A Relaxed CQ Algorithm for Solving Split Feasibility Problem
A Relaxed CQ Algorithm for Solving Split Feasibility Problem
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DOI：
10.1109/iccase.2011.5997801
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发表时间：
2011-07
期刊：
影响因子：
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通讯作者：
Jing Sun;Rudong Chen
中科院分区：
文献类型：
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作者：
Jing Sun;Rudong Chen
分裂可行性问题（SFP）是要找到一个点\(x\in C\)，使得\(Ax\in Q\)，其中\(A:H_1\rightarrow H_2\)是一个有界线性算子，并且\(C\)和\(Q\)分别是希尔伯特空间\(H_1\)和\(H_2\)的非空闭凸子集。在本文中，我们提出了一种用于解决分裂可行性问题的松弛CQ算法。该迭代算法生成一个序列\(\{x_n\}\)如下：\(x_{n + 1}=(1-\alpha_n)x_n+\alpha_nP_{C_n}(x_n-\gamma A^*(I - P_{Q_n})Ax_n)\)，\(n\geq0\)，其中\(0 < \gamma < \frac{2}{\|A\|^2}\)，\(x_0\in H_1\)，\(P_{C_n}\)和\(P_{Q_n}\)分别是到\(C_n\)和\(Q_n\)上的最近点投影。然后我们证明了CQ算法弱收敛到SFP的一个解。
The split feasibility problem(SFP) is to find a point x ∈ C such that Ax ∈ Q, where A, H/sub 1/ → H/sub 2/ is a bounded linear operator, and C and Q be nonempty closed convex subset of Hilbert space H/sub 1/ and H/sub 2/, respectively. In this paper, we proposed a relaxed CQ algorithm for solving split feasibility problem. The iterative algorithm generates a sequence fxng as follows x/sub n+1/=(1- α /sub n/)x/sub n/+ α /sub n/Pc/sub n/(x/sub n/- γ A*(I-P/sub Qn)Ax/sub n/), n ≥ 0, where 0 < γ < 2/A/sup 2/, x/sub 0/ ∈ H/sub 1/, P/sub Cn/ and P/sub Qn/ are the nearest point projections onto C/sub n/ and Q/sub n/, respectively. Then we proved that CQ algorithm converges weakly to a solution of the SFP.