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Liwei Zhang(lwzhangdlut.edu.cn) Abstract: This paper focuses on the study of the secondorder directional derivative of a symmetric matrixvalued 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 secondorder directional derivative of any eigenvalue of a matrix in Torki \cite{Tor01}; Second, we establish a formula for the (parabolic) secondorder directional derivative of the symmetric matrixvalued function. Finally, as an application, the secondorder derivative for the projection operator over the SDP cone is used to derive the formula for the secondorder tangent set of the SDP cone in Bonnans and Shapiro \cite{BS00}, which is the key for the Sigma term in the secondorder optimality conditions of nonlinear SDP problems. Keywords: the SDP cone; symmetric matrixvalued function; Category 1: Linear, Cone and Semidefinite Programming Citation: Download: [PDF] Entry Submitted: 04/24/2011 Modify/Update this entry  
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