We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton’s method and the linear conjugate gradient algorithm, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity results to be proved for convergence to points that satisfy approximate first-order and second-order optimality conditions. The complexity results match the best known results in the literature for second-order methods.

中文翻译：

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一种具有复杂性保证的 Newton-CG 算法，用于平滑无约束优化

我们考虑使用基于牛顿法和线性共轭梯度算法的迭代算法来最小化平滑非凸目标函数，并明确检测并使用目标函数的 Hessian 的负曲率方向。该算法密切跟踪 1980 年代开发的牛顿共轭梯度程序，但包括增强功能，允许证明最坏情况的复杂性结果收敛到满足近似一阶和二阶最优性条件的点。复杂性结果与二阶方法文献中最著名的结果相匹配。