报告人：魏木生 教授（上海师范大学）

时间：2018年3月14日(周三)上午11：00 -- 12：00

地点：闵行校区数学系102报告厅

主持人：潘建瑜 教授

报告简介：

In this paper, we discuss the problem of minimal rank positive semi-definite solutions to the approximation problem in the spectral norm:

find a matrix $X\ge0$, such that $\rank(X)=\min_{Y\geq0}\rank(Y)$, subject to $\|A-BXB^H\|_2=\theta\triangleq\min_{Y\geq0} \|A-BYB^H\|_2$. We propose two different methods to solve this problem. For the special case $\theta=\mu$, we establish the equivalent conditions for $\theta = \mu \triangleq\|P^\bot_BA\|_2$ by applying the Hermitian norm preserving dilation theorem and the generalized singular value decomposition (H-SVD). We characterize the expressions of the minimum rank and derive a general form of minimum rank positives emi-definite solutions to the matrix approximation problem. For $\theta>\mu$, we first transform the above mentioned problem into an equivalent problem: find a matrix $\hat{X}_{\theta}\ge0$, such that $\rank(\hat{X}_{\theta})= \min_{\hatY\geq0} \rank(\hatY)$, subject to $\|\hat{A}_{\theta} - \hat{X}_{\theta}\|_2=\theta\triangleq\min_{\hat Y\geq0}\|\hat{A}_{\theta} - \hat{Y}\|_2.$ In this case, we characterize the minimum rank positive semi-definite solutions, and derive a general form of these solutions. Furthermore, we generalize this method to the case $\theta=\mu$ and provide a procedure to determine the value of $\theta$ and propose a numerical algorithm. Three numerical examples are provided to illustrate our analysis.