Iterative methods for the split feasibility problem and the fixed point problem in Banach spaces

ABSTRACT

In this work, we suggest modifications of the self-adaptive method for solving the split feasibility problem and the fixed point problem of nonexpansive mappings in the framework of Banach spaces. Without the assumption on the norm of the operator, we prove that the sequences generated by our algorithms weakly and strongly converge to a solution of the problems. The numerical experiments are demonstrated to show the efficiency and the implementation of our algorithms.     

Convergence of Sequential Parafirmly Nonexpansive Mappings in Reflexive Banach Spaces

In this paper, we study parafirmly nonexpansive mappings with Bregman distance and weak convergence of underrelaxed sequential parafirmly nonexpansive mappings.     

Extrapolation algorithm for affine-convex feasibility problems

The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel block-iterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated over-relaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations.     

Iterative Methods of Solving Stochastic Convex Feasibility Problems and Applications

The stochastic convex feasibility problem (SCFP) is the problem of finding almost common points of measurable families of closed convex subsets in reflexive and separable Banach spaces. In this paper we prove convergence criteria for two iterative algorithms devised to solve SCFPs. To do that, we first analyze the concepts of Bregman projection and Bregman function with emphasis on the properties of their local moduli of convexity. The areas of applicability of the algorithms we present include optimization problems, linear operator equations, inverse problems, etc., which can be represented as SCFPs and solved as such. Examples showing how these algorithms can be implemented are also given.     

Feasibility and Solvability for Vector Complementarity Problems1

The purpose of this paper is to discuss the feasibility and solvability of vector complementarity problems. We prove that, under suitable conditions, the vector complementarity problem with a pseudomonotonicity assumption is solvable whenever it is strictly feasible. By strengthening the generalized monotonicity condition, we show also that the homogeneous vector complementarity problem is solvable whenever it is feasible. At last, we study the solvability of the vector complementarity problem on product spaces.The authors thank Professor Franco Giannessi for valuable comments and suggestions leading to improvements of this paper.     

集值向量变分不等式的严格可行性与可解性

本文研究集值向量变分不等式的严格可行性与可解性之间的关系．证明了如下结果:在一定条件下,只要集值向量变分不等式是严格可行的,那么其一定可解．     

Investment evaluation based on the commercial scope the production of natural gas

The annual production planning of a natural gas trading and transporting company is modelled as a linear system of (in)equalities. The model is used to quantify the increase of robustness with respect to commercial uncertainty, resulting from investments in production capacities. A novel concept is thecommercial scope, describing the set of future commercial scenarios that can be handled effectively. It is shown how relevant parts of the boundary of this set can be constructed using induced constraints. A numerical example is presented.     

凸可行问题的块迭代次梯度投影算法

本文,针对由非线性不等式系统构成的凸可行问题,提出了序列块迭代次梯度投影算法和平行块迭代次梯度投影算法.将非线性不等式系统分成若干个子系统,然后将当前迭代点在子系统各个子集上的次梯度投影的凸组合作为当前迭代点在这个子系统上的近似投影.在较弱条件下证明了两种算法的收敛性.     

凸可行问题的一种强收敛算法

无限维Hilbert空间中,解凸可行问题的平行投影算法通常是弱收敛的.本文对一般的平行投影算法进行改进,设计了一种解凸可行问题的具有强收敛性的新算法.该算法主要是在原有算法基础上引入了一个参数序列,在参数序列满足一定的控制条件下保证了算法的强收敛性.为了简单证明算法的强收敛性,我们构建了一个新的积空间,然后把原空间的这种改进平行投影算法转换为积空间中的交替投影算法.这样,改进的平行投影算法的强收敛性就可以通过交替投影算法的收敛性证明得到.     

求解分裂可行问题的一种半空间投影算法

本文研究了分裂可行问题.利用松弛投影的方法,获得了分裂可行问题最优值点,推广了已有文献中的有关结果.