| Summary: | The quasi-Newton method is used extensively in solving unconstrained
optimization problems. The most popular methods in quasi-Newton are
Broyden-Fletcher-Goldfarb-Shanno (BFGS) and Broyden Family method. The
convergence of quasi-Newton method depends on three factors which are
search direction, step size and approximation of Hessian. In this research,
three new search directions of quasi-Newton method have been introduced.
The suggested search directions are hybrid quasi-Newton method with
steepest descent method, hybrid quasi-Newton method with conjugate
gradient method and also quasi-Newton method by employing the conjugate
gradient coefficient. The three search directions are named as quasi-Newtonsteepest
descent method (ON-SD), quasi-Newton-conjugate gradient method
(ON-CG) and M-quasi-Newton method (M-ON) respectively. It is proven
theoretically that all these new search direction formulas fulfilled the sufficient
descent condition and global convergence properties. The search direction is
tested based on twenty four standard problems of unconstrained optimization.
The selected standard problems cover from the small scale to large scale
problems. Each of them is tested using three different initial points with a total
of 180 problems using Matlab 2012. The three initial points used ranges from
a point closest to the optimal point to the furthest. The effectiveness of these
proposed search direction is compared with the original quasi-Newton's
search direction. Numerical results obtained from the Matlab programming
was based on the number of iteration, number of function evaluations and
CPU-time. These results were then analyzed using performance profile by
Dolan and More. The result showed that all the proposed search direction for
quasi-Newton method indeed is effective compared to the original method.
These results also show that these methods possess global convergence
properties. It is also shown that the numerical results concur with the
theoretical proof.
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