希尔伯特空间中的广义分裂问题的算法研究

发布时间：2018-05-18 22:01

本文选题：分裂可行性问题 + 分裂等式问题　；参考：《天津工业大学》2017年硕士论文

【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]:In 1994, Censor and Elfving put forward the splitting feasibility problem. In order to solve the CQ algorithm proposed by SFPN Byrne. Censor and Segal put forward the split common fixed point problem and give the algorithm to solve this problem, it is a generalization of the splitting feasibility problem. A. Moudafi put forward the split equation SEP problem in 2012. It is also a generalization of SFP. A. Moudafi introduced the relaxed CQ algorithm and alternate CQ algorithm to solve the problem of SEP. Recently, Chen et al. Eslamian Latifan Chuangli Chang and Wang have proposed and studied the generalized splitting feasibility problem and generalized splitting equation problem in real Hilbert space. In this paper, the splitting problem in Hilbert space is studied from two aspects: the generalized split common fixed point problem and the generalized split equality problem. In order to solve the generalized split common fixed point problem, we propose a new algorithm and prove that the sequence generated by the proposed algorithm converges strongly to a common solution of the generalized split common fixed point problem. In this part we study a class of k-semi-contractive operators. As an application, quasi-nonexpansive operators and directional operators are considered in special cases. At this point, we give numerical examples and numerical results to verify the convergence rate of our algorithm. Secondly, for the generalized split equality problem, we propose and study a new algorithm and prove that the sequence generated by the proposed algorithm converges strongly to a common solution of the generalized split equality problem. In this part we study a class of directional operators. As an application, we consider some examples, such as metric projection and subgradient projection. At this point, we give some numerical results of generalized splitting equations to illustrate the convergence rate of our algorithm.
【学位授予单位】：天津工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177.1

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