Self-concordance and matrix monotonicity with applications to quantum entanglement problems

Abstract

Let V be a Euclidean Jordan algebra and Ω be a cone of invertible squares in V. Suppose that g:R+→R is a matrix monotone function on the positive semiaxis which naturally induces a function g˜:Ω→V. We show that −g˜ is compatible (in the sense of Nesterov–Nemirovski) with the standard self-concordant barrier B(x)=−lndet(x) on Ω. As a consequence, we show that for any c ∈ Ω, the functions of the form −tr(cg˜(x))+B(x) are self-concordant on Ω. In particular, the function x↦−tr(clnx) is a self-concordant barrier function on Ω. 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. 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.