Regularization Method for the Approximate Split Equality Problem in Infinite-Dimensional Hilbert Spaces
Regularization Method for the Approximate Split Equality Problem in Infinite-Dimensional Hilbert Spaces
复制标题
无限维希尔伯特空间中近似分裂等式问题的正则化方法
DOI：
10.1155/2013/813635
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发表时间：
2013-01-01
影响因子：
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通讯作者：
Ren, Yijie
中科院分区：
文献类型：
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作者：
Chen, Rudong;Li, Junlei;Ren, Yijie
在无限维Hilbert空间框架下研究了近似分裂等式问题（ASEP）。设H-1，H-2和H-3是无限维真实的Hilbert空间，H-1的C子集和H-2的Q子集是两个非空闭凸集，A：H-1 -> H-3和B：H-2 -> H-3是两个有界线性算子.在无限维Hilbert空间中的ASEP是最小化函数f（x，y）=（1/2）平行于Ax - By（2）（2）在x是C的元素和y是Q的元素。最近，Moudalle和Byrne提出了几种求解分裂等式问题的算法，并证明了它们的收敛性。注意，他们的算法在无限维希尔伯特空间中只有弱收敛。本文利用正则化方法建立了求解无穷维Hilbert空间中ASEP的一个单步迭代算法，并证明了该算法产生的序列强收敛于ASEP的最小范数解.注意，通过在ASEP中取B = I，我们恢复了近似分裂可行性问题（ASFP）。
We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let H-1, H-2, and H-3 be infinite-dimensional real Hilbert spaces, let C subset of H-1 and Q subset of H-2 be two nonempty closed convex sets, and let A : H-1 -> H-3 and B : H-2 -> H-3 be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function f(x, y) = (1/2)parallel to Ax - By parallel to(2)(2) over x is an element of C and y is an element of Q. Recently, Moudafi and Byrne had proposed several algorithms for solving the split equality problem and proved their convergence. Note that their algorithms have only weak convergence in infinite-dimensional Hilbert spaces. In this paper, we used the regularization method to establish a single-step iterative for solving the ASEP in infinite-dimensional Hilbert spaces and showed that the sequence generated by such algorithm strongly converges to the minimum-norm solution of the ASEP. Note that, by taking B = I in the ASEP, we recover the approximate split feasibility problem (ASFP).