# Concise Complexity Analyses for Trust-Region Methods

## TL;DR

This paper presents simplified complexity analyses for trust-region algorithms in unconstrained optimization, introducing a new radius update strategy that achieves second-order complexity bounds.

## Contribution

It provides concise complexity analyses for trust-region methods and proposes a novel radius update strategy to attain second-order complexity bounds.

## Key findings

- Algorithms require control over trust region radius after successful iterations.
- The new update strategy achieves second-order complexity bounds.
- Analyses identify key components needed for specific complexity guarantees.

## Abstract

Concise complexity analyses are presented for simple trust region algorithms for solving unconstrained optimization problems. In contrast to a traditional trust region algorithm, the algorithms considered in this paper require certain control over the choice of trust region radius after any successful iteration. The analyses highlight the essential algorithm components required to obtain certain complexity bounds. In addition, a new update strategy for the trust region radius is proposed that offers a second-order complexity bound.