A novel projection-based method for monotone equations with Aitken Δ2 acceleration and its application to sparse signal restoration

In this paper, a novel projection method for solving systems of monotone equations is introduced. The method, employs a search direction based on the normalized negative residual and incorporates a suitable linesearch technique to determine the step length. An accelerated variant is also developed using a vector generalization of the Aitken ${\Delta }^2$ method, enhanced with a convergence safeguard. These methods are both derivative-free and computationally inexpensive, making them suitable for large-scale problems. The global convergence of these methods is established under specific conditions, and their superior efficiency is demonstrated through numerical tests on large-scale test problems, outperforming several recent accelerated algorithms. Finally, the application of these methods to the signal restoration problem is also discussed.