Solution Of Split Feasibility Problem Research Articles

Inertial Halpern-type method for solving split feasibility and fixed point problems via dynamical stepsize in real Banach spaces

AbstractIn this paper, we introduce a modified Halpern inertial method for approximating solutions of split feasibility problem and fixed point problem of Bregman strongly nonexpansive mappings in the framework of p-uniformly convex and uniformly smooth real Banach spaces. We establish a strong convergence result for the sequence generated by our iterative scheme under some mild conditions without the computation of the operator norm. We state some consequences and present some examples to show the efficiency and implementation of our proposed method. The result discussed in this paper extends and generalizes many recent results in this direction. Our result extends and complements some related results in literature.

Iterative approximation of split feasibility problem in real Hilbert spaces

In this paper, we propose an iterative algorithm for finding solution of split feasibility problem involving a λ−strictly pseudo-nonspreading map and asymptotically nonexpansive semigroups in two real Hilbert spaces. We prove weak and strong convergence theorems using the sequence obtained from the proposed algorithm. Finally, we applied our result to solve a monotone inclusion problem and present a numerical example to support our result.

Three-Step Projective Methods for Solving the Split Feasibility Problems

In this paper, we focus on studying the split feasibility problem (SFP) in Hilbert spaces. Based on the CQ algorithm involving the self-adaptive technique, we introduce a three-step iteration process for approximating the solution of SFP. Then, the convergence results are established under mild conditions. Numerical experiments are provided to show the efficiency in signal processing. Some comparisons to various methods are also provided in this paper.

A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems

In this work, we study the split feasibility problem (SFP) in the framework of p-uniformly convex and uniformly smooth Banach spaces. We propose an iterative scheme with inertial terms for seeking the solution of SFP and then prove a strong convergence theorem for the sequences generated by our iterative scheme under implemented conditions on the step size which do not require the computation of the norm of the bounded linear operator. We finally provide some numerical examples which involve image restoration problems and demonstrate the efficiency of the proposed algorithm. The obtained result of this paper complements many recent results in this direction and seems to be the first one to investigate the SFP outside Hilbert spaces involving the inertial technique.

Convergence theorems for split feasibility problems on a finite sum of monotone operators and a family of nonexpansive mappings

In this paper, we present two iterative algorithms for approximating a solution of the split feasibility problem on zeros of a sum of monotone operators and fixed points of a finite family of nonexpansive mappings. Weak and strong convergence theorems are proved in the framework of Hilbert spaces under some mild conditions. We apply the obtained main result for the problem of finding a common zero of the sum of inverse strongly monotone operators and maximal monotone operators, for finding a common zero of a finite family of maximal monotone operators, for finding a solution of multiple sets split common null point problem, and for finding a solution of multiple sets split convex feasibility problem. Some applications of the main results are also provided.

A Viscosity Approximation Method for the Split Feasibility Problems in Hilbert Space

In this paper, the most basic idea is to apply the viscosity approximation method to study the split feasibility problem (SFP), we will be in the infinite-dimensional Hilbert space to study the problem . We defined $x_{0}\in C$ as arbitrary and $x_{n+1}=(1-\alpha_{n})P_{C}(I-\lambda_{n}A^{*}(I-P_{Q})A)x_{n}+\alpha_{n}f(x_{n})$, for $n\geq0,$ where $\{\alpha_{n}\}\subset(0,1)$. Under the proper control conditions of some parameters, we show that the sequence $\{x_{n}\}$ converges strongly to a solution of SFP. The results in this paper extend and further improve the relevant conclusions in Deepho (Deepho, J. \& Kumam, P., 2015).

Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces

In this paper, we introduce an iterative scheme for approximating a common element of the set of fixed points of left Bregman strongly nonexpansive mapping and the set of solutions of split feasibility problem. We further obtain a strong convergence result for finding a common solution of fixed point problem for left Bregman strongly nonexpansive mappings and split feasibility problem in the framework of -uniformly convex Banach spaces which are also uniformly smooth. We give an application of our result to approximating a solution of convexly constrained linear inverse problem which is also a fixed point of a left Bregman strongly nonexpansive mapping in -uniformly convex Banach spaces, which are also uniformly smooth. Finally, we give some numerical example of our result to study its efficiency and implementation. Our results complement many known related results in the literature.

Approximating Curve and Strong Convergence of the CQ Algorithm for the Split Feasibility Problem

Using the idea of Tikhonov's regularization, we present properties of the approximating curve for the split feasibility problem (SFP) and obtain the minimum-norm solution of SFP as the strong limit of the approximating curve. It is known that in the infinite-dimensional setting, Byrne's CQ algorithm (Byrne, 2002) has only weak convergence. We introduce a modification of Byrne's CQ algorithm in such a way that strong convergence is guaranteed and the limit is also the minimum-norm solution of SFP.

Robust solutions of split feasibility problem with uncertain linear operator

In this paper, we treat the split feasibility problem with uncertain linear operator (USFP). For this problem, we first reformulate it as an uncertain optimization problem (UOP) with zero optimal value, and then we introduce robust counterparts of the UOP and reformulate them as the tractable convex optimization problems. These convex optimization problems have close connection with the robust counterparts of USFP and the minimum SFPs under the appropriate conditions. In the end of this paper, we give some numerical results to illustrate the effectiveness of the robust solutions of the concerned problem.