TY - JOUR
T1 - Spectral Operators of Matrices: Semismoothness and Characterizations of the Generalized Jacobian
AU - Ding, Chao
AU - Sun, Defeng
AU - Sun, Jie
AU - Toh, Kim-Chuan
N1 - Funding Information:
∗Received by the editors October 22, 2018; accepted for publication (in revised form) December 23, 2019; published electronically February 20, 2020. https://doi.org/10.1137/18M1222235 Funding: The research of the first author was supported by the National Natural Science Foundation of China under projects 11671387, 11531014, and 11688101. The research of the second author was supported in part by a start-up research grant from the Hong Kong Polytechnic University. †Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China ([email protected]). ‡Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong (defeng. [email protected]). §Faculty of Science and Engineering, Curtin University, Bentley WA 6102, Australia (Jie.Sun@ curtin.edu.au). ¶Department of Mathematics and Institute of Operations Research and Analytics, National University of Singapore, Singapore ([email protected]).
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.
PY - 2020
Y1 - 2020
N2 - 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, ρ-order B(ouligand)-differentiability (0 < ρ≤ 1), ρ-order G-semismoothness (0 < ρ≤ 1), and characteriza- tion of generalized Jacobians.
AB - 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, ρ-order B(ouligand)-differentiability (0 < ρ≤ 1), ρ-order G-semismoothness (0 < ρ≤ 1), and characteriza- tion of generalized Jacobians.
KW - Generalized Jacobian
KW - Matrix-valued functions
KW - Semismoothness
KW - Spectral operators
UR - http://www.scopus.com/inward/record.url?scp=85084481450&partnerID=8YFLogxK
U2 - 10.1137/18M1222235
DO - 10.1137/18M1222235
M3 - Journal article
SN - 1052-6234
VL - 30
SP - 630
EP - 659
JO - SIAM Journal on Optimization
JF - SIAM Journal on Optimization
IS - 1
ER -