The thesis concerns mainly in finding the numerical solution of non-linear
unconstrained problems. We consider a well-known class of optimization methods
called the quasi-Newton methods, or variable metric methods. In particular, a class of
quasi-Newton method named Broyden's single parameter rank two method is
focussed.
We also investigate the global convergence properties for some step-length
procedures. Immediately from the investigations, a global convergence proof of the
Armijo quasi-Newton method is given.
Some preliminary modifications and numerical experiments are carried out to
gain useful numerical experiences for the improvements of the quasi-Ne"-'ton updates.We then derived two improvement techniques: the first we employ a switching
criteria between quasi-Newton Broyden-Fletcher-Goldfrab-Shanno or BFGS and
steepest descent direction and in the second we introduce a reduced trace-norm
condition BFGS update. The thesis includes results illustrating the numerical
performance of the modified methods on a chosen set of test problems.
Limitations and some possible extensions are also given to conclude this thesis
