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

The generalized Lanczos trust-region (GLTR) method is one of the most popular approaches for solving large-scale trust-region subproblem (TRS). In Jia and Wang, SIAM J. Optim., 31, 887–914 2021. Z. Jia et al. considered the convergence of this method and established some a priori error bounds on the residual and the Lagrange 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 present a non-asymptotic bound for the convergence of the Lagrange multiplier and define a factor that plays an important role in the convergence of the Lagrange multiplier. Third, we revisit the convergence of the Krylov subspace method for the cubic regularization variant of the trust-region subproblem and substantially improve the convergence result established in Jia et al., SIAM J. Matrix Anal. Appl. 43 (2022), pp. 812–839 2022 on the multiplier. Numerical experiments demonstrate the effectiveness of our theoretical results.