Abstract It is efficient to use the Hestenes–Stiefe (HS) conjugate gradient algorithm in solving large-scale complex smooth optimization problems because of its simplicity and low calculation requirements. Additionally, this algorithm has been used to simultaneously solve large-scale nonsmooth problems, nonlinear equations, and practical application. In this paper, the authors propose some modified HS conjugate gradient algorithms that not only address large-scale nonlinear equations and nonsmooth convex problems but also have the following properties. i) The algorithms portray a decreasing trait and trust-region property without any additional condition. ii) It combines the steepest descent algorithm with the conjugate gradient algorithm. iii) They employ the global convergence theory and iv) can be successfully used to solve nonlinear optimization problems and image restoration.

中文翻译：

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一些改进的Hestenes-Stiefel共轭梯度算法在图像恢复中的应用

摘要 Hestenes-Stiefe（HS）共轭梯度算法由于其简单性和计算要求低，在解决大规模复杂平滑优化问题中是有效的。此外，该算法已被用于同时解决大规模非光滑问题、非线性方程和实际应用。在本文中，作者提出了一些改进的 HS 共轭梯度算法，它们不仅可以解决大规模非线性方程和非光滑凸问题，而且还具有以下特性。i) 算法在没有任何附加条件的情况下描绘了一个递减的特征和信任区域属性。ii) 它结合了最速下降算法和共轭梯度算法。