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. In the preceding paper [6], Greenstadt derives two variable-metric methods, using a classical variational approach. Specifically, two iterative formulas are developed for updating the matrix Hk, (i.e., the inverse of the variable metric), where Hk is an approximation to the inverse Hessian G_1ixk) of the function being minimized.* Using the iteration formula Hk+x = Hk + Ek to provide revised estimates to the inverse Hessian at each step, Greenstadt solves for the correction term Ek that minimizes the norm NiEk) = Tr iWEkWEY) subject to the conditions