Modified globally convergent Polak-Ribière-Polyak conjugate gradient methods with self-correcting property for large-scale unconstrained optimization

In this paper, we propose a modified Polak-Ribière-Polyak conjugate gradient method. Different from the existent methods, a damping factor is introduced to monitor jamming and make instantaneous changes to incorrect predictions, and the resulting damping term is significantly embedded into the conjugate gradient parameter to self-adjust weight between self-correcting property and global convergence. A bigger value is preferred to control the magnitude of conjugate gradient parameter being less than that of the Fletcher-Reeves method, which is a benefit to sufficient descent condition of the search directions; also, to a certain, a dramatically shrinking value makes the parameter above to approximate instantaneously that of the Polak-Ribière-Polyak method, which stimulates the return of restarting mechanism. Under mild conditions, we show that the proposed methods converge globally. Numerical experiments support the theoretical result.