Variational quasi-Newton methods for unconstrained optimization

In this paper, we propose new members of the Broyden family of quasi-Newton methods. We develop, on the basis of well-known least-change results for the BFGS and DFP updates, a measure for the Broyden family which seeks to take into account the change in both the Hessian approximation and its inverse. The proposal is then to choose the formula which gives the least value of this measure in terms of the two parameters available, and hence to produce an update which is optimal in the sense of the given measure. Several approaches to the problem of minimizing the measure are considered, from which new updates are obtained. In particular, one approach yields a new variational result for the Davidon optimally conditioned method and another yields a reasonable modification to this method. The paper is also concerned with the possibility of estimating, in a certain sense, the size of the eigenvalues of the Hessian approximation on the basis of two available scalars. This allows one to derive further modifications to the above-mentioned methods. Comparisons with the BFGS and Davidson methods are made on a set of standard test problems that show promising results for certain new methods.