Three-step fixed-point quasi-Newtonmethods for unconstrained optimisation

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Multistep quasi-Newton methods were introduced by Ford and Moghrabi [1]. They address the problem of the unconstrained minimisation of a function f : ℝ n → ℝ whose gradient and Hessian are denoted by g and G, respectively. These methods generalised the standard construction of quasi-Newton methods and were based on employing interpolatory polynomials to utilise information from more than one previous step. In a series of papers, Ford and Moghrabi [2–5] have developed various techniques for determining the parametrisation of the interpolating curves. In [2], they introduced two-step metric-based methods which determine the set of parameter values required through measuring distances between various pairs of the iterates employed in the current interpolation. One of the most successful methods in [2] was found to be in the “fixed-point” class, in which the parametrisation of the interpolating curve is determined, at each iteration, by reference to distances measured from a fixed iterate. As suggested in [1], multistep quasi-Newton methods can be constructed for any number of steps.In this paper, we therefore extend the previous work by describing the development of some three-step methods which use the “fixed-point” approach and data derived from the latest four iterates. The experimental results provide evidence that the new methods offer a significant improvement in performance when compared with the standard BFGS method and the unit-spaced three-step method, particularly as the dimension of the test problems grows.