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Convexifying the set of matrices of bounded rank: applications to the quasiconvexification and convexification of the rank function

Optimization Letters volume 6, pages 841–849 (2012)Cite this article

Abstract

We provide an explicit description of the convex hull of the set of matrices of bounded rank, restricted to balls for the spectral norm. As applications, we deduce two relaxed forms of the rank function restricted to balls for the spectral norm: one is the quasiconvex hull of this rank function, another one is the convex hull of the rank function, thus retrieving Fazel’s theorem (Matrix rank minimization with applications, 2002).

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Authors and Affiliations

  1. Institut de mathématiques de Toulouse, Université Paul Sabatier (Toulouse III), 118, route de Narbonne, 31062, Toulouse Cedex 09, France

    Jean-Baptiste Hiriart-Urruty & Hai Yen Le

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  1. Jean-Baptiste Hiriart-Urruty
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  2. Hai Yen Le
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Correspondence to Hai Yen Le.

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Hiriart-Urruty, JB., Le, H.Y. Convexifying the set of matrices of bounded rank: applications to the quasiconvexification and convexification of the rank function. Optim Lett 6, 841–849 (2012). https://doi.org/10.1007/s11590-011-0304-4

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  • DOI: https://doi.org/10.1007/s11590-011-0304-4

Keywords

  • Rank of a matrix
  • Spectral norm
  • Trace (or nuclear) norm
  • Quasiconvex hull of a function