SIAM Journal on Optimization, Volume 31, Issue 1, Page 518-544, January 2021.
Worst-case complexity guarantees for nonconvex optimization algorithms have been a topic of growing interest. Multiple frameworks that achieve the best known complexity bounds among a broad class of first- and second-order strategies have been proposed. These methods have often been designed primarily with complexity guarantees in mind and, as a result, represent a departure from the algorithms that have proved to be the most effective in practice. In this paper, we consider trust-region Newton methods, one of the most popular classes of algorithms for solving nonconvex optimization problems. By introducing slight modifications to the original scheme, we obtain two methods---one based on exact subproblem solves and one exploiting inexact subproblem solves as in the popular “trust-region Newton-conjugate gradient” (trust-region Newton-CG) method---with iteration and operation complexity bounds that match the best known bounds for the aforementioned class of first- and second-order methods. The resulting trust-region Newton-CG method also retains the attractive practical behavior of classical trust-region Newton-CG, which we demonstrate with numerical comparisons on a standard benchmark test set.

中文翻译：

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具有强二阶复杂度保证的凸区域Newton-CG用于非凸优化

SIAM优化杂志，第31卷，第1期，第518-544页，2021年1月。
对于非凸优化算法，最坏情况下的复杂性保证已成为人们日益关注的话题。已经提出了在广泛的一阶和二阶策略中实现最广为人知的复杂性界限的多个框架。通常在设计这些方法时首先要考虑到复杂性保证，因此，与实践中最有效的算法背道而驰。在本文中，我们考虑信任区域牛顿法，这是解决非凸优化问题的最流行算法之一。通过对原始方案进行一些修改，我们获得了两种方法-一种具有精确的子问题解决方案，一种基于不精确的子问题解决方案，如流行的“信任区域牛顿共轭梯度”（trust-region Newton-CG gradient）方法那样，具有迭代和运算复杂性边界与上述一阶和二阶方法类别的最佳已知边界匹配。由此产生的信任区Newton-CG方法还保留了经典信任区Newton-CG的吸引人的实际行为，我们通过在标准基准测试集上进行数值比较来证明这一点。