Trust-region subproblem (TRS) is an important problem arising in many applications such as numerical optimization, Tikhonov regularization of ill-posed problems, and constrained eigenvalue problems. In recent decades, extensive works focus on how to solve the trust-region subproblem efficiently. To the best of our knowledge, there are few results on perturbation analysis of the trust-region subproblem. In order to fill in this gap, we focus on first-order perturbation theory of the trust-region subproblem. The main contributions of this paper are three-fold. First, suppose that the TRS is in \emph{easy case}, we give a sufficient condition under which the perturbed TRS is still in easy case. Second, with the help of the structure of the TRS and the classical eigenproblem perturbation theory, we perform first-order perturbation analysis on the Lagrange multiplier and the solution of the TRS, and define their condition numbers. Third, we point out that the solution and the Lagrange multiplier could be well-conditioned even if TRS is in {\it nearly hard case}. The established results are computable, and are helpful to evaluate ill-conditioning of the TRS problem beforehand. Numerical experiments show the sharpness of the established bounds and the effectiveness of the proposed strategies.

翻译：信赖域子问题（TRS）是数值优化、不适定问题的Tikhonov正则化以及约束特征值问题等众多应用中产生的重要问题。近几十年来，大量研究聚焦于如何高效求解信赖域子问题。然而据我们所知，关于信赖域子问题的摄动分析结果甚少。为填补这一空白，本文重点研究信赖域子问题的一阶摄动理论。本文的主要贡献有三方面：首先，假设TRS处于"易解情形"，我们给出了扰动后的TRS仍保持易解情形的充分条件；其次，借助TRS的结构与经典特征值问题摄动理论，对TRS的拉格朗日乘子及其解进行一阶摄动分析，并定义了它们的条件数；最后，我们指出即使TRS处于"近困难情形"，其解与拉格朗日乘子仍可能具有良好条件性。所建立的结果具有可计算性，有助于预先评估TRS问题的病态性。数值实验验证了所建立界限的紧致性及所提策略的有效性。