A new family of multi-step quasi-Newton algorithms for unconstrained optimization

Abstract

This work aims at ensuring smoothness of interpolation in both the iterate and the gradient spaces in the so-called multi-step quasi-Newton methods. It concentrates on deriving a variable-metric family of minimum curvature algorithms for unconstrained optimization. The derivation is based on considering a rational model, with a certain tuning parameter, where the aim is to develop a general framework that encompasses all possible two-step minimum curvature algorithms generated by appropriate parameter choices. One member of the family is tested against earlier developed algorithms of the multi-step type. Performance improvement is evident in our presented results, thus verifying the importance of the minimum curvature framework