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The method of periodic projections consists in iterating projections onto $m$closed convex subsets of a Hilbert space according to a periodic sweepingstrategy. In the presence of $m\geq 3$ sets, a long-standing question goingback to the 1960s is whether the limit cycles obtained by such a process can becharacterized as the minimizers of a certain functional. In this paper weanswer this question in the negative. Projection algorithms that minimizesmooth convex functions over a product of convex sets are also discussed.

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