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

J.-B. Baillon(Jean-Bernard.Baillon***at***univ-paris1.fr)
P. L. Combettes(plc***at***math.jussieu.fr)
R. Cominetti(rccc***at***dii.uchile.cl)

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.

Keywords: projection algorithm, inconsistent feasibility problem, cycles, periodic projections

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Infinite Dimensional Optimization

Citation:

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Entry Submitted: 02/07/2011
Entry Accepted: 02/07/2011
Entry Last Modified: 02/07/2011

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