矩阵的谱算子：半光滑性与广义雅可比矩阵的特征
Spectral operators of matrices: semismoothness and characterizations of the generalized Jacobian
摘要：
最近在[C. Ding，D.F. Sun，J. Sun和K.C.托，数学程序。{ \bf 168}，509- 531（2018）]是一类矩阵值函数，其通过将向量到向量函数应用于基础矩阵的所有特征值/奇异值来将矩阵映射到矩阵。谱算子在矩阵最优化问题（MOPs）等各种涉及矩阵的应用研究中起着至关重要的作用。本文主要研究谱算子的一些基本的一阶和二阶性质，包括Lipschitz连续性，$ρ$ -阶B（ouligand）-可微性（$0<ρ\le 1$），$ρ$ -阶G-半光滑性（$0<ρ\le 1$），以及广义雅可比矩阵的刻画.
英文摘要：
Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J. Sun, and K.C. Toh, Math. Program. {\bf 168}, 509--531 (2018)] are a class of matrix valued functions, which map matrices to matrices by applying a vector-to-vector function to all eigenvalues/singular values of the underlying matrices. Spectral operators play a crucial role in the study of various applications involving matrices such as matrix optimization problems (MOPs) {that include semidefinite programming as one of the most important example classes}. In this paper, we will study more fundamental first- and second-order properties of spectral operators, including the Lipschitz continuity, $ρ$-order B(ouligand)-differentiability ($0<ρ\le 1$), $ρ$-order G-semismoothness ($0<ρ\le 1$), and characterization of generalized Jacobians.
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