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The Second Order Directional Derivative of Symmetric Matrix-valued Functions

Liwei Zhang(lwzhang***at***dlut.edu.cn)
Ning Zhang (ningzhang_2008***at***yeah.net)
Xiantao Xiao(xtxiao***at***ldlut.edu.cn)

Abstract: This paper focuses on the study of the second-order directional derivative of a symmetric matrix-valued function of the form $F(X)=P\mbox{diag}[f(\lambda_1(X)),\cdots,f(\lambda_n(X))]P^T$. For this purpose, we first adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a matrix in Torki \cite{Tor01}; Second, we establish a formula for the (parabolic) second-order directional derivative of the symmetric matrix-valued function. Finally, as an application, the second-order derivative for the projection operator over the SDP cone is used to derive the formula for the second-order tangent set of the SDP cone in Bonnans and Shapiro \cite{BS00}, which is the key for the Sigma term in the second-order optimality conditions of nonlinear SDP problems.

Keywords: the SDP cone; symmetric matrix-valued function;

Category 1: Linear, Cone and Semidefinite Programming

Citation:

Download: [PDF]

Entry Submitted: 04/24/2011
Entry Accepted: 04/25/2011
Entry Last Modified: 04/24/2011

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