A three-term derivative-free projection algorithm for nonlinear equations with convex constraints: application to image denoising

Solving nonlinear equations with convex constraints and addressing image denoising problems are critical challenges in many fields such as signal processing, optimization, and computer vision. In this paper, we propose a novel three-term derivative-free projection algorithm to tackle these problems efficiently. The algorithm builds upon the structural framework of three-term conjugate gradient methods and introduces a carefully designed search direction that satisfies the sufficient descent property. This design ensures fast convergence while maintaining low computational complexity. The proposed algorithm is particularly suited for large-scale nonlinear equations due to its simple structure and minimal storage requirements. We establish the global convergence of the algorithm under reasonable assumptions. Numerical experiments demonstrate the superiority of the proposed algorithm, with it outperforming other methods in approximately 70.83% of test problems in terms of running time, 85% in the number of iterations, and 61.94% in the number of function evaluations. Additionally, we successfully apply the algorithm to image denoising problems, highlighting its practical applicability in real-world scenarios.