Viscosity approximation methods for a family of m-accretive mappings in reflexive Banach spaces

The purpose of this paper is to introduce a general iterative algorithm by viscosity method to approximate a common point of a finite family of m-accretive mappings in a reflexive Banach space which has a weakly continuous duality mapping. We obtain strong convergence theorems under some mild conditions imposed on parameters.      

On finite convergence of proximal point algorithms for variational inequalities

In this paper, we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem (VIP), where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations. By using this result, we obtain that the well-known proximal point algorithm possesses finite convergence if the solution set of VIP is weakly sharp. As an extension, we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g-sharpness.     

On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators

In this paper we introduce general iterative methods for finding zeros of a maximal monotone operator in a Hilbert space which unify two previously studied iterative methods: relaxed proximal point algorithm [H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math Soc. 66 (2002) 240–256] and inexact hybrid extragradient proximal point algorithm [R.S. Burachik, S. Scheimberg, B.F. Svaiter, Robustness of the hybrid extragradient proximal-point algorithm, J. Optim. Theory Appl. 111 (2001) 117–136]. The paper establishes both weak convergence and strong convergence of the methods under suitable assumptions on the algorithm parameters.     

关于鞍点问题的广义预处理HSS-SOR交替分裂迭代方法

本文研究了鞍点问题的迭代法. 在白中治,Golub和潘建瑜提出的预处理对称/反对称分裂（PHSS）迭代法的基础上,通过结合GSOR迭代格式,利用两个参数加速,提出了一种广义预处理HSS-SOR交替分裂迭代法,并研究了该方法的收敛性.数值结果表明本文所给方法是有效的.     

Convergence of consistent and inconsistent schemes for fractional diffusion problems with boundaries

This paper provides iterative construction of a common solution associated with a class of equilibrium problems and split convex feasibility problems. In particular, we are interested in the equilibrium problems defined with respect to the pseudomonotone and Lipschitz-type continuous equilibrium problem together with the generalized split null point problems in real Hilbert spaces. We propose an iterative algorithm that combines the hybrid extragradient method with the inertial acceleration method. The analysis of the proposed algorithm comprises theoretical results concerning strong convergence under suitable set of constraints and numerical results concerning the viability of the proposed algorithm with respect to various real-world applications.

     

An EM-based iterative method for solving large sparse linear systems

ABSTRACT

We propose a novel iterative algorithm for solving a large sparse linear system. The method is based on the EM algorithm. If the system has a unique solution, the algorithm guarantees convergence with a geometric rate. Otherwise, convergence to a minimal Kullback–Leibler divergence point is guaranteed. The algorithm is easy to code and competitive with other iterative algorithms.     

Iterative procedures for solutions of random operator equations in Banach spaces

We construct random iterative processes for weakly contractive and asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. It is shown that they converge to the random fixed points of these operators in the setting of Banach spaces. We also proved that an implicit random iterative process converges to the common random fixed point of a finite family of asymptotically quasi-nonexpansive random operators in uniformly convex Banach spaces.     

On the convergence of a new iterative algorithm of three infinite families of generalized nonexpansive multi-valued mappings

In this paper, we establish some weak and strong convergence theorems for a new iterative algorithm under some suitable conditions to approximate the common fixed point of three infinite families of multi-valued generalized nonexpansive mappings in a uniformly convex Banach spaces. Our results generalize and improve several previously known results of the existing literature.     

Hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense

In this paper we introduce an iterative algorithm for finding a common element of the fixed point set of an asymptotically strict pseudocontractive mapping S in the intermediate sense and the solution set of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in Hilbert space. The iterative algorithm is based on several well-known methods including the extragradient method, CQ method, Mann-type iterative method and hybrid gradient projection algorithm with regularization. We obtain a strong convergence theorem for three sequences generated by our iterative algorithm. In addition, we also prove a new weak convergence theorem by a modified extragradient method with regularization for the MP and the mapping S.     

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.     

The general iterative methods for nonexpansive mappings in Banach spaces

In this paper, we introduce a general iterative approximation method for finding a common fixed point of a countable family of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The main result extends various results existing in the current literature.     

与渐近非扩张半群不动点问题相关的分裂等式混合均衡问题及其应用

研究了与渐近非扩张半群不动点问题相关的分裂等式混合均衡问题.在等式约束下,为同时逼近两个空间中混合均衡问题和渐近非扩张半群不动点问题的公共解,借助收缩投影方法引出了一种迭代程序.在适当条件下,该迭代算法的强收敛性被证明.文末还把所得结果应用于分裂等式混合变分不等式问题和分裂等式凸极小化问题.     

Some iterative methods for nonconvex variational inequalities

It is well known that the nonconvex variational inequalities are equivalent to the fixed point problems. We use this equivalent alternative formulation to suggest and analyze a new class of two-step iterative methods for solving the nonconvex variational inequalities. We discuss the convergence of the iterative method under suitable conditions. We also introduce a new class of Wiener – Hopf equations. We establish the equivalence between the nonconvex variational inequalities and the Wiener – Hopf equations. This alternative equivalent formulation is used to suggest some iterative methods. We also consider the convergence analysis of these iterative methods. Our method of proofs is very simple compared to other techniques.     

A simultaneous projections method for linear inequalities

An iterative method for solving general systems of linear inequalities is considered. The method, a relaxed generalization of Cimmino's scheme for solving linear systems, was first suggested by Censor and Elfving. Each iterate is obtained as a convex combination of the orthogonal projections of the previous iterate on the half spaces defined by the linear inequalities. The algorithm is particularly suitable for implementation on computers with parallel processors. We prove convergence from any starting point for both consistent and nonconsistent systems (to a feasible point in the first case, and to a weighted least squares type solutions in the second).     

Banach空间一类H-增生算子的混合拟变分包含的邻近算子方程

本文在Banach空间中引入一类H-增生算子的混合拟变分包含,并提出求该变分包含问题解的邻近点法.通过H-增生算子的预解算子技术,建立了混合拟变分包含问题与邻近算子方程的等价关系,由这个等价关系得到求解邻近算子方程的迭代算法,该算法收敛于上述混合拟变分包含问题的解.     

无约束优化问题的一个下降方法

本文研究了无约束优化问题.利用当前和前面迭代点的信息以及曲线搜索技巧产生新的迭代点,得到了一个新的求解无约束优化问题的下降方法.在较弱条件下证明了算法具有全局收敛性.当目标函数为一致凸函数时,证明了算法具有线性收敛速率.初步的数值试验表明算法是有效的.     

Krylov subspace methods with deflation and balancing preconditioners for least squares problems

For solving least squares problems, the CGLS method is a typical method in the point of view of iterative methods. When the least squares problems are ill-conditioned, the convergence behavior of the CGLS method will present a deteriorated result. We expect to select other iterative Krylov subspace methods to overcome the disadvantage of CGLS. Here the GMRES method is a suitable algorithm for the reason that it is derived from the minimal residual norm approach, which coincides with least squares problems. Ken Hayami proposed BAGMRES for solving least squares problems in [\emph{GMRES Methods for Least Squares Problems, SIAM J. Matrix Anal. Appl., 31(2010)}, pp.2400-2430]. The deflation and balancing preconditioners can optimize the convergence rate through modulating spectral distribution. Hence, in this paper we utilize preconditioned iterative Krylov subspace methods with deflation and balancing preconditioners in order to solve ill-conditioned least squares problems. Numerical experiments show that the methods proposed in this paper are better than the CGLS method.     

A Hybrid Proximal-Extragradient Algorithm with Inertial Effects

In this article, we incorporate inertial terms in the hybrid proximal-extragradient algorithm and investigate the convergence properties of the resulting iterative scheme designed to find the zeros of a maximally monotone operator in real Hilbert spaces. The convergence analysis relies on extended Fejér monotonicity techniques combined with the celebrated Opial Lemma. We also show that the classical hybrid proximal-extragradient algorithm and the inertial versions of the proximal point, the forward-backward and the forward-backward-forward algorithms can be embedded into the framework of the proposed iterative scheme.     

A new simultaneous iterative method with a parameter for solving the extended split equality problem and the extended split equality fixed point problem

In this article, we first propose an extended split equality problem which is an extension of the convex feasibility problem, and then introduce a parameter w to establish the fixed point equation system. We show the equivalence of the extended split equality problem and the fixed point equation system. Based on the fixed point equation system, we present a simultaneous iterative algorithm and obtain the weak convergence of the proposed algorithm. Further, by introducing the concept of a G-mapping of a finite family of strictly pseudononspreading mappings \(\{T_{i}\}_{i = 1}^{N}\), we consider an extended split equality fixed point problem for G-mappings and give a simultaneous iterative algorithm with a way of selecting the stepsizes which do not need any prior information about the operator norms, and the weak convergence of the proposed algorithm is obtained. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms.     

Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers.