- The Second Order Directional Derivative of Symmetric Matrix-valued Functions Liwei Zhang(lwzhangdlut.edu.cn) Ning Zhang (ningzhang_2008yeah.net) Xiantao Xiao(xtxiaoldlut.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: 04/24/2011Entry Accepted: 04/25/2011Entry Last Modified: 04/24/2011Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.