Matrix Functions and Weighted Centers for Semidefinite Programming

In this paper, we develop various differentiation rules for general smooth matrix-valued functions, and for the class of matrix convex (or concave) functions first introduced by Löwner and Kraus in the 1930s. For a matrix monotone function, we present formulas for its derivatives of any order in an integral form. Moreover, for a general smooth primary matrix function, we derive a formula for all of its derivatives by means of the divided differences of the original function. As applications, we use these differentiation rules and the matrix concave function log X to study a new notion of weighted centers for Semidefinite Programming (SDP). We show that, with this definition, some known properties of weighted centers for linear programming can be extended to SDP. We also show how the derivative formulas can be used in the implementation of barrier methods for optimization problems involving nonlinear but convex matrix functions.