# The convergence of the Generalized Lanczos Trust-Region Method for the Trust-Region Subproblem

## TL;DR

This paper develops a comprehensive convergence theory for the generalized Lanczos trust-region (GLTR) method, providing a-priori bounds for key solution errors and residuals in large-scale trust-region subproblems, validated by numerical experiments.

## Contribution

It introduces the first a-priori convergence bounds for Lagrangian multipliers and residual norms in the GLTR method for large-scale trust-region subproblems.

## Key findings

- Derived a-priori bounds for Lagrangian multiplier errors
- Established convergence rates for residual norms
- Numerical results confirm the bounds' accuracy

## Abstract

Solving the trust-region subproblem (TRS) plays a key role in numerical optimization and many other applications. The generalized Lanczos trust-region (GLTR) method is a well-known Lanczos type approach for solving a large-scale TRS. The method projects the original large-scale TRS onto a $k$ dimensional Krylov subspace, whose orthonormal basis is generated by the symmetric Lanczos process, and computes an approximate solution from the underlying subspace. There have been some a-priori error bounds for the optimal solution and the optimal objective value in the literature, but no a-priori result exists on the convergence of Lagrangian multipliers involved in projected TRS's and the residual norm of approximate solution. In this paper, a general convergence theory of the GLTR method is established, and a-priori bounds are derived for the errors of the optimal Lagrangian multiplier, the optimal solution, the optimal objective value and the residual norm of approximate solution. Numerical experiments demonstrate that our bounds are realistic and predict the convergence rates of the three errors and residual norms accurately.