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

Abstract

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

arXiv comment17 pages; minor modifications in the presentation + added reference to [G. Sagnol, On the semidefinite representation of real functions applied to symmetric matrices, Linear Algebra Appl., 2013] which gives an alternative semidefinite representation of weighted matrix geometric means; v3: Fixed a mistake in Lemma 4