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

A famous result of Lieb establishes that the map (A,B) mapsto trleft[K^* A1-t K Btright] 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.