Generalized Equilibrium Problems Research Articles

The AA-Viscosity Algorithm for Fixed-Point, Generalized Equilibrium and Variational Inclusion Problems

The aim of this paper is to propose an inertial-type AA-viscosity algorithm for approximating the common solutions of the split variational inclusion problem, the generalized equilibrium problem and the common fixed-point problem of nonexpansive mappings. The strong convergence of an iterative sequence obtained through the proposed method is proved under some mild assumptions. Consequently, approximations of the solution of the split feasibility problem, the relaxed split feasibility problem, the split common null point problem and the split minimization problem are given. The applicability of our proposed algorithm has been illustrated with the help of a numerical example. Our iterative method was then compared graphically with different comparable methods in the existing literature.

Modified inertial subgradient extragradient algorithms for generalized equilibria systems with constraints of variational inequalities and fixed points

<abstract><p>In this research, we studied modified inertial composite subgradient extragradient implicit rules for finding solutions of a system of generalized equilibrium problems with a common fixed-point problem and pseudomonotone variational inequality constraints. The suggested methods consisted of an inertial iterative algorithm, a hybrid deepest-descent technique, and a subgradient extragradient method. We proved that the constructed algorithms converge to a solution of the considered problem, which also solved some hierarchical variational inequality.</p></abstract>

Common solutions of fixed point and generalized equilibrium problems using asymptotically nonexpansive mapping

In this paper, a common solution of fixed point and generalized equilibrium problems using an asymptotically nonexpansive mapping is determined via iterative approach. Further, an application and numerical example of the main result are given.

ON SPLIT GENERALIZED EQUILIBRIUM AND FIXED POINT PROBLEMS OF BREGMAN W-MAPPINGS WITH MULTIPLE OUTPUT SETS IN REFLEXIVE BANACH SPACES

In this paper, we introduce a Halpern iteration process for computing the common solution of split generalized equilibrium problem and fixed points of a countable family of Bregman W-mappings with multiple output sets in reflexive Banach spaces. We prove a strong convergence result for approximating the solutions of the aforementioned problems under some mild conditions. It is worth mentioning that the iterative algorithm employ in this article is designed in such a way that it does not require the prior knowledge of operator norm. We also provide some numerical examples to illustrate the performance of our proposed iterative method. The result discuss in this paper extends and complements many related results in literature.

A Modified Viscosity-Type Self-Adaptive Iterative Algorithm for Common Solution of Split Problems with Multiple Output Sets in Hilbert Spaces

A modified viscosity-type self-adaptive iterative algorithm is presented in this study, having a strong convergence theorem for estimating the common solution to the split generalized equilibrium problem along with the split common null point problem with multiple output sets, subject to some reasonable control sequence restrictions. The suggested algorithm and its immediate consequences are also discussed. The effectiveness of the proposed algorithm is finally demonstrated through analytical examples. The findings presented in this paper will help to consolidate, extend, and improve upon a number of recent findings in the literature.

On split generalized equilibrium problem with multiple output sets and common fixed points problem

Abstract In this article, we introduce and study the notion of split generalized equilibrium problem with multiple output sets (SGEPMOS). We propose a new iterative method that employs viscosity approximation technique for approximating the common solution of the SGEPMOS and common fixed point problem for an infinite family of multivalued demicontractive mappings in real Hilbert spaces. Under mild conditions, we prove a strong convergence theorem for the proposed method. Our method uses self-adaptive step size that does not require prior knowledge of the operator norm. The results presented in this article unify, complement, and extend several existing recent results in the literature.

Approximating solutions of the generalized modification of the system of equilibrium problems and fixed point problem of a nonexpansive mapping

The purpose of this research is to study the generalized modification of the system of equilibrium problems (GMSEP) and a lemma is established to show the property of this problem. Then, we prove a strong convergence theorem for finding a common element of the set of the solutions of the fixed points problem and the set of the solutions of the GMSEP under some suitable conditions, in which , where are coefficients in the main iteration. Moreover, we prove strong convergence theorems for finding solutions to the generalized equilibrium problem, the system of equilibrium problems, the variational inequality problem, the general system of variational inequality problems, and the minimization problem. Finally, we give two numerical examples, one of which shows the rate of convergence of the main iteration while the other shows the rate of convergence of the main iteration but the sum of coefficients equals 1.

A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

In this paper, based on the viscosity approximation method and the hybrid steepest-descent iterative method, a new implicit iterative algorithm is presented for finding the common fixed points set of a finite family of nonexpansive mappings in a reflexive Hilbert space, which is called a symmetric space. We prove that the sequence generated by this new implicit rule strongly converges to the unique solution of a class of variational inequalities under certain appropriate conditions of the parameters. Moreover, we also study the applications to a broader family of strictly pseudo-contractive mappings and generalized equilibrium problems that involve several variational inequality problems, optimization problems, and fixed-point problems. Finally, numerical results are provided to clarify the stability and effectiveness of the algorithm and to compare with some existing iterative algorithms.

Generalized Equilibrium Problems

If X is a convex subset of a topological vector space and f is a real bifunction defined on X×X, the problem of finding a point x0∈X such that f(x0,y)≥0 for all y∈X, is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call it a generalized equilibrium problem. In this paper, we study the existence of the solutions, first for generalized equilibrium problems and then for equilibrium problems. In the obtained results, apart from the bifunction f, another bifunction is introduced, the two being linked by a certain compatibility condition. The particularity of the equilibrium theorems established in the last section consists of the fact that the classical equilibrium condition (f(x,x)=0, for all x∈X) is missing. The given applications refer to the Minty variational inequality problem and quasi-equilibrium problems.

An inertial extragradient algorithm for split problems in Hilbert spaces

Abstract In this study, we propose an inertial extragradient algorithm for solving a split generalized equilibrium problem as well as a split feasibility and common fixed point problem. We demonstrate that, under certain reasonable assumptions, the sequences induced by the proposed algorithm converge strongly to a solution of the corresponding problem. In addition, with the help of a numerical example, we demonstrate the efficiency of proposed algorithm. As a result of this paper, some recent well-known results in this area have been improved, generalized, and extended.

Modified subgradient extragradient algorithms for systems of generalized equilibria with constraints

<abstract><p>In this paper, we introduce the modified Mann-like subgradient-like extragradient implicit rules with linear-search process for finding a common solution of a system of generalized equilibrium problems, a pseudomonotone variational inequality problem and a fixed-point problem of an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on the subgradient extragradient rule with linear-search process, Mann implicit iteration approach, and hybrid deepest-descent technique. Under mild restrictions, we demonstrate the strong convergence of the proposed algorithms to a common solution of the investigated problems, which is a unique solution of a certain hierarchical variational inequality defined on their common solution set.</p></abstract>

On the existence and approximation of solutions of generalized equilibrium problem on Hadamard manifolds

In this paper, we study the existence of solution of the generalized equilibrium problem (GEP) in the framework of an Hadamard manifold. Using the KKM lemma, we prove the existence of solution of the GEP and give the properties of the resolvent function associated with the problem under consideration. Furthermore, we introduce an iterative algorithm for approximating a common solution of the GEP and a fixed point problem. Using the proposed method, we obtain and prove a strong convergence theorem for approximating a solution of the GEP, which is also a fixed point of a nonexpansive mapping under some mild conditions. We give an application of our convergence result to a solution of the convex minimization problem. To illustrate the convergence of the method, we report some numerical experiments. The result in this paper extends the study of the GEP from the linear settings to the Hadamard manifolds.

Extragradient Methods for Solving Split Feasibility Problem and General Equilibrium Problem and Resolvent Operators in Banach Spaces

In this paper, we introduce a new extragradient algorithm by using generalized metric projection. We prove a strong convergence theorem for finding a common element of the solution set of split feasibility problem and the set of fixed points of relatively nonexpansive mapping and a finite family of resolvent operator and the set of solutions of an equilibrium problem.

Regularization Method for the Variational Inequality Problem over the Set of Solutions to the Generalized Equilibrium Problem

The paper is devoted to bilevel problems: variational inequality problems over the set of solutions to the generalized equilibrium problems in a Hilbert space. To solve these problems, an iterative algorithm is proposed that combines the ideas of the Tseng’s extragradient method, the inertial idea and iterative regularization. The proposed method adopts a non-monotonic stepsize rule without any line search procedure. Under suitable conditions, the strong convergence of the resulting method is obtained. Several numerical experiments are also provided to illustrate the efficiency of the proposed method with respect to certain existing ones.

General stochastic ridesharing user equilibrium problem with elastic demand

In this study, we propose a general stochastic ridesharing user equilibrium (SRUE) problem with elastic demand (ED) for urban transportation network analysis with ridesharing activities by extending the deterministic ridesharing user equilibrium (DRUE) problem, which is referred to as the general SRUE-ED problem. We develop a novel variational inequality (VI) model in terms of the Lagrange multipliers corresponding to the ride-matching constraints for the general SRUE-ED problem. The existence and uniqueness of the SRUE-ED solution are demonstrated under mild conditions. We proceed to put up a global convergent method to solve the developed VI model by incorporating the weighted cost averaging (WCA) method. The purpose of WCA method is to solve a parametric fixed-point model formulated for the general SRUE-ED problem without the ride-matching constraints. Finally, we use three numerical examples to provide practical insights into the operation and management of the ridesharing systems and to examine the effectiveness of the proposed model and solution method.

An inertial-viscosity algorithm for solving split generalized equilibrium problem and a system of demimetric mappings in Hilbert spaces

An inertial-viscosity algorithm for solving split generalized equilibrium problem and a system of demimetric mappings in Hilbert spaces

The Iterative Method for Generalized Equilibrium Problems and a Finite Family of Lipschitzian Mappings in Hilbert Spaces

In this research, we introduced the S -mapping generated by a finite family of contractive mappings, Lipschitzian mappings and finite real numbers using the results of Kangtunyakarn (2013). Then, we prove the strong convergence theorem for fixed point sets of finite family of contraction and Lipschitzian mapping and solution sets of the modified generalized equilibrium problem introduced by Suwannaut and Kangtunyakarn (2014). Finally, numerical examples are provided to illustrate our main theorem.

A New Alternative Regularization Method for Solving Generalized Equilibrium Problems

The purpose of this paper is to present a numerical method for solving a generalized equilibrium problem involving a Lipschitz continuous and monotone mapping in a Hilbert space. The proposed method can be viewed as an improvement of the Tseng’s extragradient method and the regularization method. We show that the iterative process constructed by the proposed method converges strongly to the smallest norm solution of the generalized equilibrium problem. Several numerical experiments are also given to illustrate the performance of the proposed method. One of the advantages of the proposed method is that it requires no knowledge of Lipschitz-type constants.

"Viscosity algorithm for solving split generalized equilibrium problem and fixed-point problem"

"This article considers a split generalized equilibrium problem and fixed point problem for infi- nite family of nonexpansive mapping in Hilbert space. We propose an algorithm for finding a common solution of these problems. Under mild assumptions, we establish a strong convergence theorem for the sequence gen- erated by the proposed algorithm using viscosity technique. We present the implementation of the proposed algorithm by considering some numerical illustrations and comparison of the proposed algorithm with an ex- isting algorithm in the literature."

An inertial iterative algorithm for generalized equilibrium problems and Bregman relatively nonexpansive mappings in Banach spaces

The aim of this paper is to introduce and study an inertial hybrid iterative method for solving generalized equilibrium problems involving Bregman relatively nonexpansive mappings in Banach spaces. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational example to emphasize the efficiency and relevancy of main result.