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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.

A modified Hager-Zhang conjugate gradient method with optimal choices for solving monotone nonlinear equations