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论文类型：期刊论文

发表时间：2013-09-01

发表刊物：SET-VALUED AND VARIATIONAL ANALYSIS

收录刊物：SCIE、Scopus

卷号：21

期号：3

页面范围：557-586

ISSN号：1877-0533

关键字：The SDP cone; Eigenvalue; Singular value; Symmetric matrix-valued function; Second-order directional derivative; Second-order tangent set

摘要：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.