分裂等式均衡问题与临近点算法的研究

发布时间：2018-04-13 22:01

本文选题：分裂等式均衡问题 + 渐近非扩张多值映射　；参考：《云南财经大学》2017年硕士论文

【摘要】：由于分裂可行性问题的广泛应用性,它已成为非线性泛函分析中的一个极其重要的问题,并吸引了众多学者的关注.在1994年,Censor和Elfving[1]首先提出了有限维Hilbert空间中的分裂可行性问题,它的出现为我们解决不同空间上产生的问题提供了重要的理论依据.2010年,Moudafi[43]提出了分裂公共不动点问题,该问题是分裂可行性问题和凸可行性问题的推广,Moudafi[15]在2013年提出的分裂等式不动点问题又是分裂公共不动点问题的推广.为了解决分裂等式不动点问题,Moudafi在文献[15]中引入交替CQ算法,并且得到了弱收敛定理.2013年,Kazmi和Rizvi[30]提出的分裂均衡问题同样是是分裂公共不动点问题的推广.近年来由于分裂均衡问题的广泛应用,许多学者致力于分裂均衡问题的研究[26,30,31].比如Witthayarat,Ab-dou和Cho在文献[26]中提出了一种新的解决Hilbert空间中分裂均衡问题和不动点问题的压缩方法.本文中,我们主要研究分裂等式均衡问题,为了解决Hilbert空间中的分裂等式均衡问题,在2015年,Ma, Wang, Chang和Duan在文献[42]中提出了一种新的迭代算法得到了弱收敛定理,且在半紧的条件下得到其强收敛定理.但是半紧条件比较强.由此,我们结合Witthayarat, Abdou和Cho在文献[26]中提出的关于分裂均衡问题的迭代算法,在本文中构造一种新的迭代算法,在没有半紧条件并运用压缩投影的方法下得到了分裂等式均衡问题的强收敛定理.因为Hilbert空间是个完备的内积空间,因此接下来我们将Hilbert空间上关于分裂等式均衡问题的结果推广到Banach空间上.到目前为止还没有解决在Banach空间上的分裂等式均衡问题.由此我们就想在Banach空间上研究分裂等式均衡问题,利用广义投影,构造一种新的分裂等式均衡问题的迭代算法,并得到其强收敛定理.通过临近点算法(PPA)解决优化问题的一些收敛结果已经从一般的线性空间比如欧几里得空间,Hilbert空间以及Banach空间扩展到其它各种各样的空间.目标凸函数的最小值点的研究对于分析和几何方面的研究起到了关键性作用,优化问题可以应用在计算机视图,机器学习,电子结构计算,系统平衡,以及机器人操纵等方面[50-56].最近,Chang, Wu,Wang, Wang在文献[59]中提出并研究了改进的临近点算法以解决Hilbert空间中的非延伸型多值映射的不动点问题,其条件比较强.由此,我们把以上结果推广到更为广泛的渐近非扩张多值映射的临近点算法问题,证明其强弱收敛性.本文的内容分为四部分:第一,简述分裂等式均衡问题的背景和研究现状.第二,在Hilbert空间中提出一种新的解决分裂等式均衡问题的迭代算法,并在无半紧条件下得到强收敛性定理.第三,在Banach空间中研究分裂等式均衡问题,利用广义投影,构造一种新的分裂等式均衡问题的迭代算法,并得到其强收敛定理.最后,在Hilbert空间中,将Chang, Wu, Wa.ng, Wang在文献[59]中的关于非延伸型多值映射的临近点算法推广到渐近非扩张多值映射.
[Abstract]:Due to the wide application of the split feasibility problem, it has become a nonlinear functional analysis in an extremely important issue, and has attracted the attention of many scholars. In 1994, Censor and Elfving[1] first proposed the split feasibility problem in finite dimensional Hilbert space, it provides important theoretical basis for our.2010 to solve the different space problems, Moudafi[43] put forward the split common fixed point, the problem is a generalization of the split feasibility problem and convex feasibility problem, split equation proposed by Moudafi[15] in 2013 and the fixed point problem is a generalization of the split common fixed point problem. In order to solve the separation equation of the fixed point problem, introducing Moudafi alternating algorithm in CQ in [15], and get the weak convergence theorem of.2013, Kazmi and Rizvi[30] proposed the split equilibrium problem is also split the public real Promote the point of the problem. In recent years due to the wide application of split equilibrium problems, many scholars are committed to research [26,30,31]. split equilibrium problems such as Witthayarat, Ab-dou and Cho proposed a new compression method to solve Hilbert space division equilibrium problems and fixed point problems in [26]. In this paper, we mainly study the division in order to solve the problem of equilibrium equation, split equation equilibrium problems in Hilbert space, in 2015, Ma, Wang, Chang and Duan proposed a new iterative algorithm in the literature [42] obtained the weak convergence theorem, and obtain a strong convergence theorem in semi compact conditions. But the condition is relatively strong. The SEMICOMPACT we, according to Witthayarat, about the split balance problem of iterative algorithm is proposed for Abdou and Cho in [26], to construct a new iterative algorithm in this paper, in the absence of SEMICOMPACT conditions and the use of compression The projection method has strong convergence theorems for equilibrium equation problem split. Because Hilbert space is a complete inner product space, so we will Hilbert space on split equilibrium results are extended to the Banach equation in space. So far there is no solution to split equation equilibrium problems in Banach spaces. Thus we have to study the split equation equilibrium problems in Banach space, by using the generalized projection, an iterative algorithm for splitting equation to construct a new equilibrium problem, and obtain the strong convergence theorem. The proximal point algorithm (PPA) to solve some optimization problems have convergence results from the linear space such as Euclidean space, Hilbert space and Banach space extended to all sorts of other space. The target of convex function minimum point for the study of analysis and geometry plays a key Function optimization problems can be used in computer view, machine learning, electronic structure calculation, system balance, and robot manipulation [50-56]. recently, Chang, Wu, Wang, Wang in [59] is proposed and studied the improved algorithm to solve the point near the fixed point problem of the multi valued mapping of non extension type in Hilbert spaces the condition is relatively strong. Thus, we put the above results to the proximal point algorithm is more extensive asymptotically nonexpansive Multivalued Mappings, prove its convergence. This paper is divided into four parts: first, the background and research status of the split equation for equilibrium problem simply. Second, the proposed iterative algorithm a new solution of equilibrium problems in split equation in Hilbert space, and obtain the strong convergence theorem in semi compact conditions. Third, the research division equation of equilibrium problems in Banach space, by using the generalized projection, A new iterative algorithm for splitting equational equilibrium problem is constructed, and its strong convergence theorem is obtained. Finally, in Hilbert space, we extend the proximal point algorithm of Chang, Wu, Wa.ng and Wang in the [59] extension to the asymptotically nonexpansive Multivalued Mappings.

【学位授予单位】：云南财经大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177

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