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.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.