Alternating multi-step quasi-Newton methods for unconstrained optimization

We consider multi-step quasi-Newton methods for unconstrained optimization. These methods were introduced by the authors (Ford and Moghrabi [5, 6, 8]), who showed how an interpolating curve in the variable-space could be used to derive an appropriate generalization of the Secant Equation normally employed in the construction of quasi-Newton methods. One of the most successful of these multi-step methods employs the current approximation to the Hessian to determine the parametrization of the interpolating curve and, hence, the derivatives which are required in the generalized updating formula. However, certain approximations were found to be necessary in the process, in order to reduce the level of computation required (which must be repeated at each iteration) to acceptable levels. In this paper, we show how a variant of this algorithm, which avoids the need for such approximations, may be obtained. This is accomplished by alternating, on successive iterations, a single-step and a two-step method. The results of a series of experiments, which show that the new algorithm exhibits a clear improvement in numerical performance, are reported.