Polyhedral Approximation of Spectrahedral Shadows via Homogenization

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 $$\delta $$ 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 $$\delta $$ δ diminishes. Moreover, we show that a homogeneous $$\delta $$ 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 $$\delta $$ δ -approximations of spectrahedral shadows and demonstrate it on examples.