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 the 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 ( 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 ln x ) is discussed in detail.