A relaxed extended CQ algorithm for the 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.

In this paper, we propose a CQ-type algorithm for solving the split feasibility problem (SFP) in real Hilbert spaces. The algorithm is designed such that the step-sizes are directly computed at each iteration. We will show that the sequence generated by the proposed algorithm converges in norm to the minimum-norm solution of the SFP under appropriate conditions. In addition, we give some numerical examples to verify the implementation of our method. Our result improves and complements many known related results in the literature.

In this paper, we introduce two iterative algorithms for the split feasibility problem in real Hilbert spaces by reformulating it as a fixed point equation. Under suitable conditions, weak and strong convergence theorems are established. As a consequence, we obtain weak and strong convergence iterative sequences for the split equality problem introduced by Moudafi. The efficiency of the proposed algorithms is illustrated by numerical experiments. Our results improve and extend the corresponding results announced by many others.

The split feasibility problem (SFP) in Hilbert spaces is addressed in this study using an efficient iterative approach. Under mild conditions, we prove convergence theorems for the algorithm for finding a solution to the SFP. We also present numerical examples to illustrate that the acceleration of our algorithm is effective. Our results are applied to solve image deblurring and signal recovery problems. Furthermore, we show the use of the proposed method to generate polynomiographs.

Inspired by the works of Lopez et al. [ 21 ] and the recent paper of Dang et al. [ 15 ], we devise a new inertial relaxation of the CQ algorithm for solving Split Feasibility Problems (SFP) in real Hilbert spaces. Under mild and standard conditions we establish weak convergence of the proposed algorithm. We also propose a Mann-type variant which converges strongly. The performances and comparisons with some existing methods are presented through numerical examples in Compressed Sensing and Sparse Binary Tomography by solving the LASSO problem.

The split feasibility problem SFP has received much attention due to its various applications in signal processing and image reconstruction. In this paper, we propose two inertial relaxed C Q algorithms for solving the split feasibility problem in real Hilbert spaces according to the previous experience of applying inertial technology to the algorithm. These algorithms involve metric projections onto half-spaces, and we construct new variable step size, which has an exact form and does not need to know a prior information norm of bounded linear operators. Furthermore, we also establish weak and strong convergence of the proposed algorithms under certain mild conditions and present a numerical experiment to illustrate the performance of the proposed algorithms.

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.

In this paper, we propose several new iterative algorithms to solve the split feasibility problem in the Hilbert spaces. By virtue of new analytical techniques, we prove that the iterative sequence generated by these iterative procedures converges to the solution of the split feasibility problem which is the best close to a given point. In particular, the minimum-norm solution can be found via our iteration method.

The main purpose of this paper is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove that the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of some others. MSC:90C25, 90C30, 47J25, 47H09.

In this article, we propose four alternated inertial algorithms for finding a common solution of equilibrium problems and split feasibility problems in Hilbert spaces. We present a variable step size, which is not required to know the operator norm. Furthermore, these algorithms adopt the new convex subset form by a sequence of closed balls instead of half spaces, and it is easy to calculate the projections onto these sets. We establish strong and weak convergence theorems of these algorithms under some proper assumptions and also present a numerical experiment to illustrate the performance and the advantage of the proposed algorithms.

In this paper, we introduce a new relaxed method for solving the split feasibility problem in Hilbert spaces. In our method, the projection to the halfspace is replaced by the one to the intersection of two halfspaces. We give convergence of the sequence generated by our method under some suitable assumptions. Finally, we give a numerical example for illustrating the efficiency and implementation of our algorithms in comparison with existing algorithms in the literature.

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.

Inspired by the recent work of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205–221], in this paper, we study generalized split feasibility problems (GSFPs) in the setting of Banach spaces. We propose iterative algorithms to compute the approximate solutions of such problems. The weak convergence of the sequence generated by the proposed algorithms is studied. As applications, we derive some algorithms and convergence results for some problems from nonlinear analysis, namely, split feasibility problems, equilibrium problems, etc. Our results generalize several known results in the literature including the results of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, SetValued Var. Anal., 23 (2015), 205–221].

In the present paper, we consider the varying stepsize CQ algorithm for solving the split feasibility problem in Hilbert spaces, investigate the linear convergence issue and explore an application in systems biology. In particular, we introduce a notion of bounded linear regularity property for the split feasibility problem, and use it to establish the linear convergence property for the varying stepsize CQ algorithm when using some suitable types of stepsizes, which covers most types of stepsizes used in the literature of CQ algorithms. We also provide some mild sufficient conditions for ensuring this bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize CQ algorithm for many application cases. To the best of our knowledge, this is the first work to study the linear convergence rate of CQ algorithms. In the aspect of application, we consider the gene regulatory network inference arising in systems biology, which is formulated as a group Dantzig selector and then cast into a split feasibility problem. The numerical study on gene expression data of mouse embryonic stem cell shows that the varying stepsize CQ algorithm is applicable to gene regulatory network inference in the sense that it obtains a reliable solution matching with biological standards.

This paper aims to study a new type of CQ algorithm for split feasibility problems in Hilbert spaces. We introduce a new CQ algorithm by incorporating a step that involves an inertial extrapolation term and correction terms into the basic CQ algorithm. We then obtain weak convergence results under some conditions on the iterative parameters. Furthermore, a linear convergence result is obtained when the split feasibility problem satisfies some bounded linear regularity property. Several results in the literature are recovered as special cases of our method. Our numerical experiments from image deblurring and CT image reconstruction show the superiority of our method over related methods for split feasibility problems in the literature.