We consider the Generalized Trust Region Subproblem (GTRS) of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint. A lifting of this problem recasts the GTRS as minimizing a linear objective subject to two nonconvex quadratic constraints. Our first main contribution is structural: we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil. This result may be of interest in building relaxations for nonconvex quadratic programs. Moreover, this result allows us to reformulate the GTRS as the minimization of two convex quadratic functions in the original space. Our next set of contributions is algorithmic: we present an algorithm for solving the GTRS up to an epsilon additive error based on this reformulation. We carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms. Notably, our algorithms run in linear (in the size of the input) time. Furthermore, our algorithm for computing an epsilon-optimal solution has a slightly-improved running time dependence on epsilon over the state-of-the-art algorithm. Our analysis shows that the dominant cost in solving the GTRS lies in solving a generalized eigenvalue problem -- establishing a natural connection between these problems. Finally, generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality-, interval-, and hollow- constrained variants of the GTRS. This gives the first linear-time algorithm in the literature for these variants of the GTRS.

中文翻译：

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广义信任域子问题：解复杂度和凸包结果

我们考虑在非凸二次约束上最小化非凸二次目标的广义信任区域子问题 (GTRS)。解决这个问题将 GTRS 重新定义为最小化受两个非凸二次约束的线性目标。我们的第一个主要贡献是结构性的：我们根据相关矩阵铅笔的广义特征值给出了这个非凸集的凸包的明确描述。这个结果可能对构建非凸二次程序的松弛感兴趣。此外，这个结果允许我们将 GTRS 重新表述为原始空间中两个凸二次函数的最小化。我们的下一组贡献是算法方面的：我们提出了一种算法来解决 GTRS 的问题，该算法基于此重新表述达到 epsilon 加性误差。我们仔细处理因不精确的广义特征值和特征向量计算而产生的数值问题，并为这些算法建立明确的运行时间保证。值得注意的是，我们的算法以线性（输入的大小）时间运行。此外，与最先进的算法相比，我们用于计算 epsilon 最优解的算法对 epsilon 的运行时间依赖性略有改善。我们的分析表明，解决 GTRS 的主要成本在于解决广义特征值问题——在这些问题之间建立自然联系。最后，我们的凸包结果的推广使我们能够将我们的算法及其理论保证直接应用于 GTRS 的等式、区间和空心约束变体。