Splitting methods for split feasibility problems with application to Dantzig selectors. (3rd March 2017)

• Record Type: Journal Article
• Authors: He, Hongjin; Xu, Hong-Kun
• Is Part Of: Inverse problems. Volume 33:Number 5(2017:May)
• Publication Date: 2017-03-03
• Subjects: split feasibility problem -- regularization -- alternating direction method of multipliers -- proximal point algorithm -- variational inequality -- Dantzig selector; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aa5ec5
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: The split feasibility problem (SFP), which refers to the task of finding a point that belongs to a given nonempty, closed and convex set, and whose image under a bounded linear operator belongs to another given nonempty, closed and convex set, has promising applicability in modeling a wide range of inverse problems. Motivated by the increasingly data-driven regularization in the areas of signal/image processing and statistical learning, in this paper, we study the regularized split feasibility problem (RSFP), which provides a unified model for treating many real-world problems. By exploiting the split nature of the RSFP, we shall gainfully employ several efficient splitting methods to solve the model under consideration. A remarkable advantage of our methods lies in their easier subproblems in the sense that the resulting subproblems have closed-form representations or can be efficiently solved up to a high precision. As an interesting application, we apply the proposed algorithms for finding Dantzig selectors, in addition to demonstrating the effectiveness of the splitting methods through some computational results on synthetic and real medical data sets.