Affiliation:
1. Emeritus Professor, Department of Computer Science, Rutgers University, Piscataway, NJ
2. Department of Computer Science, Rutgers University, Piscataway, NJ
Abstract
The
Convex Hull Membership
(CHM) tests whether
\( p \in conv(S) \)
, where
p
and the
n
points of
S
lie in
\( \mathbb { R}^m \)
. CHM finds applications in Linear Programming, Computational Geometry, and Machine Learning. The
Triangle Algorithm
(TA), previously developed, in
\( O(1/\varepsilon ^2) \)
iterations computes
\( p^{\prime } \in conv(S) \)
, either an
\( \varepsilon \)
-
approximate solution
, or a
witness
certifying
\( p \not\in conv(S) \)
. We first prove the equivalence of exact and approximate versions of CHM and
Spherical
-CHM, where
\( p=0 \)
and
\( \Vert v\Vert =1 \)
for each
v
in
S
. If for some
\( M \ge 1 \)
every non-witness with
\( \Vert p^{\prime }\Vert \gt \varepsilon \)
admits
\( v \in S \)
satisfying
\( \Vert p^{\prime } - v\Vert \ge \sqrt {1+\varepsilon /M} \)
, we prove the number of iterations improves to
\( O(M/\varepsilon) \)
and
\( M \le 1/\varepsilon \)
always holds. Equivalence of CHM and Spherical-CHM implies
Minimum Enclosing Ball
(MEB) algorithms can be modified to solve CHM. However, we prove
\( (1+ \varepsilon) \)
-approximation in MEB is
\( \Omega (\sqrt {\varepsilon }) \)
-approximation in Spherical-CHM. Thus, even
\( O(1/\varepsilon) \)
iteration MEB algorithms are not superior to Spherical-TA. Similar weakness is proved for MEB core sets. Spherical-TA also results a variant of the
All Vertex Triangle Algorithm
(AVTA) for computing all vertices of
\( conv(S) \)
. Substantial computations on distinct problems demonstrate that TA and Spherical-TA generally achieve superior efficiency over algorithms such as Frank–Wolfe, MEB, and LP-Solver.
Publisher
Association for Computing Machinery (ACM)
Subject
Applied Mathematics,Software
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