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Abstract
The ℓ_{0}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 ℓ_{1}method is one of the most plausible methods for solving this problem. In this paper, we develop a new weighted ℓ_{1}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 ℓ_{1} minimization. As a result, ℓ_{0}minimization can be transformed, in theory, to ℓ_{0}maximization in dual space through some weight. This theoretical result provides a basis and an incentive to develop a new weighted ℓ_{1}algorithm, which is remarkably distinct from existing sparsityseeking 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 language  English 

Pages (fromto)  5776 
Number of pages  20 
Journal  Mathematics of Operations Research 
Volume  42 
Issue number  1 
Early online date  30 Sept 2016 
DOIs  
Publication status  Published  Feb 2017 
Keywords
 duality theory
 bilevel programming
 polyhedral set
 sparsest point
 weighted ℓ1algorithm
 convex optimization
 sparsity recovery
 strict complementarity
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Dive into the research topics of 'Constructing new weighted ℓ_{1}algorithms for the sparsest points of polyhedral sets'. Together they form a unique fingerprint.Projects
 1 Finished

Foundation and Reweighted Algorithms for Sparsest Points of Convex Sets with Application to Data Processing
Zhao, Y.
Engineering & Physical Science Research Council
18/04/13 → 31/05/15
Project: Research Councils