A new minimum curvator multi-step method for unconstrained optimization

Abstract

Multistep quasi-Newton methods for optimization were derived by J. A. Ford and I. A. Moghrabi [J. Comput. Appl. Math. 50, No. 1-3, 305-323 (1994; Zbl 0807.65062)], where it was shown how an interpolation in the variable-space could be used to generate “better” Hessian approximations. The work presented by J. A. Ford and I. A. Moghrabi [Comput. Math. Appl. 31, No. 4-5, 179-186 (1996; Zbl 0874.65046)] concentrated a choice of the curve parameters that ensure a “smooth” interpolation. In this paper, we carry on with a similar idea and define a rational model with a free parameter. Our derivation of the new algorithm is based on determining some value of the parameter that minimizes the curvature in some chosen metric. It is shown how such value can be “cheaply” calculated at each iteration. Numerical comparison between the new algorithm and other multistep algorithms reveal the merits of the new approach.