A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone. Cones that are homogeneous and self-dual are called symmetric. Conic optimization problems over symmetric cones have been extensively studied, particularly in the literature on interior-point algorithms, and as the foundation of modelling tools for convex optimization. In this paper we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual.We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. Chordal sparsity guarantees that positive define matrices in the cone have zero-fill Cholesky factorizations. The stronger properties that make the cone homogeneous guarantee that the inverse Cholesky factors have the same zero-fill pattern. We describe transitive subsets of the cone automorphism groups, and important properties of the composition of log-det barriers with the automorphisms.Next, we consider extensions to linear slices of the positive semidefinite cone, and review conditions that make such cones homogeneous. An important example is the matrix norm cone, the epigraph of a quadratic-over-linear matrix function. The properties of homogeneous sparse matrix cones are shown to extend to this more general class of homogeneous matrix cones.We then give an overview of the algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation.We conclude by discussing the role of homogeneous structure in primal–dual symmetric interior-point methods, contrasting this with the well-developed algorithms for symmetric cones that exploit the strong properties of self-scaled barriers, and with symmetric primal–dual methods for general convex cones.

中文翻译：

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齐次矩阵锥的线性优化

如果凸锥的自同构群传递作用于锥的内部，则凸锥是齐次的。同质且自对偶的锥体称为对称锥体。对称圆锥上的圆锥优化问题已得到广泛研究，特别是在有关内点算法的文献中，并且作为凸优化建模工具的基础。在本文中，我们考虑了研究较少的锥体上的锥体优化问题，这些锥体是齐次但不一定是自对偶的。我们从具有给定稀疏模式的半正定对称矩阵的锥体开始。此类中的均匀锥体的特征是嵌套块箭头稀疏模式，弦稀疏模式的一个子集。弦稀疏性保证锥体中的正定义矩阵具有零填充 Cholesky 分解。使圆锥均匀的更强的特性保证了逆 Cholesky 因子具有相同的零填充模式。我们描述了锥自同构群的传递子集，以及具有自同构的 log-det 障碍组合的重要性质。接下来，我们考虑对半正定锥的线性切片的扩展，并回顾使此类锥齐次的条件。一个重要的例子是矩阵范数锥，二次线性矩阵函数的题记。齐次稀疏矩阵锥的性质被证明可以扩展到这种更一般的齐次矩阵锥。然后我们概述了由 Vinberg 和 Rothaus 提出的齐次锥的代数理论。