First-order methods for the convex hull membership problem

TL;DR

This paper reviews and compares first-order methods for the convex hull membership problem, introducing new stopping criteria and analyzing their connections, with practical experiments and applications in linear programming and image classification.

Contribution

It introduces stopping criteria for first-order methods in CHMP and reveals connections between Triangle Algorithm and Frank-Wolfe, enhancing decision-making efficiency.

Findings

Triangle Algorithm is an inexact Frank-Wolfe method based on distance duality.

Stopping criteria improve the efficiency of Frank-Wolfe and Gradient methods for CHMP.

Numerical experiments show algorithm performance varies with convex hull geometry.

Abstract

The convex hull membership problem (CHMP) consists in deciding whether a certain point belongs to the convex hull of a finite set of points, a decision problem with important applications in computational geometry and in foundations of linear programming. In this study, we review, compare and analyze first-order methods for CHMP, namely, Frank-Wolfe type methods, Projected Gradient methods and a recently introduced geometric algorithm, called Triangle Algorithm (TA). We discuss the connections between this algorithm and Frank-Wolfe, showing that TA can be interpreted as an inexact Frank-Wolfe. Despite this similarity, TA is strongly based on a theorem of alternatives known as distance duality. By using this theorem, we propose suitable stopping criteria for CHMP to be integrated into Frank-Wolfe type and Projected Gradient, specializing these methods to the membership decision problem. Interestingly, Frank-Wolfe integrated with such stopping criteria coincides with a greedy version of the Triangle Algorithm which is, in its turn, equivalent to an algorithm due to von Neumann. We report numerical experiments on random instances of CHMP, carefully designed to cover different scenarios, that indicate which algorithm is preferable according to the geometry of the convex hull and the relative position of the query point. Concerning potential applications, we present two illustrative examples, one related to linear programming feasibility problems and another related to image classification problems.