Nonlinear Conjugate Gradient Coefficients with Exact and Strong Wolfe Line Searches Techniques

Abstract

Nonlinear conjugate gradient (CG) methods are very important for solving unconstrained optimization problems. These methods have been subjected to extensive researches in terms of enhancing them. Exact and strong Wolfe line search techniques are usually used in practice for the analysis and implementation of conjugate gradient methods. For better results, several studies have been carried out to modify classical CG methods. The method of Fletcher and Reeves (FR) is one of the most well-known CG methods. It has strong convergence properties, but it gives poor numerical results in practice. The main goal of this paper is to enhance this method in terms of numerical performance via a convexity type of modification on its coefficient ${\beta }_k$ . We ensure that with this modification, the method is still achieving the sufficient descent condition and global convergence via both exact and strong Wolfe line searches. The numerical results show that this modified FR is more robust and effective.