PreprintCC BY 4.0arXiv
Spectral Self-Concordant Barriers: A Proof of a Conjecture by Tunçel
Abstract
Let Q ⊆ ℝn be a open, convex, and symmetric set and let f:Q → ℝ be a ϑ-self-concordant barrier function, that is also symmetric. We show the associated spectral function F(X)= 𝒪2 (f ∘ λ)(X) is a 𝒪2ϑ-self-concordant function, where 𝒪 = 8max{1+3ϑ, 12}, on the domain of all n × n symmetric matrices X, with eigenvalues λ(X) ∈ Q. This generalizes the relationship between the standard logarithmic barriers −∑i=1nlog xi and −log det X and gives a solution to a conjecture by L.~Tunçel, 1995.
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BibTeX
@misc{pena2026spectral,
title = {{Spectral Self-Concordant Barriers: A Proof of a Conjecture by Tunçel}},
author = {Javier Peña and Hristo Sendov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17442},
eprint = {2610.17442},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17442},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral Self-Concordant Barriers: A Proof of a Conjecture by Tunçel
AU - Javier Peña
AU - Hristo Sendov
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17442
UR - https://arxiv.org/abs/2610.17442
AB - Let Q ⊆ ℝ^n be a open, convex, and symmetric set and let f:Q → ℝ be a ϑ-self-concordant barrier function, that is also symmetric. We show the associated spectral function F(X)= 𝒪^2 (f ∘ λ)(X) is a 𝒪^2ϑ-self-concordant function, where 𝒪 = 8max{1+3ϑ, 12}, on the domain of all n × n symmetric matrices X, with eigenvalues λ(X) ∈ Q. This generalizes the relationship between the standard logarithmic barriers −∑_i=1^nlog x_i and −log det X and gives a solution to a conjecture by L.~Tunçel, 1995.
ER -