We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.

翻译：我们提出了一种极其通用的方法，用于解决一大类矩阵邻近问题（可能附带线性约束）。该方法基于将矩阵邻近问题拆分为两个嵌套优化问题：其中内层问题可以精确或低成本求解，而外层问题可重新表述为光滑实黎曼流形上的无约束优化任务。我们观察到该范式适用于文献中许多具有实际意义的矩阵邻近问题，从而揭示它们在本质上等价于黎曼优化问题。我们还证明了黎曼流形上待最小化的目标函数可能具有不连续性，因此需要正则化技术，并给出了出现这种情况的条件。最后，我们通过将该方法应用于多个具有实际意义且目前被认为计算代价高昂的矩阵邻近问题，证明了其实用性。大量数值实验表明，我们的方法通常显著优于现有算法，包括那些为特定问题专门设计的算法。