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arXiv · 1801.03216

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

Roberto Cominetti·Vera Roshchina·Andrew Williamson
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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.

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Roberto Cominetti, Vera Roshchina, Andrew Williamson. 2018-01-10. A counterexample to De Pierro's conjecture on the convergence of under-relaxed cyclic projections. https://arxiv.org/abs/1801.03216

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