The conjugate gradient method is one of the most robust algorithms to solve large-scale monotone problems due to its limited memory requirements. However, in this article, we used the modified secant equation and proposed two optimal choices for the non-negative constant of the Hager-Zhang (HZ) conjugate gradient method by minimizing the upper bound of the condition number for the HZ search direction matrix. Two algorithms for solving large-scale non-linear monotone equations that incorporate the concept of projection method are provided. Based on monotone and Lipschitz continuous assumptions, we developed the global convergence of the methods. Computational results indicate that the proposed algorithms are effective and efficient.

由于其有限的内存需求，共轭梯度法是解决大规模单调问题的最稳健的算法之一。然而，在本文中，我们使用修正的割线方程，并通过最小化 HZ 搜索方向矩阵的条件数上限，提出了 Hager-Zhang (HZ) 共轭梯度法的非负常数的两个最优选择。提供了两种求解大规模非线性单调方程的算法，它们结合了投影法的概念。基于单调和 Lipschitz 连续假设，我们开发了这些方法的全局收敛性。计算结果表明，所提出的算法是有效和高效的。