论文标题
一种具有复杂性的牛顿-MR算法，可保证非凸平的平滑不受约束优化
A Newton-MR algorithm with complexity guarantees for nonconvex smooth unconstrained optimization
论文作者
论文摘要
在本文中，我们考虑了牛顿-MR算法的变体，用于解决不受限制，平滑但非凸优化问题的变体。与绝大多数牛顿型方法不同，牛顿型方法依靠共轭梯度算法作为其各自的子问题的主要主力，牛顿MR采用了最小残留（minres）方法。最近，已经确定，量子具有固有的能力来检测非阳性曲率方向，并在检测到某些有用的单调性能。我们利用这些最近的结果，并表明我们的算法具有理想的特性，包括竞争性的一阶和二阶最差复杂性。数值示例证明了我们提出的算法的性能。
In this paper, we consider variants of Newton-MR algorithm for solving unconstrained, smooth, but non-convex optimization problems. Unlike the overwhelming majority of Newton-type methods, which rely on conjugate gradient algorithm as the primary workhorse for their respective sub-problems, Newton-MR employs minimum residual (MINRES) method. Recently, it has been established that MINRES has inherent ability to detect non-positive curvature directions as soon as they arise and certain useful monotonicity properties will be satisfied before such detection. We leverage these recent results and show that our algorithms come with desirable properties including competitive first and second-order worst-case complexities. Numerical examples demonstrate the performance of our proposed algorithms.
