges on ideas of McCormick and Ritter and Powell.

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References

  1. 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 

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

    Google Scholar 

  3. Brent, R. P.,Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.

    Google Scholar 

  4. Himmelblau, D. M.,Applied Nonlinear Programming, McGraw Hill Book Company, New York, New York, 1972.

    Google Scholar 

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

    Google Scholar 

  6. McCormick, G. P., andRitter, K.,Methods of Conjugate Directions versus Quasi-Newton Methods, Mathematical Programming, Vol. 3, pp. 101–116, 1972.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  9. Toint, P. L., andCallier, F. M.,On the Uniform Nonsingularity of Matrices of Search Directions and the Rate of Convergence in Minimization Algorithms, Journal of Optimization Theory and Applications, Vol. 23, No. 4, 1977.

  10. Toint, P. L., andCallier, F. M.,On the Accelerating Property of an Algorithm for Function Minimization without Calculating Derivatives, Facultés Universitaires de Namur, Namur, Belgium, Département de Mathématique, Report No. 75/2, 1975.

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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 encouragements. The authors also thank the reviewer for many helpful suggestions.

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Toint, P.L., Callier, F.M. On the accelerating property of an algorithm for function minimization without calculating derivatives. J Optim Theory Appl 23, 531–547 (1977). https://doi.org/10.1007/BF00933295

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