# Beyond Nonconvexity: A Universal Trust-Region Method with New Analyses

arXiv: https://arxiv.org/abs/2311.11489 · HTML: https://mlsanity.com/paper/2311.11489/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2023.

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

- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [Gradient-Normalized Smoothness for Optimization with Approximate Hessians](https://mlsanity.com/paper/2506.13710/) (2025)

## Abstract (arXiv)

The trust-region (TR) method is renowned historically for its robustness in nonconvex problems and extraordinary numerical performance, but the study of its performance in convex optimization is somehow limited. This paper complements the existing literature by presenting a universal trust-region method that simultaneously incorporates the quadratic regularization and ball constraint. In particular, we introduce a novel descent property tailored for trust-region-type algorithms, enabling us to unify and streamline the analysis for both convex and nonconvex optimization. Our method exhibits an iteration complexity of tilde O(ε-3/2) to find an ε-approximate second-order stationary point for nonconvex optimization. Meanwhile, the analysis reveals that the universal method attains an O(ε-1/2) complexity bound for convex optimization. Finally, we develop an adaptive universal method to address practical implementations. The numerical results show the effectiveness of our method in both nonconvex and convex problems.