On the finite termination of the Douglas-Rachford method for the convex feasibility problem |
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Authors: | Shin-ya Matsushita Li Xu |
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Institution: | 1. Department of Electronics and Information Systems, Akita Prefectural University, Yuri-Honjo, Japan. matsushita@akita-pu.ac.jp;3. Department of Electronics and Information Systems, Akita Prefectural University, Yuri-Honjo, Japan. |
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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. |
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Keywords: | Douglas–Rachford method convex feasibility problem finite termination Slater condition metric projection nonexpansive error bound conic feasibility problem semidefinite feasibility problem |
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