Spectral operators of matrices: semismoothness and characterizations of the generalized Jacobian

TL;DR

This paper investigates the fundamental mathematical properties of spectral operators of matrices, focusing on their differentiability, semismoothness, and Jacobian characterizations, which are essential for advanced matrix optimization techniques.

Contribution

It provides new insights into the first- and second-order properties of spectral operators, including Lipschitz continuity and semismoothness, enhancing their theoretical foundation for optimization applications.

Findings

Established conditions for Lipschitz continuity of spectral operators.

Characterized the generalized Jacobian of spectral operators.

Analyzed the semismoothness properties relevant to matrix optimization.

Abstract

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, $\rho$-order B(ouligand)-differentiability ($0<\rho\le 1$), $\rho$-order G-semismoothness ($0<\rho\le 1$), and characterization of generalized Jacobians.