SIAM Journal on Optimization, Volume 31, Issue 1, Page 887-914, January 2021.
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 sequence of lower dimensional Krylov subspaces, whose orthonormal bases are generated by the symmetric Lanczos process, and computes approximate solutions from the underlying subspaces. There have been some a priori bounds available for the errors of the approximate solutions and approximate objective values obtained by the GLTR method, but no a priori bound exists on the errors of the approximate Lagrangian multipliers and the residual norms of approximate solutions obtained by the GLTR method. In this paper, a general convergence theory of the GLTR method is established for the TRS in the easy case, showing that the a priori bounds for these four quantities are closely interrelated and the one for the computable residual norm is of crucial importance in both theory and practice as it can predict the sizes of other three uncomputable errors reliably. Numerical experiments demonstrate that our bounds are realistic and predict the convergence rates of the four quantities accurately.

中文翻译：

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信赖域子问题的广义Lanczos信赖域方法的收敛性

SIAM优化杂志，第31卷，第1期，第887-914页，2021年1月。
解决信任区域子问题（TRS）在数值优化和许多其他应用中起着关键作用。广义Lanczos信任区（GLTR）方法是解决大型TRS的著名Lanczos类型方法。该方法将原始的大规模TRS投影到一系列低维Krylov子空间上，该子空间的正交基由对称Lanczos过程生成，并从底层子空间计算近似解。对于通过GLTR方法获得的近似解和近似目标值的误差，存在一些先验边界，但是对于近似拉格朗日乘子的误差和通过GLTR获得的近似解的残差范数不存在先验边界方法。在本文中，在简单的情况下，为TRS建立了GLTR方法的通用收敛理论，表明这四个量的先验边界是紧密相关的，而可计算残差范数的先验界在理论和实践中都至关重要。可以可靠地预测其他三个不可计算误差的大小。数值实验表明我们的边界是现实的，并且可以准确地预测这四个量的收敛速度。