### Abstract

For a finite set of distinct points S = {p_{i} ε l} in ℝ^{d} there exists Î ⊆ I such that all points in Ŝ = {p_{i} i ε Î} are extreme points and conv(Ŝ) = conv(S). Since a point p_{k} ε S is extreme if and only if the inequality p^{T}_{k} y ≤ 1 is necessary with respect to the representation {y ε ℝ^{d}{pipe} p^{T}_{i} y≤ 1, i ε I} 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.

Original language | English (US) |
---|---|

Pages (from-to) | 131-142 |

Number of pages | 12 |

Journal | Journal of Interdisciplinary Mathematics |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - 2007 |

### Fingerprint

### Keywords

- Convex hull
- Extreme points
- Linear inequalities
- Probabilistic method
- Redundancy
- Solid angles

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Interdisciplinary Mathematics*,

*10*(1), 131-142. https://doi.org/10.1080/09720502.2007.10700478

**Probabilistic algorithms for extreme point identification.** / Jibrin, Shafiu; Boneh, Arnon; Caron, Richard J.

Research output: Contribution to journal › Article

*Journal of Interdisciplinary Mathematics*, vol. 10, no. 1, pp. 131-142. https://doi.org/10.1080/09720502.2007.10700478

}

TY - JOUR

T1 - Probabilistic algorithms for extreme point identification

AU - Jibrin, Shafiu

AU - Boneh, Arnon

AU - Caron, Richard J.

PY - 2007

Y1 - 2007

N2 - For a finite set of distinct points S = {pi ε l} in ℝd 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 ε S is extreme if and only if the inequality pTk y ≤ 1 is necessary with respect to the representation {y ε ℝd{pipe} pTi y≤ 1, i ε I} of the polar dual SΔ 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.

AB - For a finite set of distinct points S = {pi ε l} in ℝd 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 ε S is extreme if and only if the inequality pTk y ≤ 1 is necessary with respect to the representation {y ε ℝd{pipe} pTi y≤ 1, i ε I} of the polar dual SΔ 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.

KW - Convex hull

KW - Extreme points

KW - Linear inequalities

KW - Probabilistic method

KW - Redundancy

KW - Solid angles

UR - http://www.scopus.com/inward/record.url?scp=56049096100&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=56049096100&partnerID=8YFLogxK

U2 - 10.1080/09720502.2007.10700478

DO - 10.1080/09720502.2007.10700478

M3 - Article

VL - 10

SP - 131

EP - 142

JO - Journal of Interdisciplinary Mathematics

JF - Journal of Interdisciplinary Mathematics

SN - 0972-0502

IS - 1

ER -