Multiple-output Setting Research Articles

Mann-type approximation scheme for solving a new class of split inverse problems in Hilbert spaces

ABSTRACT In this paper, we introduce and study a new class of split inverse problems, named split hierarchical monotone variational inclusion problem with multiple output sets in real Hilbert spaces. By using the inertial technique and self-adaptive step size strategy, we propose and analyze a new Mann-type iterative method for solving the problem. The convergence analysis of the proposed iterative method under some suitable conditions is studied. Also, we show that the sequence of iterates generated by this method converges strongly to a minimum-norm solution of the problem. As theoretical applications, we apply our results to approximate the solutions of other classes of split inverse problems. Finally, we present some numerical experiments to illustrate the practical potential and advantages of our proposed method.

Variational inequalities over the solution sets of split variational inclusion problems

We study variational inequalities over the solution sets of split variational inclusion problems with multiple output sets. In order to find a solution to such problems, we introduce a new parallel iterative algorithm by combining the inertial forward–backward splitting method with the steepest descent method. Our algorithm does not depend on the norm of the transfer mappings. Some of its applications to solving split common null point, split minimum point, and split common fixed point problems are also given.

New Algorithms for Solving the Split Common Zero Point Problem in Hilbert Space

We introduce new self-adaptive algorithms for solving the split common zero point problem with multiple output sets in Hilbert space. We also apply our main results to solving split feasibility problems with multiple output sets.

Two new self-adaptive algorithms for solving split common fixed point problems with multiple output sets

In this paper, we study split common fixed point problems with multiple output sets in Hilbert spaces. For solving this problem, we improve Reich and Tuyen's algorithms [10] and propose two new self-adaptive algorithms which do not need any priori information on the norms of the transfer operators. Then, we establish weak convergence theorems for them. It can be seen from our numerical experiments that the proposed methods depict a very significant improvement in terms of the number of iterations and execution time.

On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints

Abstract In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique for accelerating the speed of convergence and a self-adaptive step size so that its implementation does not require prior knowledge of the operator norm. Under mild conditions, we obtain a strong convergence result for the proposed algorithm and obtain a consequent result, which complements several existing results in the literature. Moreover, we apply our result to study the notions of split variational inequality problem with multiple output sets with fixed point constraints and split convex minimisation problem with multiple output sets with fixed point constraints in Hilbert spaces. Finally, we present some numerical experiments to demonstrate the implementability of our proposed method.

The split common fixed point problem with multiple output sets for demicontractive mappings

In this paper, we investigate the split common split feasibility with multiple output sets when the relevant nonlinear mapping is demicontractive. To solve this problem, we propose several new methods and state their convergence theorems under some mild conditions. The split feasibility problem with numerous output sets is solved using the outcomes as an example.

A New Algorithm for the Split Feasibility Problem With Multiple Output Sets in Hilbert Spaces

In this paper, we study the split feasibility problem with multiple output sets in Hilbert spaces. We propose a new self-adaptive algorithm combing with ball-relaxation and inertial acceleration, and prove its strong convergence. Numerical simulations are provided to illustrate the effectiveness of the proposed algorithm.

Strong convergence of two regularized relaxed extragradient schemes for solving the split feasibility and fixed point problem with multiple output sets

We study the split feasibility and fixed point problem with multiple output sets in real Hilbert spaces. Using Tikhonov's regularization and viscosity approximation techniques, we propose two relaxed extragradient algorithms for finding approximate solutions to this problem and prove that the sequences they generate converge strongly. Finally, we give some consequences and an application of our results, and also present numerical examples to validate the efficiency of our algorithms.

A Relaxed Inertial Tseng’s Extragradient Method for Solving Split Variational Inequalities with Multiple Output Sets

Recently, the split inverse problem has received great research attention due to its several applications in diverse fields. In this paper, we study a new class of split inverse problems called the split variational inequality problem with multiple output sets. We propose a new Tseng extragradient method, which uses self-adaptive step sizes for approximating the solution to the problem when the cost operators are pseudomonotone and non-Lipschitz in the framework of Hilbert spaces. We point out that while the cost operators are non-Lipschitz, our proposed method does not involve any linesearch procedure for its implementation. Instead, we employ a more efficient self-adaptive step size technique with known parameters. In addition, we employ the relaxation method and the inertial technique to improve the convergence properties of the algorithm. Moreover, under some mild conditions on the control parameters and without the knowledge of the operators’ norm, we prove that the sequence generated by our proposed method converges strongly to a minimum-norm solution to the problem. Finally, we apply our result to study certain classes of optimization problems, and we present several numerical experiments to demonstrate the applicability of our proposed method. Several of the existing results in the literature in this direction could be viewed as special cases of our results in this study.

Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets

<p style='text-indent:20px;'>In this paper, we propose two new self-adaptive inertial relaxed <inline-formula><tex-math id="M2">\begin{document}$ CQ $\end{document}</tex-math></inline-formula> algorithms for solving the split feasibility problem with multiple output sets in the framework of real Hilbert spaces. The proposed algorithms involve computing projections onto half-spaces instead of onto the closed convex sets, and the advantage of the self-adaptive step size introduced in our algorithms is that it does not require the computation of operator norm. We establish and prove weak and strong convergence theorems for the iterative sequences generated by the introduced algorithms for solving the aforementioned problem. Moreover, we apply the new results to solve some other problems. Finally, we present some numerical examples to illustrate the implementation of our algorithms and compared them to some existing results.</p>

Two-step inertial method for solving split common null point problem with multiple output sets in Hilbert spaces

<abstract><p>In this paper, an algorithm with two-step inertial extrapolation and self-adaptive step sizes is proposed to solve the split common null point problem with multiple output sets in Hilbert spaces. Weak convergence analysis are obtained under some easy to verify conditions on the iterative parameters in Hilbert spaces. Preliminary numerical tests are performed to support the theoretical analysis of our proposed algorithm.</p></abstract>

A self-adaptive inertial viscosity projection algorithm for solving split feasibility problem with multiple output sets

A self-adaptive inertial viscosity projection algorithm for solving split feasibility problem with multiple output sets

Regularization methods for solving the split feasibility problem with multiple output sets in Hilbert spaces

We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem, we introduce several new iterative processes by using the Tikhonov regularization method.

On split generalized equilibrium and fixed point problems with multiple output sets in real Banach spaces

On split generalized equilibrium and fixed point problems with multiple output sets in real Banach spaces

The Split Feasibility Problem with Multiple Output Sets for Demicontractive Mappings

The Split Feasibility Problem with Multiple Output Sets for Demicontractive Mappings

Self-adaptive algorithms for solving split feasibility problem with multiple output sets

Self-adaptive algorithms for solving split feasibility problem with multiple output sets

A Self-Adaptive Step Size Algorithm for Solving Variational Inequalities with the Split Feasibility Problem with Multiple Output Sets Constraints

In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of the split feasibility problem with multiple output sets in real Hilbert spaces. The strong convergence of the proposed algorithm is proved without knowing any information of the Lipschitz and strongly monotone constants of the mapping. In addition, the implementation of the algorithm does not require the computation or estimation of the norms of the given bounded linear operators. Special cases are considered. Finally, a numerical experiment has been carried out to illustrate the proposed algorithm.

Approximation of solutions of the split minimization problem with multiple output sets and common fixed point problems in real Banach spaces

Approximation of solutions of the split minimization problem with multiple output sets and common fixed point problems in real Banach spaces

Inertial accelerated algorithms for solving split feasibility with multiple output sets in Hilbert spaces

Abstract In this paper, we propose two inertial accelerated algorithms which do not require prior knowledge of operator norm for solving split feasibility problem with multiple output sets in real Hilbert spaces. We prove weak and strong convergence results for approximating the solution of the considered problem under certain mild conditions. We also give some numerical examples to demonstrate the performance and efficiency of our proposed algorithms over some existing related algorithms in the literature.

An Inertial Accelerated Algorithm for Solving Split Feasibility Problem with Multiple Output Sets

The paper proposes an inertial accelerated algorithm for solving split feasibility problem with multiple output sets. To improve the feasibility, the algorithm involves computing of projections onto relaxed sets (half spaces) instead of computing onto the closed convex sets, and it does not require calculating matrix inverse. To accelerate the convergence, the algorithm adopts self-adaptive rules and incorporates inertial technique. The strong convergence is shown under some suitable conditions. In addition, some newly derived results are presented for solving the split feasibility problem and split feasibility problem with multiple output sets. Finally, numerical experiments illustrate that the algorithm converges more quickly than some existing algorithms. Our results extend and improve some methods in the literature.