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 Loewner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function -log X to study a new notion of weighted convex 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.

matrix convexity, matrix monotonicity, semidefinite programming
Time-Series Models; Dynamic Quantile Regressions (jel C22), Optimization Techniques; Programming Models; Dynamic Analysis (jel C61)
hdl.handle.net/1765/7025
Econometric Institute Research Papers
Report / Econometric Institute, Erasmus University Rotterdam
Erasmus School of Economics

Brinkhuis, J, Luo, Z-Q, & Zhang, S. (2005). Matrix convex functions with applications to weighted centers for semidefinite programming (No. EI 2005-38). Report / Econometric Institute, Erasmus University Rotterdam. Retrieved from http://hdl.handle.net/1765/7025