Probabilistic algorithms for extreme point identification

Abstract

For a finite set of distinct points S = {pi, i ε I} , in ℝ d there exists Î ⊆ I such that all points in Ŝ = {pi , i ε Î are extreme points and conv(Ŝ) = conv(S). Since a point pk is extreme if and only if the inequality is necessary with respect to the representation of the polar dual S Δ of S, Ŝ can be found by classifying the inequalities in the representation as necessary or redundant. Thus, the problem of finding Ŝ is polynomial. This paper shows the advantage of using probabilistic hit-and-run algorithms applied to the polar dual for the quick identification of points in Ŝ and shows how, in an application to a certain cluster analysis problem, it can be used to also identify points in S/Ŝ. Further, it shows that the hit-and-run variant known as Stand-and-Hit provides approximations for the exterior solid angles at the extreme points of S.