Separable self-concordant spectral functions and a conjecture of Tunçel

Abstract

Given a separable strongly self-concordant function ${f: \mathbb {R}^n \rightarrow \mathbb {R}}$ , we show the associated spectral function ${F(X)=(f \circ \lambda)(X)}$ is also a strongly self- concordant function. In addition, there is a universal constant ${\fancyscript {O} \leq 22}$ such that if f(x) is a separable self-concordant barrier, then ${\fancyscript {O}^2F(X)}$ is a self-concordant barrier. This generalizes the relationship between the standard logarithmic barriers ${-\sum_{i=1}^n {\rm log}\,x_i}$ and -log det X and gives a partial solution to a conjecture of L. Tunçel.