A combination of two conjugate gradient methods under a new line search with its application

A combined conjugate gradient algorithm is introduced for solving unconstrained optimization problems. In the suggested approach, the conjugate gradient parameter is defined as a combination of PRP (Polak-Ribíere-Polyak) and BRB (Rahali-Belloufi-Benzine) conjugate gradient parameters. To improve the convergence properties, we have adopted a new inexact line search technique that fits in with the suggested approach. The proposed line search technique can be useful for other gradient descent methods. We have established the existence of a step length that meets the new line search conditions. The generated descent direction and the convergence properties of the suggested approach are studied under the new line search conditions and the proposed method converges globally under mild assumptions. Our approach is evaluated on various test functions, and a comparison with similar recent algorithms is carried out. Furthermore, the proposed algorithm is applied for restoring images with different noise levels.