A New Derivation of Symmetric Positive Definite Secant Updates.
Robert B. Schnabel, J. E. Dennis, Jr. . · 1980
In this paper, we introduce a simple new set of techiques for deriving symmetric and positive definite secant updates. We use these techniques to present a simple new derivation of the BFGS update using neither matrix inverses nor weighting matrices. A related derivation is shown to generate a large class of symmetric rank-two update formulas, together with the condition for each to preserve positive definiteness. We apply our techniques to generate a new projected BFGS update, and indicate applications to the efficient implementation of secant algorithms via the Cholsky factorization. (Author)