Implicit updates in multistep quasi-Newton methods

We consider multistep quasi-Newton methods for unconstrained optimization. These methods were introduced by Ford and Moghrabi [1,2], who showed how in terpolating curves could be used to derive a generalization of the secant e quation (the relation normally employed in the construction of quasi-Newton methods). One of the most successful of these multistep methods makes use of the current approximation to the Hessian to determine the parametrizatio n of the interpolating curve in the variable-space and, hence, the generali zed updating formula. In this paper, we investigate the use of implicit upd ates to the approximate Hessian, in an attempt to determine a better parame trization of the interpolation (while avoiding the computational burden of actually carrying out the update) and, thus, improve the numerical performa nce of such algorithms. (C) 2001 Elsevier Science Ltd. All rights reserved.