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Local Linear Convergence for Alternating and Averaged Nonconvex Projections

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Abstract

The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness, for example), we prove that von Neumann’s method of “alternating projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method of “averaged projections” converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly.

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Authors and Affiliations

  1. ORIE, Cornell University, Ithaca, NY, 14853, USA

    A. S. Lewis

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    D. R. Luke

  3. INRIA Rhone-Alpes, 655 avenue de l’Europe, 38334, St Ismier Cedex, France

    J. Malick

Authors
  1. A. S. Lewis
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  2. D. R. Luke
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  3. J. Malick
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Correspondence to A. S. Lewis.

Additional information

Communicated by Michael Todd.

Research of A.S. Lewis supported in part by National Science Foundation Grant DMS-0504032.

Research of D.R. Luke supported in part by National Science Foundation Grant DMS-0712796.

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Lewis, A.S., Luke, D.R. & Malick, J. Local Linear Convergence for Alternating and Averaged Nonconvex Projections. Found Comput Math 9, 485–513 (2009). https://doi.org/10.1007/s10208-008-9036-y

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  • DOI: https://doi.org/10.1007/s10208-008-9036-y

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