On convergence of the generalized Lanczos trust-region method for trust-region subproblems

TL;DR

This paper improves the theoretical understanding of the convergence properties of the generalized Lanczos trust-region method for large-scale trust-region subproblems, providing sharper bounds and insights into the factors affecting convergence.

Contribution

The paper establishes sharper residual bounds, new convergence bounds for the solution approximation, and non-asymptotic bounds for the Lagrange multiplier, enhancing previous results.

Findings

Sharper residual upper bounds are derived.

Convergence of the approximation is independent of spectral separation.

Non-asymptotic bounds for the Lagrange multiplier are provided.

Abstract

The generalized Lanczos trust-region (GLTR) method is one of the most popular approaches for solving large-scale trust-region subproblem (TRS). Recently, Jia and Wang [Z. Jia and F. Wang, \emph{SIAM J. Optim., 31 (2021), pp. 887--914}] considered the convergence of this method and established some {\it a prior} error bounds on the residual, the solution and the Largrange multiplier. In this paper, we revisit the convergence of the GLTR method and try to improve these bounds. First, we establish a sharper upper bound on the residual. Second, we give a new bound on the distance between the approximation and the exact solution, and show that the convergence of the approximation has nothing to do with the associated spectral separation. Third, we present some non-asymptotic bounds for the convergence of the Largrange multiplier, and define a factor that plays an important role on the convergence of the Largrange multiplier. Numerical experiments demonstrate the effectiveness of our theoretical results.