Optimization Online


Constructing New Weighted l1-Algorithms for the Sparsest Points of Polyhedral Sets

Yun-Bin Zhao (y.zhao.2***at***bham.ac.uk)
Z.Q. Luo (luozzq***at***ece.umn.edu)

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.

Keywords: Polyhedral set, sparsest point, weighted ℓ1-algorithm, convex optimization, sparsity recovery, strict complementarity, duality theory, bilevel programming

Category 1: Convex and Nonsmooth Optimization (Other )

Category 2: Applications -- Science and Engineering (Basic Sciences Applications )


Download: [PDF]

Entry Submitted: 08/01/2016
Entry Accepted: 08/01/2016
Entry Last Modified: 10/01/2016

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society