Behavior of the combination of PRP and HZ methods for unconstrained optimization

Abstract

To achieve a conjugate gradient method which is strong in theory and efficient in practice for solving unconstrained optimization problem, we propose a hybridization of the Hager and Zhang (HZ) and Polak-Ribiere and Polyak (PRP) conjugate gradient methods which possesses an important property of the well known PRP method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, averting a sequence of tiny steps from happening, the new scalar \begin{document}$ \beta_k $\end{document} is obtained by convex combination of PRP and HZ under the wolfe line search we prove the sufficient descent and the global convergence. Numerical results are reported to show the effectiveness of our procedure.