Weak, strong and linear convergence of the CQ-method via the regularity of Landweber operators. (3rd March 2020)

• Record Type: Journal Article
• Authors: Cegielski, Andrzej; Reich, Simeon; Zalas, Rafał
• Is Part Of: Optimization. Volume 69:Number 3(2020)
• Page Start: 605
• Page End: 636
• Publication Date: 2020-03-03
• Subjects: CQ-method -- linear rate -- split feasibility problem; 47J25 -- 47N10 -- 49N45; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2019.1598407
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne [Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 2002;18:441–453], we define an abstract Landweber transform which applies to more general operators than the metric projection. We call the result of this transform a Landweber operator. It turns out that the Landweber transform preserves many interesting properties. For example, the Landweber transform of a (quasi/firmly) nonexpansive mapping is again (quasi/firmly) nonexpansive. Moreover, the Landweber transform of a (weakly/linearly) regular mapping is again (weakly/linearly) regular. The preservation of regularity is important because it leads to (weak/linear) convergence of many CQ-type methods.