Iterative methods for nonlinear monotone equations do not require the differentiability assumption on the residual function. This special property of the methods makes them suitable for solving large-scale nonsmooth monotone equations. In this work, we present a diagonal Polak-Ribi$ \grave{e} $re-Polyak (PRP) conjugate gradient-type method for solving large-scale nonlinear monotone equations with convex constraints. The search direction is a combine form of a multivariate (diagonal) spectral method and a modified PRP conjugate gradient method. Proper safeguards are devised to ensure positive definiteness of the diagonal matrix associated with the search direction. Based on Lipschitz continuity and monotonicity assumptions the method is shown to be globally convergent. Numerical results are presented by means of comparative experiments with recently proposed multivariate spectral Dai-Yuan-type (J. Ind. Manag. Optim. 13 (2017) 283-295) and Wei-Yao-Liu-type (Int. J. Comput. Math. 92 (2015) 2261-2272) conjugate gradient methods.

中文翻译：

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凸约束非线性单调方程的对角线PRP型投影方法

非线性单调方程的迭代方法不需要对残差函数进行微分假设。这些方法的特殊性质使其适合求解大型非光滑单调方程。在这项工作中，我们提出了对角Polak-Ribi $ \ grave {e} $ re-Polyak（PRP）共轭梯度型方法，用于求解具有凸约束的大型非线性单调方程。搜索方向是多变量（对角线光谱法和改进的PRP共轭梯度法）。设计了适当的保护措施以确保与搜索方向关联的对角矩阵的正定性。基于Lipschitz连续性和单调性假设，该方法被证明是全局收敛的。通过比较实验，使用最近提出的多元光谱Dai-Yuan型（J.Ind.Manag.Optim.13（2017）283-295）和Wei-Yao-Liu型（Int.J.Comput数学（92）（2015）2261-2272）共轭梯度法。