论文标题
光谱阴影经济衰退锥的多面体近似算法
A polyhedral approximation algorithm for recession cones of spectrahedral shadows
论文作者
论文摘要
仿射子空间与阳性半金属矩阵的锥的相交称为谱。其正交投影称为谱系阴影或投影谱。 Spectrahedra及其预测可以看作是Polyhedra的概括。本文涉及近似通过多面体锥的光谱和谱系阴影的衰退锥的问题。我们提出了两种迭代算法，以将外部和内部近似值计算为任意规定的准确性。第一种算法是针对谱图量身定制的，是由用于紧凑型凸组集的多面体近似算法得出的，并依赖于以下事实，即对经济衰退的代数描述可用。第二算法是为投影谱系设计的，不需要对经济衰退锥的代数描述，这通常更难获得。我们证明了这两种算法的正确性和有限性，并提供了数值示例。
The intersection of an affine subspace with the cone of positive semidefinite matrices is called a spectrahedron. An orthogonal projection thereof is called a spectrahedral shadow or projected spectrahedron. Spectrahedra and their projections can be seen as a generalization of polyhedra. This article is concerned with the problem of approximating the recession cones of spectrahedra and spectrahedral shadows via polyhedral cones. We present two iterative algorithms to compute outer and inner approximations to within an arbitrary prescribed accuracy. The first algorithm is tailored to spectrahedra and is derived from polyhedral approximation algorithms for compact convex sets and relies on the fact, that an algebraic description of the recession cone is available. The second algorithm is designed for projected spectrahedra and does not require an algebraic description of the recession cone, which is in general more difficult to obtain. We prove correctness and finiteness of both algorithms and provide numerical examples.
