Optimization Online


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


Download: [PDF]

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

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society