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

J. B. Baillon; P. L. Combettes; R. Cominetti

doi:10.1016/j.jfa.2011.09.002

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

J. B. Baillon, P. L. Combettes, R. Cominetti

Faculty of Engineering and Science

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

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≥. 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.

Original language	English
Pages (from-to)	400-408
Number of pages	9
Journal	Journal of Functional Analysis
Volume	262
Issue number	1
DOIs	https://doi.org/10.1016/j.jfa.2011.09.002
State	Published - 1 Jan 2012

Keywords

Alternating projections
Best approximation
Limit cycle
Von Neumann algorithm

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10.1016/j.jfa.2011.09.002

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title = "There is no variational characterization of the cycles in the method of periodic projections",

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≥. 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.",

keywords = "Alternating projections, Best approximation, Limit cycle, Von Neumann algorithm",

author = "Baillon, {J. B.} and Combettes, {P. L.} and R. Cominetti",

note = "Funding Information: The research of P.L. Combettes was supported by the Agence Nationale de la Recherche under grant ANR-08-BLAN-0294-02. The research of R. Cominetti was supported by FONDECYT No. 1100046 and Instituto Milenio Sistemas Complejos en Ingenier{\'i}a. R. Cominetti gratefully acknowledges the hospitality of Universit{\'e} Pierre et Marie Curie, where this joint work began.",

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N2 - 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.

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KW - Limit cycle

KW - Von Neumann algorithm

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

Abstract

Keywords

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