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Spectral Operators of Matrices: Semismoothness and Characterizations of the Generalized Jacobian
SIAM Journal on Optimization ( IF 2.6 ) Pub Date : 2020-02-20 , DOI: 10.1137/18m1222235
Chao Ding , Defeng Sun , Jie Sun , Kim-Chuan Toh

SIAM Journal on Optimization, Volume 30, Issue 1, Page 630-659, January 2020.
Spectral operators of matrices proposed recently in [C. Ding, D. F. Sun, J. Sun, and K. C. Toh, Math. Program., 168 (2018), pp. 509--531] 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 that include semidefinite programming as one of most important example classes. In this paper, we will study more fundamental first- and second-order properties of spectral operators, including the Lipschitz continuity, $\rho$-order B(ouligand)-differentiability ($0<\rho\le 1$), $\rho$-order G-semismoothness ($0<\rho\le 1$), and characterization of generalized Jacobians.


中文翻译:

矩阵的谱算子:广义雅可比算符的半光滑性和表征

SIAM优化杂志,第30卷,第1期,第630-659页,2020年1月。
最近在[C. 丁(Ding DF Sun),孙J.Sun和KC Toh(数学)。计划,168(2018),509--531页]是一类矩阵值函数,通过将矢量到矢量函数应用于基础矩阵的所有特征值/奇异值,将矩阵映射到矩阵。频谱算子在涉及矩阵的各种应用程序的研究中扮演着至关重要的角色,例如矩阵优化问题,其中包括半定规划作为最重要的示例类之一。在本文中,我们将研究频谱算子的更基本的一阶和二阶性质,包括Lipschitz连续性,$ \ rho $阶B(ouligand)可微性($ 0 <\ rho \ le 1 $),$ \ rho $阶G-半光滑度($ 0 <\ rho \ le 1 $),以及广义雅可比定理的表征。
更新日期:2020-02-20
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