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.