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 φ(x) = tr(g(x)) on the cone of squares of an arbitrary Euclidean Jordan algebra. We show that φ(x)− ln det(x) is a self-concordant function on the interior of the cone. We also show that − ln(t−φ(x))−ln det(x) is √ 5 3 (r+1)-self-concordant barrier on the epigraph of φ, where r is the rank of the Jordan algebra. The case φ(x) = tr(x lnx) is discussed in detail.