Year: 2026
Type: preprint
Abstract: Let $Q \subseteq \mathbb{R}^n$ be a open, convex, and symmetric set and let $f:Q \rightarrow \mathbb{R}$ be a $\vartheta$-self-concordant barrier function, that is also symmetric. We show the associated spectral function $F(X)= \mathcal{O}^2 (f \circ λ)(X)$ is a $\mathcal{O}^2\vartheta$-self-concordant function, where $\mathcal{O} = 8\max\{1+3\vartheta, 12\}$, on the domain of all $n \times n$ symmetric matrices $X$, with eigenvalues $λ(X) \in Q$. This generalizes the relationship between the standard logarithmic barriers $-\sum_{i=1}^n\log x_i$ and $-\log \det X$ and gives a solution to a conjecture by L.~Tunçel, 1995.
Source: arXiv (Cornell University)
Authors: Peña, Javier, Sendov, Hristo
Institutions: Western University, Carnegie Mellon University
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221766333