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

  • Amir Beck
  • Marc Teboulle
Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)


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.


Nonconvex affine feasibility Inverse problems Gradient projection algorithm Linear rate of convergence Scalable restricted isometry Mutual coherence of a matrix Sparse signal recovery Compressive sensing Affine rank minimization 


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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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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Industrial EngineeringTechnion Israel Institute of TechnologyHaifaIsrael

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