A Globally Convergent Improved Hybrid Method of Quasi Newton and Conjugate Gradient Method

A new algorithm which combines the Quasi-Newton (QN) method and conjugate gradient (CG) method are presented in this paper. It is denoted as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{QN}-\mathbf{WAM}+$</tex> method and tested using twenty-six different standard test functions from small scale to large scale with four different initial points. All computations are conducted through Matlab software. It is then compared with the other hybrid methods based on number of iterations and CPU times and then analyzed using performance profile. From numerical results, it shows that the new algorithm with BFGS update formula as an approximation of Hessian matrix perform well and proved to be more efficient compared to other hybrid methods. Accordingly, the global convergent is theoretically prove and fulfil sufficient descent condition.