The combination projection method for solving convex feasibility problems
type: Item
instance of: scholarly article
title: The combination projection method for solving convex feasibility problems
zbMATH Open document ID: 1468.65071
author: Songnian He
author: Qiao-Li Dong
published in: Mathematics
publication date: 2018-12-18T00:00:00Z
review text: Summary: In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some \(x^\ast \in C : = \cap_{i = 1}^m \{x \in \mathcal{H}\mid c_i(x) \leq 0 \}\), where \(m\) is a positive integer, \(\mathcal H\) is a real Hilbert space, and \(\{c_i \}_{i = 1}^m\) are convex functions defined as \(\mathcal H\). The key of the CPM is that, for the current iterate \(x^k\), the CPM firstly constructs a new level set \(H_k\) through a convex combination of some of \(\{c_i \}_{i = 1}^m\) in an appropriate way, and then updates the new iterate \(x^{k + 1}\) only by using the projection \(P_{H_k}\). We also introduce the combination relaxation projection methods (CRPM) to project onto half-spaces to make CPM easily implementable. The simplicity and easy implementation are two advantages of our methods since only one projection is used in each iteration and the projections are also easy to calculate. The weak convergence theorems are proved and the numerical results show the advantages of our methods.
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C52
zbMATH DE Number: 6994852
zbMATH Keywords: convex feasibility problem
zbMATH Keywords: projection operator
zbMATH Keywords: combination projection method
zbMATH Keywords: Hilbert space
MaRDI profile type: Publication
full work available at URL: math6110249
OpenAlex ID: W2900772434
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DOI: 10.3390/MATH6110249
description: scientific article; zbMATH DE number 6994852