An optimal parameter for the Generalized Descent Symmetrical Hestenes-Stiefel algorithm

We have proposed a Generalized Descent Symmetrical Hestenes-Stiefel algorithm [12] , GDSHS for short, which can generate sufficient descent directions for the objective function. Using the Wolfe line search conditions, the global convergence property of the method is also obtained based on the spectral analysis of the conjugate gradient iteration matrix and the Zoutendijk condition for steepest descent methods. I propose in this paper a theoretical choice to improve the performance of the GDSHS algorithm, by the use of an optimal parameter. Based on this, some descent algorithms are developed. 86 numerical experiments are presented to verify their performance and the numerical results show that the new conjugate gradient method GDSHS with the parameter c =1 , denoted GDSHS1, is competitive with GDSHS algorithms that have a parameter c chosen in the interval ] 0 , + ∞ [ .