Operator convexity along lines, self-concordance, and sandwiched Rényi entropies
Kerry He, James Saunderson, Hamza Fawzi
TL;DR
The paper addresses the challenge of constructing self-concordant barriers for epigraphs/hypographs of quantum-information trace functions, notably the sandwiched Rényi entropy. It introduces a criterion: if a function is operator convex along lines, its logarithmic barrier is self-concordant, enabling interior-point methods for optimization. The authors derive self-concordant barriers for $\Psi_α$ and related quantities in key α-ranges ($[\tfrac{1}{2},1]$ and $[1,2]$) with optimal barrier parameters, and provide simplified proofs for known results via operator-concavity along lines; they implement these barriers in QICS and demonstrate numerical efficacy on tasks like sandwiched Rényi mutual information and quantum rate-distortion. The work advances efficient optimization in quantum information theory and suggests broader extensions to other trace-function families, with open questions for $α\ge2$.
Abstract
Barrier methods play a central role in the theory and practice of convex optimization. One of the most general and successful analyses of barrier methods for convex optimization, due to Nesterov and Nemirovskii, relies on the notion of self-concordance. While an extremely powerful concept, proving self-concordance of barrier functions can be very difficult. In this paper we give a simple way to verify that the natural logarithmic barrier of a convex nonlinear constraint is self-concordant via the theory of operator convex functions. Namely, we show that if a convex function is operator convex along any one-dimensional restriction, then the natural logarithmic barrier of its epigraph is self-concordant. We apply this technique to construct self-concordant barriers for the epigraphs of functions arising in quantum information theory. Notably, we apply this to the sandwiched Rényi entropy function, for which no self-concordant barrier was known before. Additionally, we utilize our sufficient condition to provide simplified proofs for previously established self-concordance results for the noncommutative perspective of operator convex functions. An implementation of the convex cones considered in this paper is now available in our open source interior-point solver QICS.