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

详细信息    查看全文

• 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304852&_mathId=si1.gif&_user=111111111&_pii=S0024379516304852&_rdoc=1&_issn=00243795&md5=61f1c5983da1fab08dab2cead45e20d6" title="Click to view the MathML source">(A,B)↦tr[K^⁎A^1−tKB^t]class="mathContainer hidden">class="mathCode">$(A,B)\mapsto \mathrm{tr}\left[K^{⁎}{A}^{1-t}K{B}^t\right]$ is jointly concave in the pair class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304852&_mathId=si2.gif&_user=111111111&_pii=S0024379516304852&_rdoc=1&_issn=00243795&md5=df2d24d86109571d1b398987eb3a6e15" title="Click to view the MathML source">(A,B)class="mathContainer hidden">class="mathCode">$(A,B)$ of positive definite matrices, where K   is a fixed matrix and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304852&_mathId=si15.gif&_user=111111111&_pii=S0024379516304852&_rdoc=1&_issn=00243795&md5=f37be1da849e238387b2139ccadce957" title="Click to view the MathML source">t∈[0,1]class="mathContainer hidden">class="mathCode">$t\in [0,1]$. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304852&_mathId=si15.gif&_user=111111111&_pii=S0024379516304852&_rdoc=1&_issn=00243795&md5=f37be1da849e238387b2139ccadce957" title="Click to view the MathML source">t∈[0,1]class="mathContainer hidden">class="mathCode">$t\in [0,1]$. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.