Alternating Linear Minimization: Revisiting von Neumann's alternating projections

TL;DR

This paper introduces an algorithm for intersecting convex sets using only linear minimization oracles, revisiting von Neumann's classic alternating projections method.

Contribution

It extends the classical projection method to a weaker setting with linear minimization oracles, broadening its applicability.

Findings

The algorithm successfully finds points in the intersection of convex sets using linear minimization.

It generalizes von Neumann's method to scenarios lacking explicit projection operators.

The approach demonstrates effectiveness in convex set intersection problems with limited access.

Abstract

In 1933 von Neumann proved a beautiful result that one can approximate a point in the intersection of two convex sets by alternating projections, i.e., successively projecting on one set and then the other. This algorithm assumes that one has access to projection operators for both sets. In this note, we consider the much weaker setup where we have only access to linear minimization oracles over the convex sets and present an algorithm to find a point in the intersection of two convex sets.