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