# A Trust Region Method for Finding Second-Order Stationarity in Linearly Constrained Non-Convex Optimization

## TL;DR

This paper introduces a trust region algorithm designed to efficiently find second-order stationary points in linearly constrained non-convex optimization problems, improving convergence analysis without cubic regularization.

## Contribution

It presents a novel trust region method with proven convergence to second-order stationary points, applicable to general linearly constrained non-convex problems, without requiring cubic regularization.

## Key findings

- Converges to (_g, _H)-second order stationary points.
- Achieves (_g^{-3/2}, _H^{-3}) iteration complexity.
- Applicable to general linearly constrained non-convex optimization.

## Abstract

Motivated by TRACE algorithm [Curtis et al. 2017], we propose a trust region algorithm for finding second order stationary points of a linearly constrained non-convex optimization problem. We show the convergence of the proposed algorithm to (\epsilon_g, \epsilon_H)-second order stationary points in \widetilde{\mathcal{O}}(\max{\epsilon_g^{-3/2}, \epsilon_H^{-3}}) iterations. This iteration complexity is achieved for general linearly constrained optimization without cubic regularization of the objective function.