Spectral operators of matrices: Semismoothness and characterizations of the generalized jacobian

Abstract

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.