A family of variable-metric methods derived by variational means

TLDR

The new method preserves the positive‑definiteness of the approximating matrix, like the Davidon‑Fletcher‑Powell (DFP) variable‑metric method. Using Greenstadt’s variational approach, the authors derive a rank‑two variable‑metric method and specify choices for the inverse weighting matrix that directly yield the DFP and rank‑one methods. The method generates a one‑parameter family of variable‑metric methods that includes the DFP and rank‑one methods as special cases and is equivalent to Broyden’s one‑parameter family. Published in Math.

Abstract

A new rank-two variable-metric method is derived using Greenstadt’s variational approach [<italic>Math. Comp.</italic>, 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 [<italic>Math. Comp. </italic>, 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.