论文标题
等级-1 $ $ 1 $矩阵微分方程，用于结构化特征值优化
Rank-$1$ matrix differential equations for structured eigenvalue optimization
论文作者
论文摘要
提出和研究了一种解决大型结构化矩阵特征值优化问题的新方法。所考虑的优化问题类别与计算结构化的伪谱及其极端点有关，以及结构化的矩阵近感问题，例如计算结构化距离与不稳定性或奇异性。该结构可以是一般的线性结构，包括具有给定稀疏模式的大型矩阵，具有给定范围和共同范围的矩阵以及哈密顿矩阵。值得注意的是，特征值优化可以对复杂（或真实）等级-1矩阵的多种形式进行，该矩阵的储存量显着降低，在某些情况下，计算成本的情况下。该方法依赖于受约束的梯度系统以及梯度在复杂级别级别的歧管的切线上的投影-1 $矩阵。结果表明，在局部最小化的附近，此预测非常接近身份图，因此计算有利的等级-1投影系统在本地的行为就像计算上的％昂贵梯度系统一样。
A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and to structured matrix nearness problems such as computing the structured distance to instability or to singularity. The structure can be a general linear structure and includes, for example, large matrices with a given sparsity pattern, matrices with given range and co-range, and Hamiltonian matrices. Remarkably, the eigenvalue optimization can be performed on the manifold of complex (or real) rank-1 matrices, which yields a significant reduction of storage and in some cases of the computational cost. The method relies on a constrained gradient system and the projection of the gradient onto the tangent space of the manifold of complex rank-$1$ matrices. It is shown that near a local minimizer this projection is very close to the identity map, and so the computationally favorable rank-1 projected system behaves locally like the %computationally expensive gradient system.
