We study the problem of minimizing the ratio of quadratic functions with a quadratic constraint (QCRQ), which is a generation of the trust-region subproblem and covers the regularized total least square problem as a special case. In this paper, we carefully employ the bisection search method to solve the scaled Lagrangian dual of (QCRQ) as strong duality holds for the primal and dual problems. We show that our algorithm can globally solve the nonconvex optimization (QCRQ) in linear time. Numerical experiments demonstrate the computational efficiency over other semidefinite programming solvers.

中文翻译：

---

线性时间算法，用于最小化具有二次约束的二次函数的比率

我们研究了使用二次约束（QCRQ）来最小化二次函数比率的问题，这是信任区域子问题的生成，并且在特殊情况下涵盖了正规化的总最小二乘问题。在本文中，由于强对偶成立于原始问题和对偶问题，因此我们小心翼翼地采用对分搜索方法来求解缩放的拉格朗日对偶（QCRQ）。我们证明了我们的算法可以在线性时间内全局求解非凸优化（QCRQ）。数值实验证明了其在其他半定规划求解器上的计算效率。