The Second Order Directional Derivative of Symmetric Matrix-valued Functions ⁄
Liwei Zhang, Ning Zhang, Xiantao Xiao · 2011
This paper focuses on the study of the second-order direc- tional derivative of a symmetric matrix-valued function of the form F(X) = Pdiag(f(‚1(X));¢¢¢ ;f(‚n(X)))P T. For this purpose, we flrst adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a ma- trix in Torki (13); Second, we establish a formula for the (parabolic) second-order direc- tional 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 (3), which is the key for the Sigma term in the second-order optimality conditions of nonlinear SDP problems.