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

Yun Bin Zhao, Zhi Quan Luo

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

The l0-minimization problem that seeks the sparsest point of a polyhedral set is a long-standing, 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 (US)
Pages (from-to)57-76
Number of pages20
JournalMathematics of Operations Research
Volume42
Issue number1
DOIs
StatePublished - Feb 2017

Keywords

  • Bilevel programming
  • Convex optimization
  • Duality theory
  • Polyhedral set
  • Sparsest point
  • Sparsity recovery
  • Strict complementarity
  • Weighted l1-algorithm

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