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

A famous result of Lieb establishes that the map (A, B) →tr [K∗A^1−tKB^t] is jointly concave in the pair (A, B)of posi-tive 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.