Numerical results for Problem 1.

In this paper, a hybrid conjugate gradient projection method for finding solutions of constrained nonlinear equations is proposed by integrating both hyperplane projection and hybrid techniques. The key features of this method are as follows: (1) It is characterized by a low storage requirement and relies solely on function values; (2) The designed search direction ensures the sufficient descent property without the need for line search approaches; (3) Under certain reasonable assumptions, the global convergence of the method is established; (4) Experimental results demonstrate that the proposed method outperforms the two existing methods about 75.71%, 85.36%, and 86.43% of benchmark problems in terms of CPU time, the number of function evaluations, and iterations. Furthermore, it is applied to successfully solve the sparse signal restoration problems.