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Mathematics of Computation

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A family of variable-metric methods derived by variational means


Author: Donald Goldfarb
Journal: Math. Comp. 24 (1970), 23-26
MSC: Primary 65.30
DOI: https://doi.org/10.1090/S0025-5718-1970-0258249-6
MathSciNet review: 0258249
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Abstract: A new rank-two variable-metric method is derived using Greenstadt's variational approach [Math. Comp., this issue]. Like the Davidon-Fletcher-Powell (DFP) variable-metric method, the new method preserves the positive-definiteness of the approximating matrix. Together with Greenstadt's method, the new method gives rise to a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. It is equivalent to Broyden's one-parameter family [Math. Comp., v. 21, 1967, pp. 368-381]. Choices for the inverse of the weighting matrix in the variational approach are given that lead to the derivation of the DFP and rank-one methods directly.


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DOI: https://doi.org/10.1090/S0025-5718-1970-0258249-6
Keywords: Unconstrained optimization, variable-metric, variational methods, Davidon method, rank-one formulas
Article copyright: © Copyright 1970 American Mathematical Society