On Symmetric Minimum Norm Updates

The theory of the 1-inverse of a matrix is used to derive allsymmetric updates which satisfy a matrix equation. This generalsolution is then used to obtain the solution which is minimumin a symmetrically weighted Frobenius norm. The minimum normsolution can be used to derive the minimum norm updates usedin quasi-Newton methods for unconstrained optimization. This research was supported in part by NIH Grant No. AM17593.     

A Unified Derivation of Symmetric Quasi-Newton Update Formulas

In this paper a generalization of the quasi-Newton equationis considered. The symmetric solution which is minimum in aweighted Frobenius norm is presented along with the completesymmetric solution. These two solutions are then shown to containa number of symmetric quasi-Newton update formulas includingupdates which are orthogonal to a given subspace.