Minimum curvature multistep quasi-Newton methods

We consider multistep quasi-Newton methods for unconstrained optimization problems. Such methods were introduced by the authors [1–3], and employ interpolating polynomials to utilize data from the m most recent iterations. For the case m = 2, we observe that there is a free parameter (used in constructing the interpolation) which is essentially at our disposal. We propose (as a criterion for determining this parameter) the minimization of a measure of the curvature of the interpolating polynomial, thus producing a “smooth” interpolant. We show how this “minimum curvature” problem may be solved cheaply and effectively at each iteration. The performance of an algorithm which employs this technique is compared (numerically) with that of the standard BFGS method and of previously-introduced multistep methods [2,3].