A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method

Abstract

<p style='text-indent:20px;'>In this paper, we develop a three-term Polak-Ribière-Polyak conjugate gradient method, in which the search direction is close to the direction in the memoryless BFGS quasi-Newton method. The new scheme reduces to the standard Polak-Ribière-Polyak method when an exact line search is used. For any line search, the method satisfies the sufficient descent condition <inline-formula><tex-math id="M1">\begin{document}$g_{k}^{T}d_{k}≤ -{(1-\frac{1}{4}(1+\overline{t})^2})\|g_k\|^2$\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$\overline{t}∈[0,1)$\end{document}</tex-math></inline-formula> is a constant. The global convergence results of the new algorithm are established with suitable line search methods. Numerical results show that the proposed method is efficient for the unconstrained problems in the CUTEr library.