Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem

Split variational inclusion problem is an important problem, and it is a generalization of the split feasibility problem. In this paper, we present feasible algorithms for the split variational inclusion problems in Hilbert spaces, and provide convergence theorems for these algorithms. As application, we study the split feasibility problem in real Hilbert spaces. Final, numerical results are given for our main results.     

New algorithms for the split variational inclusion problems and application to split feasibility problems

ABSTRACT

In this paper, we suggest two new iterative methods for finding an element of the solution set of split variational inclusion problem in real Hilbert spaces. Under suitable conditions, we present weak and strong convergence theorems for these methods. We also apply the proposed algorithms to study the split feasibility problem. Finally, we give some numerical results which show that our proposed algorithms are efficient and implementable from the numerical point of view.     

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     

On inertial proximal algorithm for split variational inclusion problems

ABSTRACT

In this paper, a hybrid proximal algorithm with inertial effect is introduced to solve a split variational inclusion problem in real Hilbert spaces. Under mild conditions on the parameters, we establish weak convergence results for the proposed algorithm. Unlike the earlier iterative methods, we do not impose any conditions on the sequence generated by the proposed algorithm. Also, we extend our results to find a common solution of a split variational inclusion problem and a fixed-point problem. Finally, some numerical examples are given to discuss the convergence and superiority of the proposed iterative methods.     

Hybrid Moreau’s Proximal Algorithms and Convergence Theorems for Minimization Problems in Hilbert Spaces with Applications

Abstract

In this paper, motivated by Moreau’s proximal algorithm, we give several algorithms and related weak and strong convergence theorems for minimization problems under suitable conditions. These algorithms and convergence theorems are different from the results in the literatures. Besides, we also study algorithms and convergence theorems for the split feasibility problem in real Hilbert spaces. Finally, we give numerical results for our main results.     

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.     

Strong convergence of two algorithms for the split feasibility problem in Banach spaces

ABSTRACT

In this paper, we consider the split feasibility problem in Banach spaces. By converting it to an equivalent null-point problem, we propose two iterative algorithms, which are new even in Hilbert spaces. The parameter in one algorithm is chosen in such a way that no priori knowledge of the operator norms is required. It is shown that these two algorithms are strongly convergent provided that the involved Banach spaces are smooth and uniformly convex. Finally, we conduct numerical experiments to support the validity of the obtained results.     

Some subgradient extragradient type algorithms for solving split feasibility and fixed point problems

In this paper, we consider the split feasibility problem (SFP) in infinite‐dimensional Hilbert spaces and propose some subgradient extragradient‐type algorithms for finding a common element of the fixed‐point set of a strict pseudocontraction mapping and the solution set of a split feasibility problem by adopting Armijo‐like stepsize rule. We derive convergence results under mild assumptions. Our results improve some known results from the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Extragradient and linesearch methods for solving split feasibility problems in Hilbert spaces

Utilizing the Tikhonov regularization method and extragradient and linesearch methods, some new extragradient and linesearch algorithms have been introduced in the framework of Hilbert spaces. In the presented algorithms, the convexity of optimization subproblems is assumed, which is weaker than the strong convexity assumption that is usually supposed in the literature, and also, the auxiliary equilibrium problem is not used. Some strong convergence theorems for the sequences generated by these algorithms have been proven. It has been shown that the limit point of the generated sequences is a common element of the solution set of an equilibrium problem and the solution set of a split feasibility problem in Hilbert spaces. To illustrate the usability of our results, some numerical examples are given. Optimization subproblems in these examples have been solved by FMINCON toolbox in MATLAB.     

Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces

In this paper, we introduce a modified relaxed projection algorithm and a modified variable-step relaxed projection algorithm for the split feasibility problem in infinite-dimensional Hilbert spaces. The weak convergence theorems under suitable conditions are proved. Finally, some numerical results are presented, which show the advantage of the proposed algorithms.     

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.

     

Accelerated projection-based forward-backward splitting algorithms for monotone inclusion problems

In this paper, based on inertial and Tseng''s ideas, we propose two projection-based algorithms to solve a monotone inclusion problem in infinite dimensional Hilbert spaces. Solution theorems of strong convergence are obtained under the certain conditions. Some numerical experiments are presented to illustrate that our algorithms are efficient than the existing results.     

Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces

The split feasibility problem is to find a point x^? with the property that x^?∈ C and Ax^?∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces X and Y, respectively, and A is a bounded linear operator from X to Y. The split feasibility problem models inverse problems arising from phase retrieval problems and the intensity-modulated radiation therapy. In this paper, we introduce a new inertial relaxed CQ algorithm for solving the split feasibility problem in real Hilbert spaces and establish weak convergence of the proposed CQ algorithm under certain mild conditions. Our result is a significant improvement of the recent results related to the split feasibility problem.

     

A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.     

求解两块可分离凸极小化问题的惯性交替方向乘子法

The alternating direction method of multipliers(ADMM)is a widely used method for solving many convex minimization models arising in signal and image processing.In this paper,we propose an inertial ADMM for solving a two-block separable convex minimization problem with linear equality constraints.This algorithm is obtained by making use of the inertial Douglas-Rachford splitting algorithm to the corresponding dual of the primal problem.We study the convergence analysis of the proposed algorithm in infinite-dimensional Hilbert spaces.Furthermore,we apply the proposed algorithm on the robust principal component analysis problem and also compare it with other state-of-the-art algorithms.Numerical results demonstrate the advantage of the proposed algorithm.     

分裂可行问题(SFP)的投影算法

本文探讨了分裂可行问题(SFP)的投影算法．我们先定义了(SFP)的反问题(ISFP), 然后利用正交投影性质,给出了(SFP)与(ISFP)及某些投影不动点问题之间的关系．随后, 给出了求解(SFP)的几种正交投影算法,其中包括精确和不精确投影格式．基于变分不等式中投影算法收敛性的证明思路和分裂可行问题的特有形式,证明了这几种算法的收敛性．最后通过几个算例对讨论的方法进行了初步比较．     

GENERAL SPLIT FEASIBILITY PROBLEMS FOR TWO FAMILIES OF NONEXPANSIVE MAPPINGS IN HILBERT SPACES

The purpose of this article is to introduce a general split feasibility problems for two families of nonexpansive mappings in Hilbert spaces. We prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known 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.     

Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications

ABSTRACT

In convex optimization, numerous problems in applied sciences can be modelled as the split variational inclusion problem (SVIP). In this connection, we aim to design new and efficient proximal type algorithms which are based on the inertial technique and the linesearches terminology. We then discuss its convergence under some suitable conditions without the assumption on the operator norm. We also apply our main result to the split minimization problem, the split feasibility problem, the relaxed split feasibility problem and the linear inverse problem. Finally, we provide some numerical experiments and comparisons to these problems. The obtained result mainly improves the recent results investigated by Chuang.     

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.