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arXiv · 1810.09856

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

Chao Ding·Defeng Sun·Jie Sun·Kim-Chuan Toh
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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.

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Chao Ding, Defeng Sun, Jie Sun, Kim-Chuan Toh. 2018-10-22. Spectral operators of matrices: semismoothness and characterizations of the generalized Jacobian. https://arxiv.org/abs/1810.09856

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