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The optimal barrier parameter of the three-dimensional power cone
Abstract
We determine the optimal logarithmically homogeneous self-concordant barrier parameter for the three-dimensional power cone. Although several self-concordant barriers for this cone have been constructed, its optimal barrier parameter had remained unknown. For a power cone with exponent c∈(0,1), we prove that the optimal parameter is 3/(1+min{c,1−c}). Except at c=1/2, this value is strictly smaller than the parameter 3−2min{c,1−c} of Chares' conjectured barrier. At c=1/2, the optimal parameter equals $2$, in agreement with the optimal barrier parameter of the second-order cone. The proof combines a sharp lower bound for arbitrary logarithmically homogeneous self-concordant barriers with an explicit barrier attaining the bound.
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BibTeX
@misc{zhang2026optimal,
title = {{The optimal barrier parameter of the three-dimensional power cone}},
author = {Rui-Jin Zhang and Wenhao Fu and Yu-Hong Dai},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06404},
eprint = {2611.06404},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06404},
}
RIS
TY - UNPB
TI - The optimal barrier parameter of the three-dimensional power cone
AU - Rui-Jin Zhang
AU - Wenhao Fu
AU - Yu-Hong Dai
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06404
UR - https://arxiv.org/abs/2611.06404
AB - We determine the optimal logarithmically homogeneous self-concordant barrier parameter for the three-dimensional power cone. Although several self-concordant barriers for this cone have been constructed, its optimal barrier parameter had remained unknown. For a power cone with exponent c∈(0,1), we prove that the optimal parameter is 3/(1+min{c,1−c}). Except at c=1/2, this value is strictly smaller than the parameter 3−2min{c,1−c} of Chares' conjectured barrier. At c=1/2, the optimal parameter equals $2$, in agreement with the optimal barrier parameter of the second-order cone. The proof combines a sharp lower bound for arbitrary logarithmically homogeneous self-concordant barriers with an explicit barrier attaining the bound.
ER -