Implicit updates in multistep quasi-Newton methods

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We consider multistep quasi-Newton methods for unconstrained optimization. These methods were introduced by Ford and Moghrabi [1,2], who showed how interpolating curves could be used to derive a generalization of the secant equation (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 parametrization of the interpolating curve in the variable-space and, hence, the generalized updating formula. In this paper, we investigate the use of implicit updates to the approximate Hessian, in an attempt to determine a better parametrization of the interpolation (while avoiding the computational burden of actually carrying out the update) and, thus, improve the numerical performance of such algorithms.