齐次锥的面结构及其在齐次弦性中的应用
Faces of homogeneous cones and applications to homogeneous chordality
AI总结：
本文阐明齐次锥的面结构，证明每个面可由自同构映射到主面，并利用广义Cholesky分解算法构造，进而证明齐次锥投影暴露，分析齐次弦性PSD矩阵锥的面结构及其对补全问题的影响。
摘要：
一个凸锥 $\mathcal{K}$ 被称为齐次的，如果其自同构群在其相对内部上传递作用。齐次锥的重要例子包括对称锥以及满足由齐次弦图给出的稀疏模式的正半定（PSD）矩阵锥。本文的目标是阐明齐次锥的面结构，使其像PSD矩阵的面一样透明。我们证明，齐次锥 $\mathcal{K}$ 的每个面都能被 $\mathcal{K}$ 的一个自同构映射到其有限多个所谓主面之一。此外，构造这样的自同构可以通过利用广义Cholesky分解以算法方式完成。在其他推论中，我们证明了齐次锥是投影暴露的，这加强了先前的最佳结果即它们是顺从的。利用我们的结果，我们将仔细分析满足齐次弦性的PSD矩阵锥的面结构，并讨论对相应的PSD补全问题族的影响。
英文摘要：
A convex cone $\mathcal{K}$ is said to be homogeneous if its group of automorphisms acts transitively on its relative interior. Important examples of homogeneous cones include symmetric cones and cones of positive semidefinite (PSD) matrices that follow a sparsity pattern given by a homogeneous chordal graph. Our goal in this paper is to elucidate the facial structure of homogeneous cones and make it as transparent as the faces of the PSD matrices. We prove that each face of a homogeneous cone $\mathcal{K}$ is mapped by an automorphism of $\mathcal{K}$ to one of its finitely many so-called principal faces. Furthermore, constructing such an automorphism can be done algorithmically by making use of a generalized Cholesky decomposition. Among other consequences, we give a proof that homogeneous cones are projectionally exposed, which strengthens the previous best result that they are amenable. Using our results, we will carefully analyze the facial structure of cones of PSD matrices satisfying homogeneous chordality and discuss consequences for the corresponding family of PSD completion problems.
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