SIAM Journal on Optimization, Volume 30, Issue 1, Page 915-932, January 2020.
In this paper, we provide the first provable linear-time (in terms of the number of nonzero entries of the input) algorithm for approximately solving the generalized trust region subproblem (GTRS) of minimizing a quadratic function over a quadratic constraint under some regularity condition. Our algorithm is motivated by and extends a recent linear-time algorithm for the trust region subproblem by Hazan and Koren [Math. Program., 158 (2016), pp. 363--381]. However, due to the nonconvexity and noncompactness of the feasible region, such an extension is nontrivial. Our main contribution is to demonstrate that under some regularity condition, the optimal solution is in a compact and convex set and lower and upper bounds of the optimal value can be computed in linear time. Using these properties, we develop a linear-time algorithm for the GTRS.

中文翻译：

---

广义信赖域子问题的线性时间算法

SIAM优化杂志，第30卷，第1期，第915-932页，2020年1月。
在本文中，我们提供了第一种可证明的线性时间（根据输入的非零项的数量）算法，用于近似求解在某些规则性条件下使二次约束上的二次函数最小化的广义信任区域子问题（GTRS） 。我们的算法是由Hazan和Koren [Math.Am.Sci。计划，158（2016），363--381页。然而，由于可行区域的非凸性和非紧致性，这种扩展是不平凡的。我们的主要贡献是证明在某些规则性条件下，最优解是在一个紧致的凸集内，并且可以在线性时间内计算出最优值的上下边界。利用这些属性，我们为GTRS开发了线性时间算法。