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

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• 作者：Hamza Fawzi^a ; ^{h.fawzi@damtp.cam.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; James Saunderson^b ; ^{james.saunderson@monash.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：90C22 ; 47A63 ; 81P45
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：513
• 期：Complete
• 页码：240-263
• 全文大小：485 K

文摘

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