Split Equilibrium Problem Research Articles

Inertial Extragradient Methods for Solving Split Equilibrium Problems

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms.

A new extragradient algorithm with adaptive step-size for solving split equilibrium problems

He (J. Inequal. Appl. 2012:Article ID 162 2012) introduced the proximal point CQ algorithm (PPCQ) for solving the split equilibrium problem (SEP). However, the PPCQ converges weakly to a solution of the SEP and is restricted to monotone bifunctions. In addition, the step-size used in the PPCQ is a fixed constant μ in the interval (0, frac{1}{ | A |^{2} } ). This often leads to excessive numerical computation in each iteration, which may affect the applicability of the PPCQ. In order to overcome these intrinsic drawbacks, we propose a robust step-size { mu _{n} }_{n=1}^{infty } which does not require computation of | A | and apply the adaptive step-size rule on { mu _{n} }_{n=1}^{infty } in such a way that it adjusts itself in accordance with the movement of associated components of the algorithm in each iteration. Then, we introduce a self-adaptive extragradient-CQ algorithm (SECQ) for solving the SEP and prove that our proposed SECQ converges strongly to a solution of the SEP with more general pseudomonotone equilibrium bifunctions. Finally, we present a preliminary numerical test to demonstrate that our SECQ outperforms the PPCQ.

An extragradient algorithm for the split equilibrium problems without prior knowledge of operator norm

An extragradient algorithm for the split equilibrium problems without prior knowledge of operator norm

Iterative algorithms of common solutions for a hierarchical fixed point problem, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces

In this paper, we suggest and analyze an iterative algorithm to approximate a common solution of a hierarchical fixed point problem for nonexpansive mappings, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces. Under some suitable conditions imposed on the sequences of parameters, we prove that the sequence generated by the proposed iterative method converges strongly to a common element of the solution set of these three kinds of problems. The results obtained here extend and improve the corresponding results of the relevant literature.

Iterative solution of split equilibrium and fixed point problems in real Hilbert spaces

In this article, we introduce a hybrid iteration involving inertial-term for split equilibrium problem and fixed point for a finite family of asymptotically strictly pseudocontractive mappings. We prove that the sequence converges strongly to a solution of split equilibrium problem and a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings. The results proved extend and improve recent results of Chang et al. [S. S. Chang, H. W. J. Lee, C. K. Chan, L. Wang, L. J. Qin, Appl. Math. Comput., \(\bf 219\) (2013), 10416--10424], Dewangan et al. [R. Dewangan, B. S. Thakur, M. Postolache, J. Inequal. Appl., \(\bf 2014\) (2014), 11 pages], and many others.

An iterative scheme for split equality equilibrium problems and split equality hierarchical fixed point problem

This paper deals with a split equality equilibrium problem for pseudomonotone bifunctions and a split equality hierarchical fixed point problem for nonexpansive and quasinonexpansive mappings. We suggest and analyze an iterative scheme where the stepsizes do not depend on the operator norms, the so-called simultaneous projected subgradient-proximal iterative scheme for approximating a common solution of the split equality equilibrium problem and the split equality hierarchical fixed point problem. Further, we prove a weak convergence theorem for the sequences generated by this scheme. Furthermore, we discuss some consequences of the weak convergence theorem. We present a numerical example to justify the main result.

Inertial algorithm with self-adaptive step size for split common null point and common fixed point problems for multivalued mappings in Banach spaces

ABSTRACT In this article we introduce an inertial iterative scheme with self-adaptive step size for finding a common solution of split common null point problem for a finite family of maximal monotone operators and fixed point problem for a finite family of multivalued demicontractive mappings between a Banach space and Hilbert space. Strong convergence result is obtained for the proposed algorithm. The self-adaptive step size ensures no requirement for a prior knowledge or estimate of the norm of the operator. The inertial term introduced in the algorithm is efficient, it helps to avoid imposing some strong conditions usually used for inertial-type algorithms by many authors. We give some applications of our results to game theory, split equilibrium and minimum-norm problems. Numerical experiment is also presented to demonstrate the efficiency of our proposed method as well as comparing with other existing method in the literature. Our results improve and generalize many well known results in this direction in the literature.

An inertially constructed forward\u2013backward splitting algorithm in Hilbert spaces

In this paper, we develop an iterative algorithm whose architecture comprises a modified version of the forward–backward splitting algorithm and the hybrid shrinking projection algorithm. We provide theoretical results concerning weak and strong convergence of the proposed algorithm towards a common solution of the fixed point problem associated to a finite family of demicontractive operators, the split equilibrium problem and the monotone inclusion problem in Hilbert spaces. Moreover, we compute a numerical experiment to show the efficiency of the proposed algorithm. As a consequence, our results improve various existing results in the current literature.

An Iterative Method for a Common Solution of a Combination of the Split Equilibrium Problem, a Finite Family of Nonexpansive Mapping and a Combination of Variational Inequality Problem

The present paper aims to deal with an iterative algorithm for finding common solution of the combination of the split equilibrium problem and a finite family of non-expansive mappings and the combination of variational inequality problem in the setting of real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution to these problems. A numerical example is presented to illustrate the proposed method and convergence result. The results and method presented in this paper generalize, extend and unify some known results in the literatures.

Strong Convergence Results of Split Equilibrium Problems and Fixed Point Problems

In this paper, we investigate the split equilibrium problem and fixed point problem in Hilbert spaces. We propose an iterative scheme for solving such problem in which the involved equilibrium bifunctionsfandgare pseudomonotone and monotone, respectively, and the operatorsSandTare all pseudocontractive. We show that the suggested scheme converges strongly to a solution of the considered problem.

A New Study on Halpern and Nonconvex Combination Algorithm for Nonlinear Mappings in Banach Spaces with Applications

In this paper, we introduce a Halpern algorithm and a nonconvex combination algorithm to approximate a solution of the split common fixed problem of quasi- ϕ -nonexpansive mappings in Banach space. In our algorithms, the norm of linear bounded operator does not need to be known in advance. As the application, we solve a split equilibrium problem in Banach space. Finally, some numerical examples are given to illustrate the main results in this paper and compare the computed results with other ones in the literature. Our results extend and improve some recent ones in the literature.

Approximation of common solutions of fixed point problem for α-hemicontractive mapping, split equilibrium and variational inequality problems

ln this paper, a Halpern-type algorithm for approximating a common fixed point of multivalued α-hemicontractive mappings and a set of solutions of split equilibrium and variational inequality problems is constructed. Strong convergence of the sequence generated by the algorithm is proved in the setting real Hilbert spaces. Our results improved and generalised the results of Meche and Zegeye [2] in particular and some recent results in Literature.

The projection method with inertial extrapolation for solving split equilibrium problems in Hilbert spaces

The projection method with inertial extrapolation for solving split equilibrium problems in Hilbert spaces

An iterative algorithm for the system of split mixed equilibrium problem

AbstractIn this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

Iterative algorithms for split equilibrium problems of monotone operators and fixed point problems of pseudo-contractions

In this paper, we investigate the split equilibrium problems and fixed point problems in Hilbert spaces. We provide a unified framework for solving such problem in which the involved equilibrium bifunctions f and g are pseudomonotone and monotone, respectively, and the operators S and T are pseudocontractive. We suggest an iterative algorithm for solving the split problem and demonstrate its weak convergence.

A Self-Adaptive Shrinking Projection Method with an Inertial Technique for Split Common Null Point Problems in Banach Spaces

In this paper, we present a new self-adaptive inertial projection method for solving split common null point problems in p-uniformly convex and uniformly smooth Banach spaces. The algorithm is designed such that its convergence does not require prior estimate of the norm of the bounded operator and a strong convergence result is proved for the sequence generated by our algorithm under mild conditions. Moreover, we give some applications of our result to split convex minimization and split equilibrium problems in real Banach spaces. This result improves and extends several other results in this direction in the literature.

A self-adaptive extragradient–CQ method for a class of bilevel split equilibrium problem with application to Nash Cournot oligopolistic electricity market models

Van Quy (Optimization 68(4):753–771, 2018) established an extragradient–CQ algorithm for solving a class of bilevel split equilibrium problem. The step-size in the algorithm requires computation of a certain matrix norm which is costly. Moreover, bilevel problems often possess huge number of constraints and possibly require a robust and adaptive step-size algorithm to withstand failure in instances of real-world problems with large data sizes. In this paper, we propose a self-adaptive step-size extragradient–CQ algorithm for solving the same problem without prior knowledge of operator norm and provide a numerical example to demonstrate the effectiveness of our method.

Modified Krasnoselski\u2013Mann iterative method for hierarchical fixed point problem and split mixed equilibrium problem

In this paper, we introduce a modified Krasnoselski–Mann type iterative method for capturing a common solution of a split mixed equilibrium problem and a hierarchical fixed point problem of a finite collection of k-strictly pseudocontractive nonself-mappings. Many of the algorithms for solving the split mixed equilibrium problem involve a step size which depends on the norm of a bounded linear operator. Since the computation of the operator norm is very difficult, we formulate our iterative algorithm in such a way that the implementation of the proposed algorithm does not require any prior knowledge of operator norm. Weak convergence results are established under mild conditions. We also establish strong convergence results for a certain class of hierarchical fixed point and split equilibrium problem. Our results generalize some important results in the recent literature.

Approximation results for split equilibrium problems and fixed point problems of nonexpansive semigroup in Hilbert spaces

In this paper, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of a nonexpansive semigroup in real Hilbert spaces. The weak and strong convergence characteristics of the proposed algorithm are investigated by employing suitable control conditions in such a setting of spaces. As a consequence, we provide a simplified analysis of various existing results concerning the extragradient method in the current literature. We also provide a numerical example to strengthen the theoretical results and the applicability of the proposed algorithm.

The Split Various Variational Inequalities Problems for Three Hilbert Spaces

There are many methods for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. They proved the strong convergence theorem. Many split feasibility problems are generated in real Hillbert spaces. The open problem is proving a strong convergence theorem of three Hilbert spaces with different methods from the lasted method. In this research, a new split variational inequality in three Hilbert spaces is proposed. Important tools which are used to solve classical problems will be developed. The convergence theorem for finding a common element of the set of solution of such problems and the sets of fixed-points of discontinuous mappings has been proved.