Trust-Region Newton-Cg With Strong Second-Order Complexity Guarantees For Nonconvex Optimization

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.