Metric-Based Parameterizations for Multi-Step Unconstrained Optimization

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We consider multi-step quasi-Newton methods for unconstrained optimization. These methods were introduced by Ford and Moghrabi (Appl. Math., vol. 50, pp. 305–323, 1994s Optimization Methods and Software, vol. 2, pp. 357–370, 1993), 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 multi-step methods makes use of the current approximation to the Hessian to determine the parameterization of the interpolating curve in the variable-space and, hence, the generalized updating formula. In this paper, we investigate new parameterization techniques to the approximate Hessian, in an attempt to determine a better Hessian approximation at each iteration and, thus, improve the numerical performance of such algorithms.