节点文献
希尔伯特空间中的广义分裂问题的算法研究
【作者】 孙涛；
【导师】 陈汝栋；
【作者基本信息】 天津工业大学 ， 数学， 2017， 硕士
【摘要】 1994年,Censor和Elfving提出了分裂可行性问题SFP.为了解决SFP,Byrne提出了 CQ算法.Censor和Segal提出了分裂公共不动点问题并且给出解决这个问题的算法,它是分裂可行性问题的推广.2012年,A.Moudafi提出了分裂等式SEP问题,它也是SFP的推广.A.Moudafi为了解决SEP介绍了松弛的CQ算法和交替的 CQ 算法.最近,Chen et al.,Eslamian、Latif、Chuang、Chang 和Wang提出和研究了在实希尔伯特空间上的广义分裂可行性问题和广义分裂等式问题.本文主要从广义分裂公共不动点问题、广义分裂等式问题两个方面对希尔伯特空间中的分裂问题进行研究.首先为了解决广义分裂公共不动点问题,我们提出了一个新的算法,同时证明了由提出的算法产生的序列强收敛到广义分裂公共不动点问题的一个公共解.这部分中我们研究的是一类k-半压缩算子.作为一个应用,特殊的情况中拟非扩张算子和方向算子被考虑.这时,我们给出了数值例子和数值结果来验证了我们算法的收敛速度.其次,对于广义分裂等式问题,我们提出和研究了一个新的算法,同时证明了由提出的算法产生的序列强收敛到广义分裂等式问题的一个公共解.这部分中我们研究的是一类方向算子.作为一个应用,我们考虑一些例子,如:度量投影和次梯度投影.这时,我们给出了一些广义分裂等式数值结果说明了我们算法的收敛速率.
【Abstract】 The split feasibility problem SFP was first introduced in 1994 by Censor and Elfving.To solve the SFP,an algorithm called the CQ-algorithm was proposed by Byrne.Split common fixed point problem SCFP was introduced by Censor and Se-gal and proposed an algorithm for solving such the problem,which also generalizes the split feasibility problem.In 2012,A.Moudafi proposed the split equality prob-lem,which also generalizes the split feasibility problem.A.Moudafi introduced the alternating CQ-algorithms and the relaxed alternating CQ-algorithms for solv-ing split feasibility problem.Recently,Chen et al.,Eslamian and Latif,Chuang and Chang and Wang introduced and studied general split feasibility problem and general split equality problem in real Hilbert spaces.In this article we studied the split problem of Hilbert mainly from two as-pects:general split common fixed point problem,general split equality problem.Firstly,we propose a new iterative algorithm for solving general split common fixed point problem,in this aspect we study a family of k-demicontractive operators.As an application,Particular cases of directed operator and quasi-nonexpansive op-erator are also considered.Then,we gave an example and numerical results to illustrate the efficiency of our algorithms.Secondly,we propose a new iterative algorithm for solving general split equality problem,in this aspect we study a fam-ily of directed operators.As an application,we consider some examples of general split equality problems and fixed point problem.Then,we give several numerical results of general split common fixed point problems to demonstrate the efficiency of the proposed algorithm.