Operator convexity along lines, self-concordance, and sandwiched Rényi entropies
本文给出一个简单方法，通过算子凸性验证凸非线性约束的自然对数障碍函数是自和谐的，并应用于量子信息理论中的夹心Rényi熵函数，首次为其构造了自和谐障碍函数。
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 .