On the Approximation of Unbounded Convex Sets by Polyhedra

TL;DR

This paper introduces a new method for approximating unbounded convex sets with polyhedra, extending existing compact set results, and presents an algorithm for spectrahedra with proven correctness.

Contribution

It proposes the ($oldsymbol{ ext{ε,δ}}$)-approximation concept for unbounded convex sets and develops an algorithm for spectrahedra, filling a gap in the literature.

Findings

($ ext{ε,δ}$)-approximations are effective for unbounded convex sets

The algorithm for spectrahedra is correct and finite

Basic properties of ($ ext{ε,δ}$)-approximations are established

Abstract

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the connections between existing results before introducing a new notion of polyhedral approximation called ($\varepsilon,\delta$)-approximation that integrates the unbounded case in a meaningful way. Some basic results about ($\varepsilon,\delta$)- approximations are proven for general convex sets. In the last section an algorithm for the computation of ($\varepsilon,\delta$)-approximations of spectrahedra is presented. Correctness and finiteness of the algorithm are proven.