A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem
A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem
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DOI：
10.1088/0266-5611/22/6/007
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发表时间：
2006-12-01
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
变量Krasnosel‘skii-Mann算法通过公式x(n+1)=(1-α(N))x(N)+α(N)T(N)x(N)产生一个序列{x(N)}，其中{α(N)}是[0，1]中的一个序列，{T-n}是一个非扩张映射序列。在相当一般的Banach空间中，我们将证明生成的序列{x(N)}弱收敛。这个结果被用来解决分裂可行性问题，即寻找一个点x，其性质是x是C的元素，Ax是Q的元素，其中C和Q分别是Hilbert空间H-1和H-2的闭凸子集，A是从H-1到H-2的有界线性算子。最近由检查员等人提出的多集分裂可行性问题被描述为：找到一个点x是布尔元，C-N(i=1)i使得Ax是布尔元，(M)(j=1)q(J)，其中N和M是正整数，{C-1，…，C-N}和{Q(1)，…，Q(M)}分别是H-1和H-2的闭凸子集，A也是从H-1到H-2的有界线性算子。本文的目的之一是在无限维希尔伯特空间的框架下引入更多的迭代算法来解决这一问题。
A variable Krasnosel'skii-Mann algorithm generates a sequence {x(n)} via the formula x(n+1) = (1 - alpha(n))x(n) + alpha(n)T(n)x(n), where {alpha(n)} is a sequence in [0, 1] and {T-n} is a sequence of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence {x(n)} generated converges weakly. This result is used to solve the split feasibility problem which 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 closed convex subsets of Hilbert spaces H-1 and H-2, respectively, and A is a bounded linear operator from H-1 to H-2. The multiple-set split feasibility problem recently introduced by Censor et al is stated as finding a point x is an element of boolean AND C-N(i=1)i such that Ax is an element of boolean AND(M)(j=1) Q(j), where N and M are positive integers, {C-1,..., C-N} and {Q(1),..., Q(M)} are closed convex subsets of H-1 and H-2, respectively, and A is again a linear bounded operator from H-1 to H-2. One of the purposes of this paper is to introduce more iterative algorithms that solve this problem in the framework of infinite-dimensional Hilbert spaces.