## Abstract

A famous result of Lieb establishes that the map (

*A, B*) →tr [*K∗A*] is jointly concave in the pair (^{1−t}KB^{t}*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.Original language | English |
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Pages (from-to) | 240-263 |

Number of pages | 24 |

Journal | Linear Algebra and Its Applications |

Volume | 513 |

DOIs | |

Publication status | Published - Jan 2017 |

## Keywords

- Lieb's concavity theorem
- Linear matrix inequalities
- Matrix convexity
- Matrix geometric means
- Semidefinite optimization