Polyhedral approximation of spectrahedral shadows via homogenization
arXiv:2305.16909 · doi:10.1007/s10957-023-02363-5
Abstract
This article is concerned with the problem of approximating a not necessarily bounded spectrahedral shadow, a certain convex set, by polyhedra. By identifying the set with its homogenization the problem is reduced to the approximation of a closed convex cone. We introduce the notion of homogeneous δ-approximation of a convex set and show that it defines a meaningful concept in the sense that approximations converge to the original set if the approximation error δ diminishes. Moreover, we show that a homogeneous δ-approximation of the polar of a convex set is immediately available from an approximation of the set itself under mild conditions. Finally, we present an algorithm for the computation of homogeneous δ-approximations of spectrahedral shadows and demonstrate it on examples.
23 pages, 3 figures; adds simplified version of Proposition 3.4, minor changes to bibliography