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.

翻译：广义Lanczos信赖域（GLTR）方法是求解大规模信赖域子问题（TRS）最常用的方法之一。近期，Jia和Wang [Z. Jia and F. Wang, \emph{SIAM J. Optim., 31 (2021), pp. 887--914}]研究了该方法的收敛性，并建立了残量、解和拉格朗日乘子的一些先验误差界。本文重新审视GLTR方法的收敛性，并试图改进这些界。首先，我们建立了残量的更紧上界。其次，给出了近似解与精确解之间距离的新界，并证明近似解的收敛性与相关的谱分离无关。第三，提出了拉格朗日乘子收敛的一些非渐近界，并定义了一个对拉格朗日乘子收敛起重要作用的因子。数值实验验证了我们理论结果的有效性。