# An Inexact Regularized Newton Framework with a Worst-Case Iteration Complexity of O(ε-3/2) for Nonconvex Optimization

arXiv: https://arxiv.org/abs/1708.00475 · HTML: https://mlsanity.com/paper/1708.00475/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Aug 2017.

## Built on by (most cited first, 2 of 2)

- [Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching](https://mlsanity.com/paper/2305.18379/) (2023)
- [A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees](https://mlsanity.com/paper/2207.05697/) (2022)

## Abstract (arXiv)

An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes O(ε-3/2) iterations to drive the norm of the gradient of the objective function below a prescribed positive real number ε and can take O(ε-3) iterations to drive the leftmost eigenvalue of the Hessian of the objective above -ε. The proposed algorithm is a general framework that covers a wide range of techniques including quadratically and cubically regularized Newton methods, such as the Adaptive Regularisation using Cubics (ARC) method and the recently proposed Trust-Region Algorithm with Contractions and Expansions (TRACE). The generality of our method is achieved through the introduction of generic conditions that each trial step is required to satisfy, which in particular allow for inexact regularized Newton steps to be used. These conditions center around a new subproblem that can be approximately solved to obtain trial steps that satisfy the conditions. A new instance of the framework, distinct from ARC and TRACE, is described that may be viewed as a hybrid between quadratically and cubically regularized Newton methods. Numerical results demonstrate that our hybrid algorithm outperforms a cublicly regularized Newton method.