Numerical results for Problem 5.

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.本文结合超平面投影与混合技术，提出一种用于求解约束非线性方程组的混合共轭梯度投影方法。该方法的核心特性如下：(1) 存储需求低，且仅依赖函数值进行计算；(2) 所设计的搜索方向可保证充分下降性，无需借助线搜索方法；(3) 在合理的假设条件下，可证明该方法具备全局收敛性；(4) 实验结果表明，在CPU运行时间、函数评估次数与迭代次数三个指标下，该方法在约75.71%、85.36%及86.43%的基准测试问题上优于两种现有方法。此外，该方法可成功应用于稀疏信号恢复问题的求解。