Results
Title

A POLYHEDRAL APPROXIMATION ALGORITHM FOR RECESSION CONES OF SPECTRAHEDRAL SHADOWS.

Authors

DÖRFLER, DANIEL; LOHNE, ANDREAS

Abstract

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 spec-trahedron. 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.

Subjects

APPROXIMATION theory; SUBSPACES (Mathematics); MATRICES (Mathematics); ORTHOGRAPHIC projection; ITERATIVE methods (Mathematics)

Publication

Journal of Nonlinear & Variational Analysis, 2024, Vol 8, Issue 4, p549

ISSN

2560-6921

Publication type

Academic Journal

DOI

10.23952/jnva.8.2024.4.05