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Mathematics of Computation

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Minimum norm symmetric quasi-Newton updates restricted to subspaces

Author: Robert B. Schnabel
Journal: Math. Comp. 32 (1978), 829-837
MSC: Primary 65K10; Secondary 65F30
MathSciNet review: 492041
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Abstract: The Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno updates have been the two most successful quasi-Newton updates for a variety of applications. One reason offered in explanation is that they constitute, in an appropriate norm and metric, the minimum norm change to the matrix, or its inverse, being approximated which preserves symmetry and obeys the quasi-Newton equation. Recent methods have reason to consider updates restricted to certain subspaces. In this paper we derive the general minimum norm symmetric quasi-Newton updates restricted to such subspaces. In the same appropriate norm and metric, the minimum norm change update to the matrix or its inverse is shown to be, respectively, the rank-two update which is a particular projection of the DFP or BFGS onto this subspace.

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Keywords: Quasi-Newton updates, minimum norm change updates
Article copyright: © Copyright 1978 American Mathematical Society

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