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Mathematics of Computation

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Variations on variable-metric methods. (With discussion)


Author: J. Greenstadt
Journal: Math. Comp. 24 (1970), 1-22
MSC: Primary 65.30
DOI: https://doi.org/10.1090/S0025-5718-1970-0258248-4
MathSciNet review: 0258248
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Abstract: In unconstrained minimization of a function $ f$, the method of Davidon-FletcherPowell (a "variable-metric" method) enables the inverse of the Hessian $ H$ of $ f$ to be approximated stepwise, using only values of the gradient of $ f$. It is shown here that, by solving a certain variational problem, formulas for the successive corrections to $ H$ can be derived which closely resemble Davidon's. A symmetric correction matrix is sought which minimizes a weighted Euclidean norm, and also satisfies the "DFP condition." Numerical tests are described, comparing the performance (on four "standard" test functions) of two variationally-derived formulas with Davidon's. A proof by Y. Bard, modelled on Fletcher and Powell's, showing that the new formulas give the exact $ H$ after $ N$ steps, is included in an appendix.


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DOI: https://doi.org/10.1090/S0025-5718-1970-0258248-4
Keywords: Variable-metric, Davidon method, unconstrained minimization, variational method
Article copyright: © Copyright 1970 American Mathematical Society