A Hybrid Conjugate Gradient Method for Unconstrained Optimization with Application in Image Restoration

Hybrid conjugate gradient methods are considered as an efficient family of conjugate gradient (CG) methods used to solve unconstrained optimization problems. In this paper, on account of the outstanding performance of the PRP (Polak–Ribière–Polyak) conjugate gradient method and its exceptional numerical computational stability, we propose a hybrid conjugate gradient method for solving unconstrained optimization problems. By combining two PRP-type directions via convex combination, the proposed search direction dynamically adjusts to gradient change rates and satisfies the sufficient descent property. Under mild conditions, the global convergence of the proposed method is established. Numerical computations are presented to display the efficacy of the proposed algorithm compared to some existing algorithms. It is indicated that the proposed method is more effective in dealing with non-convex optimization problems. Finally, the applicability of the proposed method is shown in image restoration problems with noise, and preliminary experimental results demonstrate its effectiveness compared to some other methods.