Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces
Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces
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DOI：
10.1088/0266-5611/26/10/105018
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发表时间：
2010-10-01
期刊：
影响因子：
2.1
通讯作者：
Xu, Hong-Kun
中科院分区：
文献类型：
--
作者：
Xu, Hong-Kun
拆分可行性问题（SFP） （Censor and Elfving 1994 number）。算法8 221-39)的目的是找到一个点x*，其性质是x*是C的一个元素，Ax*是Q的一个元素，其中C和Q分别是实数Hilbert空间H(1)和H(2)的非空闭凸子集，a是从H(1)到H(2)的有界线性算子。SFP模型由相位恢复问题引起的逆问题（Censor and Elfving 1994）。算法8 221-39)和调强放射治疗（Censor et al 2005逆问题21 2071-84）。本文讨论了在无穷维希尔伯特空间中求解SFP问题的迭代方法。Byrne的CQ算法（2002 Inverse Problems 18 441- 53,2004 Inverse Problems 20 10320）确实是梯度-投影算法在凸极小化中的一个特例，在无限维情况下一般具有弱收敛性。我们将主要使用不动点算法来研究SFP。引入了一种松弛的CQ算法，该算法只涉及到半空间上的投影，从而使算法具有可实现性。引入正则化和迭代算法求解SFP的最小范数解。
The split feasibility problem (SFP) (Censor and Elfving 1994 Numer. Algorithms 8 221-39) is to find a point x* with the property that x* is an element of C and Ax* is an element of Q, where C and Q are the nonempty closed convex subsets of the real Hilbert spaces H(1) and H(2), respectively, and A is a bounded linear operator from H(1) to H(2). The SFP models inverse problems arising from phase retrieval problems (Censor and Elfving 1994 Numer. Algorithms 8 221-39) and the intensity-modulated radiation therapy (Censor et al 2005 Inverse Problems 21 2071-84). In this paper we discuss iterative methods for solving the SFP in the setting of infinite-dimensional Hilbert spaces. The CQ algorithm of Byrne (2002 Inverse Problems 18 441-53, 2004 Inverse Problems 20 10320) is indeed a special case of the gradient-projection algorithm in convex minimization and has weak convergence in general in infinite-dimensional setting. We will mainly use fixed point algorithms to study the SFP. A relaxed CQ algorithm is introduced which only involves projections onto half-spaces so that the algorithm is implementable. Both regularization and iterative algorithms are also introduced to find the minimum-norm solution of the SFP.