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

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A famous result of Lieb establishes that the map (A, B) →tr [K∗A1−tKBt] 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.
Original languageEnglish
Pages (from-to)240-263
Number of pages24
JournalLinear Algebra and Its Applications
Publication statusPublished - Jan 2017


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

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