A counterexample to De Pierro's conjecture on the convergence of under-relaxed cyclic projections

TL;DR

This paper disproves de Pierro's conjecture by providing a counterexample where the under-relaxed cyclic projection method's limit cycles do not converge to a least squares solution.

Contribution

The paper constructs a specific example of three convex sets in three-dimensional space that invalidates the general convergence of the under-relaxed cyclic projection method.

Findings

Counterexample shows non-convergence in certain cases

Disproves the general validity of de Pierro's conjecture

Highlights limitations of the cyclic projection method

Abstract

The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the $\varepsilon$-under-relaxed cyclic projection method converge when $\varepsilon\downarrow 0$ towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in $\mathbb{R}^3$ for which the $\varepsilon$-under-relaxed cycles do not converge.