Matrix monotonicity and self-concordance:how to handle quantum entropy in optimization problems

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.

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Preprint,GRIPS,August 2014

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