A theoretical and computational investigation of a generalized Polak-Ribière algorithm for unconstrained optimization

In this thesis, a new conjugate gradient type method for unconstrained minimization is proposed and its theoretical and computational properties investigated. This generalized Polak-Ribiere method is based on the study of the effects of inexact line searches on conjugate gradient methods. It uses search directions which are parallel to the Newton direction of the restriction of the objective function on a two dimensional subspace spanned by the current gradient and a suitably chosen direction in the span of the previous search direction and the current gradient. It is shown that the GPR method (as it is called) has excellent convergence properties under very simple conditions. An algorithm for the new method is formulated and various implementations of this algorithm are tested. The results show that the GPR algorithm is very efficient in terms of number of iterations as well as computational labour and has modest computer storage requirements. The thesis also explores extensions of the GPR algorithm by considering multi-term restarting procedures. Further generalization of the GPR method based on (m + 1)-dimensional Newton methods is also studied. Optimized software for the implementation of the GPR algorithm is developed for general purpose use. By considering standard test problems, the superiority of the proposed software over some readily available library software and over the straight-forward Polak-Ribiere algorithm is shown. Software and user interfaces together with a simple numerical example and some more practical examples are described for the guidance of the user.