A Nonlinear Model for Function-value Multi-step Methods

We develop a framework (employing scaling functions) for the construction of multi-step quasi-Newton methods (for unconstrained optimization) which utilize values of the objective function. The methods are constructed via interpolants of the m+ 1 most recent iterates / gradient evaluations, and possess a free parameter which introduces an additional degree of flexibility. This permits the interpolating polynomials to assimilate information (in the form of function-values) which is readily available at each iteration. This information is incorporated in updating the Hessian approximation at each iteration, in an attempt to accelerate convergence. We concentrate on a specific example from the general family of methods, corresponding to a particular choice of the scaling function, and from it derive three new algorithms. The relative numerical performance of these methods is assessed, and the most successful of them is then compared with the standard BFGS method and with an earlier algori..