Self-Dual Polyhedral Cones and Their Slack Matrices

IF 1.5 2区 数学 Q2 MATHEMATICS, APPLIED SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-07-26 DOI:10.1137/22m1519869
João Gouveia, Bruno F. Lourenço
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Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 3, Page 1096-1121, September 2023.
Abstract. We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the underlying cone is irreducible, then the corresponding PSD slacks are not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of DNN matrices, which correspond to a family of extreme rays not previously described. More surprisingly, we show that, unless the cone is simplicial, PSD slacks not only fail to be completely positive matrices but they also lie outside the cone of completely positive semidefinite matrices. Finally, we show how one can use semidefinite programming to probe the existence of self-dual cones with given combinatorics. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes.
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自对偶多面体锥及其松弛矩阵
SIAM矩阵分析与应用学报,第44卷,第3期,1096-1121页,2023年9月。摘要。分析了自对偶多面体锥,证明了其松弛矩阵的几个性质。特别地,我们证明了自对偶等价于正半定松弛的存在性。除此之外,我们证明了如果底层锥是不可约的,那么相应的PSD松弛不仅是双重非负矩阵(DNN),而且是DNN矩阵的锥的极限射线,它对应于先前未描述的极限射线族。更令人惊讶的是,我们证明,除非锥是简单的,否则PSD松弛不仅不能是完全正矩阵,而且它们也位于完全正半定矩阵的锥之外。最后,我们展示了如何用半定规划来探讨给定组合的自对偶锥的存在性。我们的结果给出了多面体锥,但我们也讨论了一些后果负自极性多面体。
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来源期刊
CiteScore
2.90
自引率
6.70%
发文量
61
审稿时长
6-12 weeks
期刊介绍: The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.
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