Cyclic algorithms for split feasibility problems in Hilbert spaces

The split common fixed point problem (SCFPP) is equivalently converted to a common fixed point problem of a finite family of class-T operators. This enables us to introduce new cyclic algorithms to solve the SCFPP and the multiple-set split feasibility problem.     

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.

     

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.     

Strong convergence result for proximal split feasibility problem in Hilbert spaces

In this paper we describe and analyse new computational technique for solving proximal split feasibility problem (SFP) using a modified proximal split feasibility algorithm. The two convex and lower semi-continuous objective functions are assumed to be non-smooth. Some application to SFP are given. We demonstrate the computational efficiency of the proposed algorithm with nontrivial numerical experiments. We also compare our method with other relevant methods in the literature in terms of convergence, stability, efficiency and implementation with our illustrative numerical examples.     

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.     

Bi-extrapolated subgradient projection algorithm for solving multiple-sets split feasibility problem

This paper deals with a bi-extrapolated subgradient projection algorithm by intro- ducing two extrapolated factors in the iterative step to solve the multiple-sets split feasibility problem. The strategy is intend to improve the convergence. And its convergence is proved un- der some suitable conditions. Numerical results illustrate that the bi-extrapolated subgradient projection algorithm converges more quickly than the existing algorithms.     

An efficient simultaneous method for the constrained multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) captures various applications arising in many areas. Recently, by introducing a function measuring the proximity to the involved sets, Censor et al. proposed to solve the MSFP via minimizing the proximity function, and they developed a class of simultaneous methods to solve the resulting constrained optimization model numerically. In this paper, by combining the ideas of the proximity function and the operator splitting methods, we propose an efficient simultaneous method for solving the constrained MSFP. The efficiency of the new method is illustrated by some numerical experiments.     

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.     

Perturbed projections and subgradient projections for the multiple-sets split feasibility problem

We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case. We further derive and analyze a perturbed projection method for the multiple-sets split feasibility problem and, additionally, furnish alternative proofs to two known results.     

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.     

A new simultaneous subgradient projection algorithm for solving a multiple-sets split feasibility problem

In this paper, we present a simultaneous subgradient algorithm for solving the multiple-sets split feasibility problem. The algorithm employs two extrapolated factors in each iteration, which not only improves feasibility by eliminating the need to compute the Lipschitz constant, but also enhances flexibility due to applying variable step size. The convergence of the algorithm is proved under suitable conditions. Numerical results illustrate that the new algorithm has better convergence than the existing one.     

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.     

A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings

In this paper, by using Bregman distance, we introduce a new iterative process involving products of resolvents of maximal monotone operators for approximating a common element of the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings and the solution set of the multiple-sets split feasibility problem and common zeros of maximal monotone operators. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Finally, we mention several corollaries and two applications of our algorithm.     

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.     

Implicit and explicit algorithms for solving the split feasibility problem

Very recently, Dang and Gao (Inverse Probl 27:015007, 2011) introduced a KM-CQ algorithm with strong convergence for the split feasibility problem. In this paper, we will continue to consider the split feasibility problem. We present two algorithms. First, we introduce an implicit algorithm. Consequently, by discretizing the continuous implicit algorithm, we obtain an explicit algorithm. Under some weaker conditions, we show the strong convergence of presented algorithms to some solution of the split feasibility problem which solves some special variational inequality. As special cases, we obtain two algorithms which converge strongly to the minimum norm solution of the split feasibility problem. Results obtained in this paper include the corresponding results of Dang and Gao (2011) and extend a recent result of Wang and Xu (J Inequalities Appl 2010, doi:10.1155/2010/102085).     

Algorithms for nonexpansive self-mappings with application to the constrained multiple-set split convex feasibility fixed point problem in Hilbert spaces

In this article we devise two iteration schemes for approximating common fixed points of a finite family of nonexpansive mappings and establish the corresponding strong convergence theorem for the sequence generated by any one of our algorithms. Then we apply our results to approximate a solution of the so-called constrained multiple-set convex feasibility fixed point problem for firmly nonexpansive mappings which covers the multiple-set convex feasibility problem in the literature. In particular, our algorithms can be used to approximate the zero point problem of maximal monotone operators, and the equilibrium problem. Furthermore, the unique minimum norm solution can be obtained through our algorithms for each mentioned problem.     

多集分裂等式问题的逐次松弛投影算法

多集分裂等式问题是分裂可行性问题的拓展问题,在图像重建、语言处理、地震探测等实际问题中具有广泛的应用.为了解决这个问题,提出了逐次松弛投影算法,设计了变化的步长,使其充分利用当前迭代点的信息且不需要算子范数的计算,证明了算法的弱收敛性.数值算例验证了算法在迭代次数与运行时间等方面的优越性.