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

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 ⩾ 3 sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can b...

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Bibliographic Details
Published in:Journal of functional analysis 2012, Vol.262 (1), p.400-408
Main Authors: Baillon, J.-B., Combettes, P.L., Cominetti, R.
Format: Article
Language:English
Subjects:
Alternating projections
Best approximation
Limit cycle
Von Neumann algorithm
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Summary: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 ⩾ 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 for minimizing smooth convex functions over a product of convex sets are also discussed.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2011.09.002