On the Second-order Directional Derivatives of Singular Values of Matrices and Symmetric Matrix-valued Functions |
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Authors: | Liwei Zhang Ning Zhang Xiantao Xiao |
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Affiliation: | 1. Institute of Operations Research and Control Theory, School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
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Abstract: | The (parabolic) second-order directional derivatives of singular values of matrices and symmetric matrix-valued functions induced by real-valued functions play important roles in studying second-order optimality conditions for different types of matrix cone optimization problems. We propose a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a symmetric matrix in Torki (Nonlinear Anal 46:1133–1150 2001), from which a formula for the second-order directional derivative of any singular value of a matrix is established. We demonstrate a formula for the second-order directional derivative of the symmetric matrix-valued function. As applications, the second-order derivative for the projection operator over the SDP cone is derived and used to get the second-order tangent set of the SDP cone in Bonnans and Shapiro (2000), and the tangent cone and the second-order tangent set of the epigraph of the nuclear norm are given as well. |
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