On the Approximation of Unbounded Convex Sets by Polyhedra
本文研究无界凸集的多面体逼近，提出一种新的(ε,δ)逼近概念，并给出计算谱面体(ε,δ)逼近的算法，证明其正确性和有限性。
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.