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Journal of Optimization Theory and Applications

, Volume 172, Issue 1, pp 70–101 | Cite as

On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery

  • Jean Charles Gilbert
Article
  • 496 Downloads

Abstract

This paper first proposes another proof of the necessary and sufficient conditions of solution uniqueness in 1-norm minimization given recently by H. Zhang, W. Yin, and L. Cheng. The analysis avoids the need of the surjectivity assumption made by these authors and should be mainly appealing by its short length (it can therefore be proposed to students exercising in convex optimization). In the second part of the paper, the previous existence and uniqueness characterization is extended to the recovery problem where the L1 norm is substituted by a polyhedral gauge. In addition to present interest for a number of practical problems, this extension clarifies the geometrical aspect of the previous uniqueness characterization. Numerical techniques are proposed to compute a solution to the polyhedral gauge recovery problem in polynomial time and to check its possible uniqueness by a simple linear algebra test.

Keywords

Basis pursuit Convex polyhedral function Gauge recovery L1 minimization Minkowski function Optimality conditions Sharp minimum Solution existence and uniqueness 

Mathematics Subject Classification

65K05 90C05 90C25 90C46 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.INRIA ParisParis Cedex 12France

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