An $O(1/\varepsilon)$-Iteration Triangle Algorithm for A Convex Hull Membership

A fundamental problem in linear programming, machine learning, and computational geometry is the {\it Convex Hull Membership} (CHM): Given a point $p$ and a subset $S$ of $n$ points in $\mathbb{R}^m$, is $p \in conv(S)$? The {\it Triangle Algorithm} (TA) computes $p' \in conv(S)$ so that, either $\Vert p'- p \Vert \leq \varepsilon R$, $R= \max \{\Vert p -v \Vert: v\in S\}$; or $p'$ is a {\it witness}, i.e. the orthogonal bisector of $pp'$ separates $p$ from $conv(S)$. By the {\it Spherical}-CHM we mean a CHM, where $p=0$, $\Vert v \Vert=1$, $\forall v \in S$. First, we prove the equivalence of exact and approximate versions of CHM and Spherical-CHM. On the one hand, this makes it possible to state a simple $O(1/\varepsilon^2)$ iteration TA, each taking $O(n+m)$ time. On the other hand, using this iteration complexity we prove if for each $p' \in conv(S)$ with $\Vert p \Vert > \varepsilon$ that is not a witness there is $v \in S$ with $\Vert p' - v \Vert \geq \sqrt{1+ \varepsilon}$, the iteration complexity of TA reduces to $O(1/\varepsilon)$. This matches complexity of Nesterov's fast-gradient method. The analysis also suggests a strategy for when the property does not hold at an iterate. Lastly, as an application of TA, we show how to solve strict LP feasibility as a dual of CHM. In summary, TA and the Spherical-CHM provide a convenient geometric setting for efficient solution to large-scale CHM and related problems, such as computing all vertices of $conv(S)$.