Numerical results for Problem 3.

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.

本文提出了一种融合超平面投影与混合技术的混合共轭梯度投影方法，用于求解约束非线性方程组。该方法的核心特性如下：(1) 存储需求低，且仅依赖函数值即可开展计算；(2) 所设计的搜索方向可确保充分下降性，无需采用线搜索方法；(3) 在若干合理假设下，该方法的全局收敛性得以证明；(4) 实验结果表明，在所测试的基准问题中，相较于两种现有方法，该方法在CPU运行时间、函数评估次数与迭代次数三项指标上分别提升约75.71%、85.36%与86.43%。此外，该方法可成功应用于稀疏信号恢复问题的求解。