A three-term cg projected-based method with control factor and its inertial version for constrained nonlinear equations

Narushima et al . [ SIAM J. Optim . 21 (2011) 212–230] proposed a three-term search direction that guarantees the sufficient descent condition. Building upon this work, we introduce a control factor to refine the search direction, which not only ensures the sufficient descent property but also possesses the trust-region feature, independent of the specific line search strategy. Furthermore, we extend the modified search direction to solve constrained nonlinear equations by employing a hyperplane projection technique. To enhance efficiency, we integrate an inertial acceleration mechanism, leading to an inertial variant of the proposed method. We rigorously establish the global convergence of the inertial method, even in the absence of Lipschitz continuity in the underlying mapping. Additionally, under appropriate conditions, we prove that the method achieves a linear convergence rate. Numerical experiments validate the effectiveness of our methods, demonstrating their superior performance in solving constrained nonlinear equations and restoring blurred images corrupted by Gaussian noise.