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On the second-order directional derivatives of singular values of matrices and symmetric matrix-valued functions. (English) Zbl 1301.90090
Summary: 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 [M. Torki, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 46, No. 8, 1133–1150 (2001; Zbl 0993.15007)], 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 [J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer Series in Operations Research. New York, NY: Springer (2000; Zbl 0966.49001)], and the tangent cone and the second-order tangent set of the epigraph of the nuclear norm are given as well.

MSC:
90C30 Nonlinear programming
15A18 Eigenvalues, singular values, and eigenvectors
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