The Trust Region Subproblem is a fundamental optimization problem that takes a pivotal role in Trust Region Methods. However, the problem, and variants of it, also arise in quite a few other applications. In this article, we present a family of iterative Riemannian optimization algorithms for a variant of the Trust Region Subproblem that replaces the inequality constraint with an equality constraint, and converge to a global optimum. Our approach uses either a trivial or a non-trivial Riemannian geometry of the search-space, and requires only minimal spectral information about the quadratic component of the objective function. We further show how the theory of Riemannian optimization promotes a deeper understanding of the Trust Region Subproblem and its difficulties, e.g., a deep connection between the Trust Region Subproblem and the problem of finding affine eigenvectors, and a new examination of the so-called hard case in light of the condition number of the Riemannian Hessian operator at a global optimum. Finally, we propose to incorporate preconditioning via a careful selection of a variable Riemannian metric, and establish bounds on the asymptotic convergence rate in terms of how well the preconditioner approximates the input matrix.

信任域子问题是一个基本的优化问题，在信任域方法中起着关键作用。然而，这个问题及其变体也出现在很多其他应用程序中。在本文中，我们针对信任区域子问题的变体提出了一系列迭代黎曼优化算法，该算法用等式约束代替不等式约束，并收敛到全局最优值。我们的方法使用搜索空间的平凡或非平凡黎曼几何，并且仅需要有关目标函数的二次分量的最少光谱信息。我们进一步展示了黎曼优化理论如何促进对信赖域子问题及其困难的更深入理解，例如：信托区域子问题与寻找仿射特征向量问题之间的深层联系，以及根据全局最优处的黎曼海塞算子的条件数对所谓的硬案例进行新的检查。最后，我们建议通过仔细选择变量黎曼度量来合并预处理，并根据预处理器逼近输入矩阵的程度建立渐近收敛率的界限。