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

A famous result of Lieb establishes that the map (A,B)↦tr[K⁎A1−tKBt](A,B)↦tr[K⁎A1−tKBt] is jointly concave in the pair (A,B)(A,B) of positive definite matrices, where K   is a fixed matrix and t∈[0,1]t∈[0,1]. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational t∈[0,1]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.