Accelerated Hager-Zhang type projection scheme for monotone equations with applications

This paper proposed a new iterative method for solving nonlinear monotone equations with convex constraint and its applications in sparse signal reconstruction and image de-blurring problems. The method can be viewed as an improved adaptation of the generalized Hager-Zhang conjugate gradient method for unconstrained optimization. Unlike the latter which only converged globally for strongly convex functions when the Hager-Zhang parameter ${\theta }_k$ lies in the interval $\left(\frac{1}{4},+\mathrm{\infty }\right)$, the new method exhibited this attribute for nonlinear monotone and Lipschitz continuous functions without restriction for ${\theta }_k$ under a more relaxed condition. The derivative-free structure of the method made it suitable for both smooth and non-smooth problems. Numerical experiments on benchmark test problems demonstrated the method's superior performance compared to some state-of-the-art algorithms. Furthermore, the algorithm was successfully applied to sparse signal recovery and image de-blurring problems in compressed sensing, confirming its practical effectiveness.