Lieb's concavity theorem, matrix geometric means, and semidefinite optimization

Abstract

A famous result of Lieb establishes that the map (A,B)↦tr[K^* A^{1−t}KB^{t}] is jointly concave in the pair (A,B) of positive definite matrices, where K is a fixed matrix and t∈[0,1]. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational t∈[0,1]. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.