On the use of function-values in unconstrained optimisation (Q1262703)

By the use of a nonlinear model for the gradient of the objective function along a chosen direction it is shown how information in the form of function values may be utilised in optimization methods. The presented algorithmic outline provides a framework into which most ``quasi-Newton'' methods (for example, the DFP method, the BFGS method and the Hoshino method) may be fitted. The numerical experiments indicate that such an approach may lead to improvements in the performance of the BFGS algorithm, at the cost of the solution of a simple nonlinear equation in one variable at each iteration.