The Minimum Euclidean-Norm Point on a Convex Polytope: Wolfe's Combinatorial Algorithm is Exponential

TL;DR

This paper demonstrates that Wolfe's combinatorial algorithm for finding the minimum Euclidean-norm point on a convex polytope can take exponential time, and links the problem to strongly-polynomial linear programming.

Contribution

It provides the first exponential-time example for Wolfe's method and establishes a reduction from linear programming to the minimum norm point problem.

Findings

Wolfe's method can be exponential in the worst case.

Linear programming reduces to the minimum norm point problem in strongly-polynomial time.

The minimum Euclidean-norm point problem is computationally complex.

Abstract

The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the most practical algorithms for submodular function minimization. We present the first example that Wolfe's method takes exponential time. Additionally, we improve previous results to show that linear programming reduces in strongly-polynomial time to the minimum norm point problem over a simplex.