On the finite termination of the Douglas-Rachford method for the convex feasibility problem. (1st November 2016)

• Record Type: Journal Article
• Authors: Matsushita, Shin-ya; Xu, Li
• Is Part Of: Optimization. Volume 65:Number 11(2016)
• Page Start: 2037
• Page End: 2047
• Publication Date: 2016-11-01
• Subjects: Douglas–Rachford method -- convex feasibility problem -- finite termination -- Slater condition -- metric projection -- nonexpansive -- error bound -- conic feasibility problem -- semidefinite feasibility problem; 47J25 -- 47H09 -- 90C22 -- 90C08 -- 93B40; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2016.1209674
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : In this paper we apply the Douglas–Rachford (DR) method to solve the problem of finding a point in the intersection of the interior of a closed convex cone and a closed convex set in an infinite-dimensional Hilbert space. For this purpose, we propose two variants of the DR method which can find a point in the intersection in a finite number of iterations. In order to analyse the finite termination of the methods, we use some properties of the metric projection and a result regarding the rate of convergence of fixed point iterations. As applications of the results, we propose the methods for solving the conic and semidefinite feasibility problems, which terminate at a solution in a finite number of iterations.