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Worst-case complexity guarantees for nonconvex optimization algorithms havebeen a topic of growing interest. Multiple frameworks that achieve the bestknown complexity bounds among a broad class of first- and second-orderstrategies have been proposed. These methods have often been designed primarilywith complexity guarantees in mind and, as a result, represent a departure fromthe algorithms that have proved to be the most effective in practice. In thispaper, we consider trust-region Newton methods, one of the most popular classesof algorithms for solving nonconvex optimization problems. By introducingslight modifications to the original scheme, we obtain two methods -- one basedon exact subproblem solves and one exploiting inexact subproblem solves as inthe popular "trust-region Newton-Conjugate-Gradient" (trust-region Newton-CG)method -- with iteration and operation complexity bounds that match the bestknown bounds for the aforementioned class of first- and second-order methods.The resulting trust-region Newton-CG method also retains the attractivepractical behavior of classical trust-region Newton-CG, which we demonstratewith numerical comparisons on a standard benchmark test set.

Trust-Region Newton-CG with Strong Second-Order Complexity Guarantees for Nonconvex Optimization