On the semidefinite representation of real functions applied to symmetric matrices

Abstract

We present a new semidefinite representation for the trace of a real function f applied to symmetric matrices, when a semidefinite representation of the convex (or concave) function f is known. Our construction is intuitive, and yields a representation that is more compact than the previously known one. We also show with the help of matrix geometric means and a Riemannian metric over the set of positive definite matrices that for a rational exponent p in the interval (0,1], the matrix X raised to p is the largest element of a set represented by linear matrix inequalities. This result further generalizes to the case of the matrix A♯pB, which is the point of coordinate p on the geodesic from A to B. We give numerical results for a problem inspired from the theory of experimental designs, which show that the new semidefinite programming formulation can yield an important speed-up factor.