A Successive Projection Algorithm for Solving the Multiple-Sets Split Feasibility Problem

The multiple-sets split feasibility problem (MSFP) arises in many areas and it can be unified as a model for many inverse problems where the constraints are required on the solutions in the domain of a linear operator as well as in the operator's range. Some existing algorithms, in order to get the suitable step size, need to compute the largest eigenvalue of the related matrix, estimate the Lipschitz constant, or use some step-size search scheme, which usually requires many inner iterations. In this article, we introduce a successive projection algorithm for solving the multiple-sets split feasibility problem. In each iteration of this algorithm, the step size is directly computed, which is not needed to compute the largest eigenvalue of the matrix or estimate the Lipschitz constant. It also does not need any step-size search scheme. Its theoretical convergence results are also given.     

基于CT图像重建的多重集合分裂可行性问题应用分析

为了较好地应用CQ算法解决稀疏角度CT 图像重建的问题,提出了一种新的实时的分块逐次混合算法．首先将稀疏角度CT 图像重建的重建问题转化成分裂可行性问题．其次,通过分析非空闭凸集C和Q的不同的定义,在N维实空间中分别针对不同的CQ算法给出了7种不同的实现方案．通过试验,分别对不同算法及其方案的重建精度和收敛速度进行了对比分析,并对多重集合分裂可行性问题算法中约束权因子的选取及其对输出的影响进行了研究,从而给出了CQ算法在稀疏角度CT图像重建问题中应用的最佳凸集定义方案．以此为基础,给出了所提出算法的最佳实现方案．试验结果表明,该算法收敛速度快,重建精度高,为多重集合分裂可行性问题及其改进算法在该重建问题上的应用提供了参考．     

Remark on the Successive Projection Algorithm for the Multiple-Sets Split Feasibility Problem

Recently, assuming that the metric projection onto a closed convex set is easily calculated, Liu et al. (Numer. Func. Anal. Opt. 35:1459–1466, 2014) presented a successive projection algorithm for solving the multiple-sets split feasibility problem (MSFP). However, in some cases it is impossible or needs too much work to exactly compute the metric projection. The aim of this remark is to give a modification to the successive projection algorithm. That is, we propose a relaxed successive projection algorithm, in which the metric projections onto closed convex sets are replaced by the metric projections onto halfspaces. Clearly, the metric projection onto a halfspace may be directly calculated. So, the relaxed successive projection algorithm is easy to implement. Its theoretical convergence results are also given.     

Modified Alternating Direction Methods for the Modified Multiple-Sets Split Feasibility Problems

In this paper, we propose two new multiple-sets split feasibility problem models and new solution methods. The first model is more separable than the original one, which enables us to apply a modified alternating direction method with parallel steps to solve it. Then, to overcome the difficulty of computing projections onto the constraint sets, a special version of this modified method with the strategy of projection onto half-space is given. The second model consists in finding a least Euclidean norm solution of the multiple-sets split feasibility problem, for which we provide another modified alternating direction method. Numerical results presented at the last show the efficiency of our methods.     

Hilbert空间中分裂可行性问题的改进Halpern迭代和黏性逼近算法

在无限维Hilbert空间中,提出了求解分裂可行性问题（SFP）的改进Halpern迭代和黏性逼近算法,证明了当参数满足一定条件时,由给定算法生成的序列强收敛到分裂可行性问题的一个解.这些结论推广了Deepho和Kumam近年来的一些结果     

Hilbert空间中求解带约束的分裂公共不动点问题的三种迭代算法

该文的目的是研究带约束的分裂公共不动点问题.建立和分析了求解带约束的分裂公共不动点问题的三种新的迭代算法.建立了三种迭代算法的强收敛性结果,这些结果改进并推广了某些作者的相关结论.     

Hilbert空间中求解分裂可行问题CQ算法的强收敛性

在Hilbert空间中,为了研究分裂可行问题迭代算法的强收敛性,提出了一种新的CQ算法.首先利用CQ算法构造了一个改进的Halpern迭代序列； 然后通过把分裂可行问题转化为算子不动点， 在较弱的条件下， 证明了该序列强收敛到分裂可行问题的一个解. 推广了Wang和Xu的有关结果.     

Iterative Methods for Triple Hierarchical Variational Inequalities in Hilbert Spaces

In this paper, we consider a variational inequality with a variational inequality constraint over a set of fixed points of a nonexpansive mapping called triple hierarchical variational inequality. We propose two iterative methods, one is implicit and another one is explicit, to compute the approximate solutions of our problem. We present an example of our problem. The convergence analysis of the sequences generated by the proposed methods is also studied.     

Hilbert空间中分裂一般变分不等式组的迭代算法

引入并研究实Hilbert空间中一类分裂变分不等式组.以投影算子为工具,提出了解这类分裂变分不等式组的迭代算法并证明了算法的收敛性.所得结果推广并改进了本领域以前的一些结果.     

Simultaneous Subgradient Algorithms for the Generalized Split Variational Inclusion Problem in Hilbert Spaces

In this article, we study the generalized split variational inclusion problem. For this purpose, motivated by the projected Landweber algorithm for the split equality problem, we first present a simultaneous subgradient extragradient algorithm and give related convergence theorems for the proposed algorithm. Next, motivated by the alternating CQ-algorithm for the split equality problem, we propose another simultaneous subgradient extragradient algorithm to study the general split variational inclusion problem. As applications, we consider the split equality problem, split feasibility problem, split variational inclusion problem, and variational inclusion problem in Hilbert spaces.     

On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.     

Hilbert空间上新的变分不等式问题和不动点问题的粘性迭代算法

本文在Hilbert空间上引入了一个新的粘性迭代算法,找到了关于两个逆强单调算子的变分不等式问题的解集与非扩张映射的不动点集的公共元.通过修改的超梯度算法,得到了强收敛定理,也给出了一个数值例子.所得结果改进了许多最新结果.     

Iterative Methods for Equilibrium and Fixed Point Problems for Nonexpansive Semigroups in Hilbert Spaces

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup (T(s)) _s≥0 in Hilbert spaces. We prove that both approaches converge strongly to a common element z of the set of the equilibrium points and the set of common fixed points of (T(s)) _s≥0. Such common element z is the unique solution of a variational inequality, which is the optimality condition for a minimization problem.     

Hilbert空间中均衡问题的迭代解(英文)

研究的均衡问题(EP)是指:找x∈C,使F(x,y)≥0,y∈C.文中构造了Hilbert空间中均衡问题解的迭代序列{xn}并证明了存在均衡解x~,使得{xn}弱收敛于x~(n→+∞),所得结果包含Moudafi的已有结果作为特殊情形.     

Hilbert空间中涉及一类新的具误差迭代程序的平衡问题的黏性逼近法

In this paper, we introduce an iterative scheme with error by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. A strong convergence theorem is given, which generalizes all the results obtained by S.Takahashi and W.Takahashi in 2007. In addition, some of the methods applied in this paper improve those of S.Takahashi and W.Takahashi.     

广义平衡问题与非扩张映象不动点问题公解的一个新迭代方法

In this paper,we introduce a new iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems and the set of fixed points for nonexpansive mappings in Hilbert space.Under suitable conditions,some strong convergence theorems are proved.Our results extend and improve some recent results.     

Viscosity Approximation Methods for a Class of Generalized Split Feasibility Problems with Variational Inequalities in Hilbert Space

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.     

The Stochastic Reflection Problem in Hilbert Spaces

We construct a Markov process X associated with the stochastic reflection problem on a closed convex subset with non empty interior and smooth boundary in a Hilbert space, as a solution to a random convex control problem. The transition semigroup corresponding to X is exactly that defined by the Kolmogorov equation with Neumann homogeneous boundary conditions (see [3 Barbu , V. , Da Prato , G. , Tubaro , L. ( 2011 ). Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II . Ann. Inst. H. Poincaré 4 : 699 – 724 .[Crossref] , [Google Scholar]]).     

Stochastic Regularization Method for Backward Cauchy Problem in Banach Spaces

We consider a stochastic regularization method for solving the backward Cauchy problem in Banach spaces. An order of convergence is obtained on sourcewise representative elements.