Title: 
Matrix Convex Functions With Applications to
Weighted Centers for Semidefinite Programming

Authors:
Jan Brinkhuis
Econometric Institute
Erasmus University
The Netherlands

Zhi-Quan Luo
Department of Electrical and Computer Engineering
University of Minnesota
USA

Shuzhong Zhang
Department of Systems Engineering and Engineering Management
The Chinese University of Hong Kong
Shatin, Hong Kong


Abstract:
In this paper, we develop various calculus rules for general
smooth matrix-valued functions and for the class of matrix convex
(or concave) functions first introduced by L\"owner and Kraus in 1930s. 
Then we use these calculus rules and the matrix convex function
$-\log X$ to study a new notion of weighted centers for
semidefinite programming (SDP) and show that, with this
definition, some known properties of weighted centers for linear
programming can be extended to SDP. We also show how the calculus
rules for matrix convex functions can be used in the
implementation of barrier methods for optimization problems
involving nonlinear matrix functions.