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

• 成果类型: Article
• 署名作者: Zhao, Yun-Bin; Luo, Zhi-Quan
• 署名单位: University of Birmingham; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2016.0791
• 发表日期: 2017
• 页码: 57-76
• 关键词: signal recovery; thresholding algorithm; systems; EQUATIONS; sparsity

摘要

The l(0)-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 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 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.