Constructing new weighted ℓ1-algorithms for the sparsest points of polyhedral sets

Yun-Bin Zhao, Zhi-Quan Luo

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8 Citations (Scopus)
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Abstract

The ℓ0-minimization problem that seeks the sparsest point of a polyhedral set is a longstanding challenging problem in the fields of signal and image processing, numerical linear algebra and mathematical optimization. The weighted ℓ1-method is one of the most plausible methods for solving this problem. In this paper, we develop a new weighted ℓ1-method through the strict complementarity theory of linear programs. More specifically, we show that locating the sparsest point of a polyhedral set can be achieved by seeking the densest possible slack variable of the dual problem of weighted ℓ1 minimization. As a result, ℓ0-minimization can be transformed, in theory, to ℓ0-maximization in dual space through some weight. This theoretical result provides a basis and an incentive to develop a new weighted ℓ1-algorithm, which is remarkably distinct from existing sparsity-seeking methods. The weight used in our algorithm is computed via a certain convex optimization instead of being determined locally at an iterate. The guaranteed performance of this algorithm is shown under some conditions, and the numerical performance of the algorithm has been demonstrated by empirical simulations.
Original languageEnglish
Pages (from-to)57-76
Number of pages20
JournalMathematics of Operations Research
Volume42
Issue number1
Early online date30 Sept 2016
DOIs
Publication statusPublished - Feb 2017

Keywords

  • duality theory
  • bilevel programming
  • polyhedral set
  • sparsest point
  • weighted ℓ1-algorithm
  • convex optimization
  • sparsity recovery
  • strict complementarity

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