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.

使用基于牛顿方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，在 Hessian 矩阵的负曲率方向上显式检测和利用，可以证明该算法与 1980 年代开发的 Newton-conjugate 梯度程序非常接近，但包括了增强功能，以证明收敛到满足近似一阶和二阶最优条件的点的最坏情况复杂度结果，这些复杂度结果与文献中已知的二阶方法的最佳结果相匹配。