Title of Paper:On the Second-order Directional Derivatives of Singular Values of Matrices and Symmetric Matrix-valued Functions

Hits:

Date of Publication:2013-09-01

Journal:SET-VALUED AND VARIATIONAL ANALYSIS

Included Journals:SCIE、Scopus

Volume:21

Issue:3

Page Number:557-586

ISSN No.:1877-0533

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

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.