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.