A characterization theorem and an algorithm for a convex hull problem

Abstract

Given $$S= \{v_1, \dots , v_n\} \subset {\mathbb {R}}^m$$ and $$p \in {\mathbb {R}}^m$$ , testing if $$p \in conv(S)$$ , the convex hull of $$S$$ , 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: $$p$$ lies in $$conv(S)$$ if and only if for each $$p' \in conv(S)$$ there exists a pivot, $$v_j \in S$$ satisfying $$d(p',v_j) \ge d(p,v_j)$$ . Equivalently, $$p \not \in conv(S)$$ if and only if there exists a witness, $$p' \in conv(S)$$ whose Voronoi cell relative to $$p$$ contains $$S$$ . A witness separates $$p$$ from $$conv(S)$$ and approximate $$d(p, conv(S))$$ to within a factor of two. Next, we describe the Triangle Algorithm: given $$\epsilon \in (0,1)$$ , an iterate, $$p' \in conv(S)$$ , and $$v \in S$$ , if $$d(p, p') < \epsilon d(p,v)$$ , it stops. Otherwise, if there exists a pivot $$v_j$$ , it replace $$v$$ with $$v_j$$ and $$p'$$ with the projection of $$p$$ onto the line $$p'v_j$$ . Repeating this process, the algorithm terminates in $$O(mn \min \{ \epsilon ^{-2}, c^{-1}\ln \epsilon ^{-1} \})$$ arithmetic operations, where $$c$$ is the visibility factor, a constant satisfying $$c \ge \epsilon ^2$$ and $$\sin (\angle pp'v_j) \le 1/\sqrt{1+c}$$ , over all iterates $$p'$$ . In particular, the geometry of the input data may result in efficient complexities such as $$O(mn \root t \of {\epsilon ^{-2}} \ln \epsilon ^{-1})$$ , $$t$$ a natural number, or even $$O(mn \ln \epsilon ^{-1})$$ . Additionally, (i) we prove a strict distance duality and a related minimax theorem, resulting in more effective pivots; (ii) describe $$O(mn \ln \epsilon ^{-1})$$ -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.