scholarly journals Relaxed Inertial Tseng’s Type Method for Solving the Inclusion Problem with Application to Image Restoration

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 818 ◽  
Author(s):  
Jamilu Abubakar ◽  
Poom Kumam ◽  
Abdulkarim Hassan Ibrahim ◽  
Anantachai Padcharoen
Keyword(s):  
Lipschitz Constant ◽  
Monotone Mapping ◽  
Splitting Method ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Variable Step Size ◽  
Step Size ◽  
Type Method

The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the inertial extrapolation step. The proposed method follows from time explicit discretization of a dynamical system. A weak convergence of the iterates generated by the method involving monotone operators is given. Moreover, the iterative scheme uses a variable step size, which does not depend on the Lipschitz constant of the underlying operator given by a simple updating rule. Furthermore, the proposed algorithm is modified and used to derive a scheme for solving a split feasibility problem. The proposed schemes are used in solving the image deblurring problem to illustrate the applicability of the proposed methods in comparison with the existing state-of-the-art methods.

scholarly journals New Inertial Relaxed CQ Algorithms for Solving Split Feasibility Problems in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Haiying Li ◽  
Yulian Wu ◽  
Fenghui Wang
Keyword(s):  
Hilbert Spaces ◽  
Split Feasibility Problem ◽  
Linear Operators ◽  
Feasibility Problem ◽  
Variable Step Size ◽  
Step Size ◽  
Bounded Linear ◽  
Variable Step ◽  
The Split Feasibility Problem ◽  
Split Feasibility

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.

scholarly journals Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 773 ◽  
Author(s):  
Mihai Postolache ◽  
Ashish Nandal ◽  
Renu Chugh
Keyword(s):  
Fixed Point ◽  
Splitting Method ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Special Cases ◽  
Implicit Rule

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward–backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

scholarly journals The Split Feasibility Problem and Its Solution Algorithm

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Biao Qu ◽  
Binghua Liu
Keyword(s):  
Signal Processing ◽  
Real World ◽  
Lipschitz Constant ◽  
Split Feasibility Problem ◽  
Solution Algorithm ◽  
Feasibility Problem ◽  
Step Size ◽  
Convergence Results ◽  
The Split Feasibility Problem ◽  
Split Feasibility

The split feasibility problem arises in many fields in the real world, such as signal processing, image reconstruction, and medical care. In this paper, we present a solution algorithm called memory gradient projection method for solving the split feasibility problem, which employs a parameter and two previous iterations to get the next iteration, and its step size can be calculated directly. It not only improves the flexibility of the algorithm, but also avoids computing the largest eigenvalue of the related matrix or estimating the Lipschitz constant in each iteration. Theoretical convergence results are established under some suitable conditions.

scholarly journals An Extrapolated Iterative Algorithm for Multiple-Set Split Feasibility Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yazheng Dang ◽  
Yan Gao
Keyword(s):  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Convex Sets ◽  
Lipschitz Constant ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Step Size ◽  
Fixed Constant ◽  
The Split Feasibility Problem ◽  
Split Feasibility

The multiple-set split feasibility problem (MSSFP), as a generalization of the split feasibility problem, is to find a point in the intersection of a family of closed convex sets in one space such that its image under a linear transformation will be in the intersection of another family of closed convex sets in the image space. Censor et al. (2005) proposed a method for solving the multiple-set split feasibility problem (MSSFP), whose efficiency depends heavily on the step size, a fixed constant related to the Lipschitz constant of∇p(x)which may be slow. In this paper, we present an accelerated algorithm by introducing an extrapolated factor to solve the multiple-set split feasibility problem. The framework encompasses the algorithm presented by Censor et al. (2005). The convergence of the method is investigated, and numerical experiments are provided to illustrate the benefits of the extrapolation.

An efficient variable step-size rational Falkner-type method for solving the special second-order IVP

2016 ◽  
Vol 291 ◽  
pp. 39-51 ◽  
Author(s):  
Higinio Ramos ◽  
Gurjinder Singh ◽  
V. Kanwar ◽  
Saurabh Bhatia
Keyword(s):  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Type Method ◽  
Variable Step

Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces

2021 ◽  
pp. 2250106
Author(s):  
A. A. Mebawondu ◽  
L. O. Jolaoso ◽  
H. A. Abass ◽  
O. K. Narain
Keyword(s):  
Hilbert Spaces ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Inertial Method ◽  
Split Feasibility

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

scholarly journals An inertial iterative scheme for solving variational inclusion with application to Nash-Cournot equilibrium and image restoration problems

2021 ◽  
Vol 37 (3) ◽  
pp. 361-380
Author(s):  
JAMILU ABUBAKAR ◽  
◽  
POOM KUMAM ◽  
ABOR ISA GARBA ◽  
MUHAMMAD SIRAJO ABDULLAHI ◽  
...  
Keyword(s):  
Variational Inequality ◽  
Image Restoration ◽  
Split Feasibility Problem ◽  
Variational Inclusion ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Relaxation Techniques ◽  
Cournot Equilibrium ◽  
Monotone Equations ◽  
Variational Inclusion Problem

Variational inclusion is an important general problem consisting of many useful problems like variational inequality, minimization problem and nonlinear monotone equations. In this article, a new scheme for solving variational inclusion problem is proposed and the scheme uses inertial and relaxation techniques. Moreover, the scheme is self adaptive, that is, the stepsize does not depend on the factorial constants of the underlying operator, instead it can be computed using a simple updating rule. Weak convergence analysis of the iterates generated by the new scheme is presented under mild conditions. In addition, schemes for solving variational inequality problem and split feasibility problem are derived from the proposed scheme and applied in solving Nash-Cournot equilibrium problem and image restoration. Experiments to illustrate the implementation and potential applicability of the proposed schemes in comparison with some existing schemes in the literature are presented.

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