论文标题
一个新的约束优化模型，用于解决非对称随机反向特征值问题
A New Constrained Optimization Model for Solving the Nonsymmetric Stochastic Inverse Eigenvalue Problem
论文作者
论文摘要
随机逆特征值问题旨在从其光谱中重建随机矩阵。尽管存在有关特殊设置的解决方案存在的大量文献，但到目前为止，只有很少的数值解决方案方法。最近，Zhao等。 （2016年）提出了一个限制的优化模型，对所谓的同谱矩阵的流形，并改编了修改后的polak-ribière-polyak-polyak conjugate梯度方法，以使该歧管的几何形状。但是，并非每个随机矩阵都是同一光谱，也是Zhao等人的模型。是基于这样的假设：对于每个随机矩阵，存在一个（可能不同的）等光谱，随机矩阵，具有相同的频谱。我们不知道文献中有这样的结果，但会发现该主张至少以3 $ 3 $矩阵而言是正确的。在本文中，我们建议通过考虑与同一光谱不同的矩阵来扩展上述模型，仅通过与块对角线矩阵乘以$ 2 \ times $ 2 \ times times times times 2 $ block，其中$ sl $ sl（2）$，其中块的数量由复杂偶联的eigenvalues的成对的块数量给出。每个随机矩阵都可以以这种形式编写，这不是同一矩阵的形式。我们证明了我们的模型具有最小化器，并展示了Polak-Ribière-Polyak con轭梯度方法如何在相应的更通用的歧管上起作用。我们通过数值示例证明了新的，更通用的方法的性能与Zhao等人的示例相似。
The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. While there exists a large literature on the existence of solutions for special settings, there are only few numerical solution methods available so far. Recently, Zhao et al. (2016) proposed a constrained optimization model on the manifold of so-called isospectral matrices and adapted a modified Polak-Ribière-Polyak conjugate gradient method to the geometry of this manifold. However, not every stochastic matrix is an isospectral one and the model from Zhao et al. is based on the assumption that for each stochastic matrix there exists a (possibly different) isospectral, stochastic matrix with the same spectrum. We are not aware of such a result in the literature, but will see that the claim is at least true for $3 \times 3$ matrices. In this paper, we suggest to extend the above model by considering matrices which differ from isospectral ones only by multiplication with a block diagonal matrix with $2 \times 2$ blocks from the special linear group $SL(2)$, where the number of blocks is given by the number of pairs of complex-conjugate eigenvalues. Every stochastic matrix can be written in such a form, which was not the case for the form of the isospectral matrices. We prove that our model has a minimizer and show how the Polak-Ribière-Polyak conjugate gradient method works on the corresponding more general manifold. We demonstrate by numerical examples that the new, more general method performs similarly as the one from Zhao et al.
