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Mathematics of Computation

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Conditioning of quasi-Newton methods for function minimization

Author: D. F. Shanno
Journal: Math. Comp. 24 (1970), 647-656
MSC: Primary 90.58
MathSciNet review: 0274029
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Abstract: Quasi-Newton methods accelerate the steepest-descent technique for function minimization by using computational history to generate a sequence of approximations to the inverse of the Hessian matrix. This paper presents a class of approximating matrices as a function of a scalar parameter. The problem of optimal conditioning of these matrices under an appropriate norm as a function of the scalar parameter is investigated. A set of computational results verifies the superiority of the new methods arising from conditioning considerations to known methods.

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Keywords: Function minimization, quasi-Newton methods, variable metric methods, gradient search, steepest-descent methods, stability of search methods, conditioning of search methods, Hessian matrix, inverse approximations
Article copyright: © Copyright 1970 American Mathematical Society