A relaxed alternating CQ-algorithm for convex feasibility problems
A relaxed alternating CQ-algorithm for convex feasibility problems
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DOI：
10.1016/j.na.2012.11.013
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发表时间：
2013-03-01
影响因子：
1.4
通讯作者：
Moudafi, Abdellatif
中科院分区：
文献类型：
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作者：
Moudafi, Abdellatif
设H-1，H-2，H-3是真实的Hilbert空间，H-1的C子集，H-2的Q子集是两个非空闭凸水平集，A：H-1 -> H-3，B：H-2 -> H-3是两个有界线性算子.我们的兴趣是解决以下新的凸可行性问题找到x是C的元素，y是Q的元素，使得Ax = By，（1.1）这允许变量x和y之间存在不对称和部分关系。本文提出并研究了一种松弛交替CQ算法（RACQA）的收敛性，证明了由这种算法产生的序列弱收敛于（1.1）的解。RACQA的有趣之处在于我们只需要半空间上的投影，从而使放松的CQ算法可实现。注意，通过在（1.1）中取B = I，我们恢复了最初在Censor和Elfving（1994）[13]中引入的分裂凸可行性问题，并在后来的调强放射治疗中使用（Censor等人（2006）[11]）。我们还恢复了放松CQ算法介绍的杨（2004）[8]通过特殊化两个B和一个给定的参数。(C)2012爱思唯尔有限公司保留所有权利。
Let H-1, H-2 H-3 be real Hilbert spaces, let C subset of H-1, Q subset of H-2 be two nonempty closed convex level sets, let A : H-1 -> H-3, B : H-2 -> H-3 be two bounded linear operators. Our interest is in solving the following new convex feasibility problemFind x is an element of C, y is an element of Q such that Ax = By, (1.1)which allows asymmetric and partial relations between the variables x and y. In this paper, we present and study the convergence of a relaxed alternating CQ-algorithm (RACQA) and show that the sequences generated by such an algorithm weakly converge to a solution of (1.1). The interest of RACQA is that we just need projections onto half-spaces, thus making the relaxed CQ-algorithm implementable. Note that, by taking B = I, in (1.1), we recover the split convex feasibility problem originally introduced in Censor and Elfving (1994) [13] and used later in intensity-modulated radiation therapy (Censor et al. (2006) [11]). We also recover the relaxed CQ-algorithm introduced by Yang (2004) [8] by particularizing both B and a given parameter. (C) 2012 Elsevier Ltd. All rights reserved.