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Conditions for variable-metric algorithms to be conjugate-gradient algorithms

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Abstract

Variable-metric algorithms have played an important role in unconstrained optimization theory. This paper presents a sufficiency condition on the sequence of metrics in a variable-metric algorithm that will make it a conjugate-gradient algorithm. The Huang class of algorithms (Ref. 1) and the class of self-scaling variable-metric algorithms by Oren (Ref. 2) all satisfy the condition. This paper also includes a discussion of the behavior of algorithms that meet the condition on nonquadratic functions.

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References

  1. Huang, H. Y.,Unified Approach to Quadratically Convergent Algorithms for Function Minimization, Journal of Optimization Theory and Applications, Vol. 5, No. 6, 1970.

  2. Oren, S. S.,Self-Scaling Variable-Metric Algorithms without Line Search for Unconstrained Minimization, Mathematics of Computation, Vol. 27, No. 124, 1973.

  3. Dixon, L. C. W.,Variable-Metric Algorithms: Necessary and Sufficient Conditions for Identical Behavior of Nonquadratic Functions, Journal of Optimization Theory and Applications, Vol. 10, No. 1, 1972.

  4. Hestenes, M. R., andStiefel, E.,Method of Conjugate Gradients for Solving Linear Systems, Journal of Research of the National Bureau of Standards, Vol. 49, No. 6, 1952.

  5. Davidon, W. C.,Variable-Metric Method for Minimization, Argonne National Laboratory, Report No. ANL-5990, 1959.

  6. Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, No. 2, 1963.

  7. Huang, H. Y., andLevy, A. V.,Numerical Experiments on Quadratically Convergent Algorithms for Function Minimization, Journal of Optimization Theory and Applications, Vol. 6, No. 3, 1970.

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Authors and Affiliations

  1. Mathematics Department, Central Michigan University, Mount Pleasant, Michigan

    D. G. McDowell (Associate Professor)

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  1. D. G. McDowell
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Communicated By H. Y. Huang

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McDowell, D.G. Conditions for variable-metric algorithms to be conjugate-gradient algorithms. J Optim Theory Appl 41, 439–450 (1983). https://doi.org/10.1007/BF00935363

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  • DOI: https://doi.org/10.1007/BF00935363

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