An algorithmic separating hyperplane theorem and its applications

Research output: Contribution to journalArticle

Abstract

We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets K,K of the Euclidean space intersect, and when they are disjoint. The theorem, referred as distance duality, is distinct from classical separation theorems. Next, utilizing the theorem we develop a substantially generalized and stronger version of the Triangle Algorithm, originally designed for the convex hull membership problem. If δ =d(K,K ), the Euclidean distance between the sets, ρ the maximum of their diameters, and ε a prescribed tolerance, the Triangle Algorithm approximates δ , or induces a separating hyperplane, or approximates optimal supporting hyperplanes. Specifically, it computes (p,p )∈K×K satisfying any of the following conditions desired: (1) d(p,p )≤εd(p,v), v∈K, or d(p,p )≤εd(p ,v ), v ∈K (when δ =0); (2) the orthogonal bisector of pp separates K from K ; (3) d(p,p )−δ ≤εd(p,p ) (when δ >0); (4) the supporting hyperplanes orthogonal to pp satisfy δ −d(H,H )≤εd(p,p ). The corresponding number of iterations to solve these are, O(1∕ε 2 ), O(ρ 2 ∕δ 2 ), and O(ρ 2 ∕δ 2 ε) for the last two, all independent of K,K . The complexity in each iteration of the first two tasks is computing for a given pair of iterates (p,p )∈K×K a pivot, i.e. v∈K with d(p,v)≥d(p ,v), or v ∈K with d(p ,v )≥d(p,v ). For the last two the complexity of each iteration is either computing a pivot, or supporting hyperplanes (H,H ) orthogonal to pp . In the worst-case the complexity of each iteration is solving a linear program over K or K . Special cases include when K and K are convex hulls of finite sets, or polytopes described as the intersection of halfspaces. The corresponding problems include, linear and quadratic programming, SVM and more. In separate work we describe computational comparison between Triangle Algorithm, Frank–Wolfe, Gilbert's Algorithm, SMO, and more. Future work includes extensions to unbounded convex sets, non-Euclidean norms, also combinatorial and NP-complete problems.

Original languageEnglish (US)
Pages (from-to)59-82
Number of pages24
JournalDiscrete Applied Mathematics
Volume256
DOIs
StatePublished - Mar 15 2019

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Hyperplane
Iteration
Triangle
Theorem
Pivot
Convex Hull
Quadratic programming
Bisector
Separation Theorem
Linear programming
Approximate Algorithm
Computing
Computational complexity
Euclidean Distance
Polytopes
Quadratic Programming
Intersect
Linear Program
Iterate
Convex Sets

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

@article{6515b6a7058e4045b97a2f1d031be022,
title = "An algorithmic separating hyperplane theorem and its applications",
abstract = "We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets K,K ′ of the Euclidean space intersect, and when they are disjoint. The theorem, referred as distance duality, is distinct from classical separation theorems. Next, utilizing the theorem we develop a substantially generalized and stronger version of the Triangle Algorithm, originally designed for the convex hull membership problem. If δ ∗ =d(K,K ′ ), the Euclidean distance between the sets, ρ ∗ the maximum of their diameters, and ε a prescribed tolerance, the Triangle Algorithm approximates δ ∗ , or induces a separating hyperplane, or approximates optimal supporting hyperplanes. Specifically, it computes (p,p ′ )∈K×K ′ satisfying any of the following conditions desired: (1) d(p,p ′ )≤εd(p,v), v∈K, or d(p,p ′ )≤εd(p ′ ,v ′ ), v ′ ∈K ′ (when δ ∗ =0); (2) the orthogonal bisector of pp ′ separates K from K ′ ; (3) d(p,p ′ )−δ ∗ ≤εd(p,p ′ ) (when δ ∗ >0); (4) the supporting hyperplanes orthogonal to pp ′ satisfy δ ∗ −d(H,H ′ )≤εd(p,p ′ ). The corresponding number of iterations to solve these are, O(1∕ε 2 ), O(ρ ∗ 2 ∕δ ∗ 2 ), and O(ρ ∗ 2 ∕δ ∗ 2 ε) for the last two, all independent of K,K ′ . The complexity in each iteration of the first two tasks is computing for a given pair of iterates (p,p ′ )∈K×K ′ a pivot, i.e. v∈K with d(p,v)≥d(p ′ ,v), or v ′ ∈K ′ with d(p ′ ,v ′ )≥d(p,v ′ ). For the last two the complexity of each iteration is either computing a pivot, or supporting hyperplanes (H,H ′ ) orthogonal to pp ′ . In the worst-case the complexity of each iteration is solving a linear program over K or K ′ . Special cases include when K and K ′ are convex hulls of finite sets, or polytopes described as the intersection of halfspaces. The corresponding problems include, linear and quadratic programming, SVM and more. In separate work we describe computational comparison between Triangle Algorithm, Frank–Wolfe, Gilbert's Algorithm, SMO, and more. Future work includes extensions to unbounded convex sets, non-Euclidean norms, also combinatorial and NP-complete problems.",
author = "Bahman Kalantari",
year = "2019",
month = "3",
day = "15",
doi = "10.1016/j.dam.2018.05.009",
language = "English (US)",
volume = "256",
pages = "59--82",
journal = "Discrete Applied Mathematics",
issn = "0166-218X",
publisher = "Elsevier",

}

An algorithmic separating hyperplane theorem and its applications. / Kalantari, Bahman.

In: Discrete Applied Mathematics, Vol. 256, 15.03.2019, p. 59-82.

Research output: Contribution to journalArticle

TY - JOUR

T1 - An algorithmic separating hyperplane theorem and its applications

AU - Kalantari, Bahman

PY - 2019/3/15

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N2 - We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets K,K ′ of the Euclidean space intersect, and when they are disjoint. The theorem, referred as distance duality, is distinct from classical separation theorems. Next, utilizing the theorem we develop a substantially generalized and stronger version of the Triangle Algorithm, originally designed for the convex hull membership problem. If δ ∗ =d(K,K ′ ), the Euclidean distance between the sets, ρ ∗ the maximum of their diameters, and ε a prescribed tolerance, the Triangle Algorithm approximates δ ∗ , or induces a separating hyperplane, or approximates optimal supporting hyperplanes. Specifically, it computes (p,p ′ )∈K×K ′ satisfying any of the following conditions desired: (1) d(p,p ′ )≤εd(p,v), v∈K, or d(p,p ′ )≤εd(p ′ ,v ′ ), v ′ ∈K ′ (when δ ∗ =0); (2) the orthogonal bisector of pp ′ separates K from K ′ ; (3) d(p,p ′ )−δ ∗ ≤εd(p,p ′ ) (when δ ∗ >0); (4) the supporting hyperplanes orthogonal to pp ′ satisfy δ ∗ −d(H,H ′ )≤εd(p,p ′ ). The corresponding number of iterations to solve these are, O(1∕ε 2 ), O(ρ ∗ 2 ∕δ ∗ 2 ), and O(ρ ∗ 2 ∕δ ∗ 2 ε) for the last two, all independent of K,K ′ . The complexity in each iteration of the first two tasks is computing for a given pair of iterates (p,p ′ )∈K×K ′ a pivot, i.e. v∈K with d(p,v)≥d(p ′ ,v), or v ′ ∈K ′ with d(p ′ ,v ′ )≥d(p,v ′ ). For the last two the complexity of each iteration is either computing a pivot, or supporting hyperplanes (H,H ′ ) orthogonal to pp ′ . In the worst-case the complexity of each iteration is solving a linear program over K or K ′ . Special cases include when K and K ′ are convex hulls of finite sets, or polytopes described as the intersection of halfspaces. The corresponding problems include, linear and quadratic programming, SVM and more. In separate work we describe computational comparison between Triangle Algorithm, Frank–Wolfe, Gilbert's Algorithm, SMO, and more. Future work includes extensions to unbounded convex sets, non-Euclidean norms, also combinatorial and NP-complete problems.

AB - We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets K,K ′ of the Euclidean space intersect, and when they are disjoint. The theorem, referred as distance duality, is distinct from classical separation theorems. Next, utilizing the theorem we develop a substantially generalized and stronger version of the Triangle Algorithm, originally designed for the convex hull membership problem. If δ ∗ =d(K,K ′ ), the Euclidean distance between the sets, ρ ∗ the maximum of their diameters, and ε a prescribed tolerance, the Triangle Algorithm approximates δ ∗ , or induces a separating hyperplane, or approximates optimal supporting hyperplanes. Specifically, it computes (p,p ′ )∈K×K ′ satisfying any of the following conditions desired: (1) d(p,p ′ )≤εd(p,v), v∈K, or d(p,p ′ )≤εd(p ′ ,v ′ ), v ′ ∈K ′ (when δ ∗ =0); (2) the orthogonal bisector of pp ′ separates K from K ′ ; (3) d(p,p ′ )−δ ∗ ≤εd(p,p ′ ) (when δ ∗ >0); (4) the supporting hyperplanes orthogonal to pp ′ satisfy δ ∗ −d(H,H ′ )≤εd(p,p ′ ). The corresponding number of iterations to solve these are, O(1∕ε 2 ), O(ρ ∗ 2 ∕δ ∗ 2 ), and O(ρ ∗ 2 ∕δ ∗ 2 ε) for the last two, all independent of K,K ′ . The complexity in each iteration of the first two tasks is computing for a given pair of iterates (p,p ′ )∈K×K ′ a pivot, i.e. v∈K with d(p,v)≥d(p ′ ,v), or v ′ ∈K ′ with d(p ′ ,v ′ )≥d(p,v ′ ). For the last two the complexity of each iteration is either computing a pivot, or supporting hyperplanes (H,H ′ ) orthogonal to pp ′ . In the worst-case the complexity of each iteration is solving a linear program over K or K ′ . Special cases include when K and K ′ are convex hulls of finite sets, or polytopes described as the intersection of halfspaces. The corresponding problems include, linear and quadratic programming, SVM and more. In separate work we describe computational comparison between Triangle Algorithm, Frank–Wolfe, Gilbert's Algorithm, SMO, and more. Future work includes extensions to unbounded convex sets, non-Euclidean norms, also combinatorial and NP-complete problems.

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