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arXiv · 1512.03401

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

Hamza Fawzi·James Saunderson
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Abstract

A famous result of Lieb establishes that the map $(A,B) \mapsto \text{tr}\left[K^* A^{1-t} K B^t\right]$ 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.

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Hamza Fawzi, James Saunderson. 2015-12-10. Lieb's concavity theorem, matrix geometric means, and semidefinite optimization. https://arxiv.org/abs/1512.03401

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