On the Approximation of Unbounded Convex Sets by Polyhedra
Daniel Dörfler
Abstract
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 $$\left( \varepsilon , \delta \right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>ε</mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ</mml:mi> </mml:mfenced> </mml:math> -approximation that integrates the unbounded case in a meaningful way. Some basic results about $$\left( \varepsilon , \delta \right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>ε</mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ</mml:mi> </mml:mfenced> </mml:math> -approximations are proved for general convex sets. In the last section, an algorithm for the computation of $$\left( \varepsilon , \delta \right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>ε</mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ</mml:mi> </mml:mfenced> </mml:math> -approximations of spectrahedra is presented. Correctness and finiteness of the algorithm are proved.