There is no variational characterization of the cycles in the method of periodic projections

TL;DR

This paper proves that for three or more convex sets, the limit cycles of the periodic projection method cannot be characterized as minimizers of any functional, answering a long-standing open question.

Contribution

It provides a negative answer to the open question about variational characterization of cycles in the method of periodic projections for multiple sets.

Findings

Limit cycles for m ≥ 3 sets are not variational minimizers.

The paper discusses projection algorithms for convex functions over product sets.

Answers a question dating back to the 1960s about the nature of these cycles.

Abstract

The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers of a certain functional. In this paper we answer this question in the negative. Projection algorithms that minimize smooth convex functions over a product of convex sets are also discussed.