On the Second-order Directional Derivatives of Singular Values of Matrices and Symmetric Matrix-valued Functions
On the Second-order Directional Derivatives of Singular Values of Matrices and Symmetric Matrix-valued Functions
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DOI：
10.1007/s11228-013-0237-4
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发表时间：
2013-04
影响因子：
1.6
通讯作者：
Liwei Zhang;Ning Zhang;X. Xiao
中科院分区：
文献类型：
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作者：
Liwei Zhang;Ning Zhang;X. Xiao
由实值函数诱导的矩阵奇异值和对称矩阵函数的（抛物）二阶方向导数在研究不同类型的矩阵锥优化问题的二阶最优性条件中起着重要的作用.本文提出了一种直接推导Torki（Nonlinear Anal 46：1133-1150 2001）中对称矩阵任意特征值的二阶方向导数公式的方法，并由此建立了矩阵任意奇异值的二阶方向导数公式.给出了对称矩阵值函数的二阶方向导数的一个公式。作为应用，导出了Bonnans和Shapiro（2000）中SDP锥上投影算子的二阶导数，并利用它得到了SDP锥的二阶切集，同时给出了核范数上图的切锥和二阶切集.
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.