A polyhedral approximation algorithm for recession cones of spectrahedral shadows

TL;DR

This paper introduces two iterative algorithms for approximating the recession cones of spectrahedra and their projections using polyhedral cones, with proven correctness and numerical validation.

Contribution

It presents novel algorithms tailored for spectrahedra and their projections to efficiently approximate recession cones with arbitrary accuracy.

Findings

Algorithms are proven correct and finite.

Numerical examples demonstrate effectiveness.

Applicable to spectrahedra and projected spectrahedra.

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 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.