A family of variable-metric methods derived by variational means

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