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A Linearly Convergent Algorithm for Solving a Class of Nonconvex/Affine Feasibility Problems

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Book cover Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 49))

Abstract

We introduce a class of nonconvex/affine feasibility problems (NCF), that consists of finding a point in the intersection of affine constraints with a nonconvex closed set. This class captures some interesting fundamental and NP hard problems arising in various application areas such as sparse recovery of signals and affine rank minimization that we briefly review. Exploiting the special structure of (NCF), we present a simple gradient projection scheme which is proven to converge to a unique solution of (NCF) at a linear rate under a natural assumption explicitly given defined in terms of the problem’s data.

AMS 2010 Subject Classification: 90C30, 90C26

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Acknowledgements

We thank two anonymous referees for their useful comments and suggestions. This research was partially supported by the Israel Science Foundation under ISF Grant 489-06.

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Authors and Affiliations

  1. Department of Industrial Engineering, Technion Israel Institute of Technology, Haifa, 32000, Israel

    Amir Beck

Authors
  1. Amir Beck
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  2. Marc Teboulle
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Corresponding author

Correspondence to Amir Beck .

Editor information

Editors and Affiliations

  1. Okanagan Campus, Department of Mathematics and Statistic, University of British Columbia, Kelowna, V1V 1V7, British Columbia, Canada

    Heinz H. Bauschke

  2. School of Mathematics & Statistics, Division of Information Technology, University of South Australia, Mawson Lakes Blvd., Mawson Lakes, 5095, South Australia, Australia

    Regina S. Burachik

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Place Jussieu 4, Paris, 75005, France

    Patrick L. Combettes

  4. Dept. Physics, Lab. Atomic & Solid State, Cornell University, Clark Hall, Ithaca, 14853-2501, New York, USA

    Veit Elser

  5. Institut für Numerische und Angewandte M, Universität Göttingen, Lotzestr. 16-18, Göttingen, 37073, Germany

    D. Russell Luke

  6. Faculty of Mathematics, Dept. Combinatorics & Optimization, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada

    Henry Wolkowicz

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Beck, A., Teboulle, M. (2011). A Linearly Convergent Algorithm for Solving a Class of Nonconvex/Affine Feasibility Problems. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_3

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