A truncated three-term derivative-free projection method and its inertial-relaxed improvement for nonlinear monotone equations

We propose a new truncated three-term derivative-free projection method and its inertial-relaxed improvement, both designed for solving nonlinear monotone equations. The search direction is enhanced by incorporating a flexible non-zero vector and a truncated structure, which ensure sufficient descent and trust region properties automatically, without requiring additional conditions. Furthermore, the proposed methods utilize a newly developed self-adaptive line search to determine the step size. The theoretical convergence of the inertial-relaxed method is established. To assess the effectiveness of the proposed methods, we conduct comparative benchmark tests using ten nonlinear equations. Finally, the potential practicality and effectiveness of the methods are demonstrated by applying them to sparse signal restoration problems.