Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies

TL;DR

This paper introduces a simple method to verify self-concordance of barrier functions using operator convexity, enabling new barriers for quantum information functions like sandwiched Rényi entropy, with practical implementation in an open-source solver.

Contribution

The paper presents a novel criterion linking operator convexity along lines to self-concordance of barriers, simplifying proofs and enabling new barrier constructions for quantum information functions.

Findings

Verified self-concordance of logarithmic barriers via operator convexity.

Constructed barriers for quantum information functions, including sandwiched Rényi entropy.

Provided simplified proofs for existing self-concordance results.

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 R\'enyi 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.