- Matrix monotonicity and self-concordance:how to handle quantum entropy in optimization problems Leonid Faybusovich(lfaybusogmail.com) Takashi Tsuchiya(tsuchiyasun312.ism.ac.jp) Abstract: Let $g$ be a continuously differentiable function whose derivative is matrix monotone on positive semi-axis. Such a function induces a function $\phi (x)=tr(g(x))$ on the cone of squares of an arbitrary Euclidean Jordan algebra. We show that $\phi (x) -\ln \det(x)$ is a self-concordant function on the interior of the cone. We also show that $-\ln (t-\phi (x))-\ln \det (x)$ is $\sqrt{\frac{5}{3}}(r+1)$-self-concordant barrier on the epigraph of $\phi ,$ where $r$ is the rank of the Jordan algebra. The case $\phi (x)=tr(x\ln x)$ is discussed in detail. Keywords: quantum entropy,matrix monotonicity,self-concordance Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Linear, Cone and Semidefinite Programming Citation: Preprint,GRIPS,August 2014 Download: [PDF]Entry Submitted: 08/28/2014Entry Accepted: 08/28/2014Entry Last Modified: 08/28/2014Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.