Least Change Secant Updates for Quasi-Newton Methods

In many problems involving the solution of a system of nonlinear equations, it is necessary to keep an approximation to the Jacobian matrix which is updated at each iteration. Computational experience indicates that the best updates are those that minimize some reasonable measure of the change to the current Jacobian approximation subject to the new approximation obeying a secant condition and perhaps some other approximation properties such as symmetry. In this paper we extend the affine case of a theorem of Cheney and Goldstein on proximity maps of convex sets to show that a generalization of the symmetrization technique of Powell always generates least change updates. This generalization has such broad applicability that we obtain an easy unified derivation of all the most successful updates. Furthermore, our techniques apply to interesting new cases such as when the secant condition might be inconsistent with some essential approximation property like sparsity. We also offer advice on how to choose the properties which are to be incorporated into the approximations and how to choose the measure of changes to be minimized.