Let $V$ be an Euclidean Jordan algebra and $\Omega$ be a cone of invertible squares in $V$. Suppose that $g:\mathbb{R}_{+} \to \mathbb{R}$ is a matrix monotone function on the positive semiaxis which naturally induces a function $\tilde{g}: \Omega \to V$. We show that $-\tilde{g}$ is compatible (in the sense of Nesterov-Nemirovski) with the standard self-concordant barrier $B(x) = -\ln\det(x)$ on $\Omega$. As a consequence, we show that for any $c \in \Omega$, the functions of the form $-\tr(c\tilde{g}(x)) + B(x)$ are self-concordant on $\Omega$. In particular, the function $x \mapsto -\tr(c\ln x)$ is a self-concordant barrier function on $\Omega$. Using these results, we apply a long-step path-following algorithm developed in [L.~Faybusovich and C.~Zhou Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions. \emph{Comput Optim Appl}, 72(3):769-795, 2019] to a number of important optimization problems arising in quantum information theory. Results of numerical experiments and comparisons with existing methods are presented.
Citation
preprint, University of Notre Dame, April, 2019
Article
View Self-Concordance and Matrix Monotonicity with Applications to Quantum Entanglement Problems