A characterization theorem and an algorithm for a convex hull problem

Bahman Kalantari ()

Annals of Operations Research, 2015, vol. 226, issue 1, 349 pages

Abstract: Given $$S= \{v_1, \dots , v_n\} \subset {\mathbb {R}}^m$$ S = { v 1 , ⋯ , v n } ⊂ R m and $$p \in {\mathbb {R}}^m$$ p ∈ R m , testing if $$p \in conv(S)$$ p ∈ c o n v ( S ) , the convex hull of $$S$$ S , is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean distance duality, distinct from classical separation theorems such as Farkas Lemma: $$p$$ p lies in $$conv(S)$$ c o n v ( S ) if and only if for each $$p' \in conv(S)$$ p ′ ∈ c o n v ( S ) there exists a pivot, $$v_j \in S$$ v j ∈ S satisfying $$d(p',v_j) \ge d(p,v_j)$$ d ( p ′ , v j ) ≥ d ( p , v j ) . Equivalently, $$p \not \in conv(S)$$ p ∉ c o n v ( S ) if and only if there exists a witness, $$p' \in conv(S)$$ p ′ ∈ c o n v ( S ) whose Voronoi cell relative to $$p$$ p contains $$S$$ S . A witness separates $$p$$ p from $$conv(S)$$ c o n v ( S ) and approximate $$d(p, conv(S))$$ d ( p , c o n v ( S ) ) to within a factor of two. Next, we describe the Triangle Algorithm: given $$\epsilon \in (0,1)$$ ϵ ∈ ( 0 , 1 ) , an iterate, $$p' \in conv(S)$$ p ′ ∈ c o n v ( S ) , and $$v \in S$$ v ∈ S , if $$d(p, p') > \epsilon d(p,v)$$ d ( p , p ′ ) > ϵ d ( p , v ) , it stops. Otherwise, if there exists a pivot $$v_j$$ v j , it replace $$v$$ v with $$v_j$$ v j and $$p'$$ p ′ with the projection of $$p$$ p onto the line $$p'v_j$$ p ′ v j . Repeating this process, the algorithm terminates in $$O(mn \min \{ \epsilon ^{-2}, c^{-1}\ln \epsilon ^{-1} \})$$ O ( m n min { ϵ - 2 , c - 1 ln ϵ - 1 } ) arithmetic operations, where $$c$$ c is the visibility factor, a constant satisfying $$c \ge \epsilon ^2$$ c ≥ ϵ 2 and $$\sin (\angle pp'v_j) \le 1/\sqrt{1+c}$$ sin ( ∠ p p ′ v j ) ≤ 1 / 1 + c , over all iterates $$p'$$ p ′ . In particular, the geometry of the input data may result in efficient complexities such as $$O(mn \root t \of {\epsilon ^{-2}} \ln \epsilon ^{-1})$$ O ( m n ϵ - 2 t ln ϵ - 1 ) , $$t$$ t a natural number, or even $$O(mn \ln \epsilon ^{-1})$$ O ( m n ln ϵ - 1 ) . Additionally, (i) we prove a strict distance duality and a related minimax theorem, resulting in more effective pivots; (ii) describe $$O(mn \ln \epsilon ^{-1})$$ O ( m n ln ϵ - 1 ) -time algorithms that may compute a witness or a good approximate solution; (iii) prove generalized distance duality and describe a corresponding generalized Triangle Algorithm; (iv) prove a sensitivity theorem to analyze the complexity of solving LP feasibility via the Triangle Algorithm. The Triangle Algorithm is practical and competitive with the simplex method, sparse greedy approximation and first-order methods. Finally, we discuss future work on applications and generalizations. Copyright Springer Science+Business Media New York 2015

Keywords: Convex hull; Linear programming; Duality; Approximation algorithms; Gilbert’s algorithm; Frank–Wolfe algorithm; Minimax (search for similar items in EconPapers)
Date: 2015
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://hdl.handle.net/10.1007/s10479-014-1707-2 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:annopr:v:226:y:2015:i:1:p:301-349:10.1007/s10479-014-1707-2

Ordering information: This journal article can be ordered from
http://www.springer.com/journal/10479

DOI: 10.1007/s10479-014-1707-2

Access Statistics for this article

Annals of Operations Research is currently edited by Endre Boros

More articles in Annals of Operations Research from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().