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

Abstract

Let g be a continuously differentiable function whose derivative is matrix monotone on the positive semi-axis. Such a function induces a function $\varphi (x)=\mathrm{tr}\, \big (g(x)\big )$ on the cone of squares of an arbitrary Euclidean Jordan algebra. We show that $\varphi (x) - \ln \det (x)$ is a self-concordant function on the interior of the cone. We also show that $-\ln (t-\varphi (x))-\ln \det (x)$ is $(r+1)$-self-concordant barrier on the epigraph of $\varphi ,$ where r is the rank of the Jordan algebra. The case $\phi (x)=\mathrm{tr(x\ln x)}$ is discussed in detail.