An improved PRP conjugate gradient algorithm with a damping factor within an automatic restarting strategy

In this paper, we present an improved Polak–Ribière–Polyak (PRP) conjugate gradient algorithm, in which the search direction satisfies the sufficient descent condition with strong Wolfe conditions. A “circuit breaker” mechanism is suggested in the monitoring and pre-warning system concerned with a poor approximation to the scalar β${}_k^{PRP}$. The resulting instinct self-correcting property is satisfied at each iteration, which can be viewed as the inheritance and development of the original PRP method. Meanwhile, several auxiliary functions are introduced, which achieve twin goals of manipulating the magnitude of conjugate gradient parameter and making swift changes to incorrect predictions if jamming occurs in actual computation. Under mild conditions, we establish the global convergence of the new algorithms and its variants for the general function. Numerical results show our algorithms are practical and effective for the test problems. Finally, the preliminary numerical experiments are given to indicate the efficiency of the proposed methods for image restoration.