1,637 research outputs found
A Characterization Theorem and An Algorithm for A Convex Hull Problem
Given and , testing if , the convex hull of , is a fundamental
problem in computational geometry and linear programming. First, we prove a
Euclidean {\it distance duality}, distinct from classical separation theorems
such as Farkas Lemma: lies in if and only if for each there exists a {\it pivot}, satisfying . Equivalently, if and only if there exists a
{\it witness}, whose Voronoi cell relative to contains
. A witness separates from and approximate to
within a factor of two. Next, we describe the {\it Triangle Algorithm}: given
, an {\it iterate}, , and , if
, it stops. Otherwise, if there exists a pivot
, it replace with and with the projection of onto the
line . Repeating this process, the algorithm terminates in arithmetic operations, where
is the {\it visibility factor}, a constant satisfying and
, over all iterates . Additionally,
(i) we prove a {\it strict distance duality} and a related minimax theorem,
resulting in more effective pivots; (ii) describe -time algorithms that may compute a witness or a good
approximate solution; (iii) prove {\it generalized distance duality} and
describe a corresponding generalized Triangle Algorithm; (iv) prove a {\it
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.Comment: 42 pages, 17 figures, 2 tables. This revision only corrects minor
typo
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