Equilibrium Problems In Hilbert Spaces Research Articles

Tseng-type splitting projection algorithms for equilibrium problems in Hilbert spaces

In this work, based on the subgradient-type projection method considered by Santos and Scheimberg (Optimization, 2017), we propose some Tseng-type splitting algorithms for solving equilibrium problems whose bifunction is decomposed into the sum of two bifunctions in Hilbert spaces. We establish the weak and strong convergence of the proposed algorithm under standard assumptions. Numerical results in Nash–Cournot oligopolistic electricity market equilibrium model and comparisons with other related methods demonstrate the effectiveness of the proposed algorithm.

Convergence analysis for solving the split equality equilibrium problem in Hilbert spaces

Convergence analysis for solving the split equality equilibrium problem in Hilbert spaces

Viscosity-regularization iterative methods for solving equilibrium problems in Hilbert space

The paper concerns a new iterative method for approximating a solution of an equilibrium problem involving a monotone and Lipschitz-type bifunction in a Hilbert space. Our method combines the proximal mapping with a regularization technique. When compared with known extragradient methods, the proposed method is seen to have a simpler and more elegant structure. A strong convergence theorem is established under some appropriate conditions imposed on the control parameters. Several experiments are performed in order to illustrate the numerical effectiveness of our new algorithm in comparison with that of existing methods.

A linesearch projection algorithm for solving equilibrium problems without monotonicity in Hilbert spaces

<p style='text-indent:20px;'>We propose a linesearch projection algorithm for solving non-monotone and non-Lipschitzian equilibrium problems in Hilbert spaces. It is proved that the sequence generated by the proposed algorithm converges strongly to a solution of the equilibrium problem under the assumption that the solution set of the associated Minty equilibrium problem is nonempty. Compared with existing methods, we do not employ Fejér monotonicity in the strategy of proving the convergence. This comes from projecting a fixed point instead of the current point onto a subset of the feasible set at each iteration. Moreover, employing an Armijo-linesearch without subgradient has a great advantage in CPU-time. Some numerical experiments demonstrate the efficiency and strength of the presented algorithm.</p>

Fast inertial extragradient algorithms for solving non-Lipschitzian equilibrium problems without monotonicity condition in real Hilbert spaces

Combining inertial-type methods with projection extragradient methods, we propose three inertial extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces. Under the assumption that the solution set of the associated Minty equilibrium problem is nonempty, we establish the weak and strong convergence of the proposed algorithms, respectively. The convergence is guaranteed without any monotonicity and Lipschitz-type continuity of the equilibrium bifunction. Some numerical experiments illustrate that the presented algorithms are faster than the existing projection extragradient ones.

One-Step iterative method for bilevel equilibrium problem in Hilbert space

One-Step iterative method for bilevel equilibrium problem in Hilbert space

Inertial Projection Algorithm for Solving Split Best Proximity Point and Mixed Equilibrium Problems in Hilbert Spaces

The primary goal of this paper is to present and study an inertial projection algorithm for solving the split best proximity and mixed equilibrium problems. We find a solution of the best proximity problem in such a way that its image under a bounded linear operator is the solution of the mixed equilibrium problem under the setting of real Hilbert spaces. We construct an iterative algorithm for the proposed problem and prove a weak convergence theorem. Moreover, we deduce some consequences from the main convergence result. Finally, a numerical experiment is presented to demonstrate the convergence analysis of our algorithm. The methodology and results presented in this work improve and unify some previously published findings in this field.

Regularization iterative method of bilevel form for equilibrium problems in Hilbert spaces

We study the regularization method for a monotone equilibrium problem and propose a new iterative method for solving the problem with a Lipschitz‐type condition in a Hilbert space. The method is designed by the proximal mapping incorporated with regularization terms. The method uses variable stepsizes which are taken by simple rules without a linesearch procedure. The strong convergence of iterative sequences generated by the method is established under some suitable conditions imposed on control parameters. It turns out that the obtained asymptotic solution by the method is the solution of an equilibrium problem whose constraint is the solution set of the considered equilibrium problem. The computational effectiveness of the method is illustrated by several numerical examples.

MK viscosity approximation for fixed points and equilibrium problems in Hilbert spaces

MK viscosity approximation for fixed points and equilibrium problems in Hilbert spaces

Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems

In this paper, we introduce a new hybrid inertial accelerated algorithm with a line search technique for solving fixed point problems for demimetric mapping and split equilibrium problems in Hilbert spaces. The algorithm is inspired by Tseng’s extragradient method and the viscosity method. Then, we establish and prove the strong convergence theorem under proper conditions. Furthermore, we also give a numerical example to support the main results. The main results are new and the proofs are relatively simple and different from those in early and recent literature.

Solving common nonmonotone equilibrium problems using an inertial parallel hybrid algorithm with Armijo line search with applications to image recovery

In this work, we modify the inertial hybrid algorithm with Armijo line search using a parallel method to approximate a common solution of nonmonotone equilibrium problems in Hilbert spaces. A weak convergence theorem is proved under some continuity and convexity assumptions on the bifunction and the nonemptiness of the common solution set of Minty equilibrium problems. Furthermore, we demonstrate the quality of our inertial parallel hybrid algorithm by using image restoration, as well as its superior efficiency when compared with previously considered parallel algorithms.

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.

Modified proximal-like extragradient methods for two classes of equilibrium problems in Hilbert spaces with applications

The objective of this study is to introduce two new extragradient methods to solve equilibrium problems involving two distinct classes of bifunction. Based on the pseudomonotonicity hypothesis and the specific Lipschitz-type cost bifunction condition, we have shown a weak convergence theorem for the first proposed method. The bifunction is strongly pseudomonotone in the second method, but the step-size rule does not depend on the strongly pseudomonotone constant and Lipschitz-type constants. In contrast, the first convergence result, we prove a strong convergence theorem with the use of a particular class of variable step-size rule sequences. To confirm the validity of the proposed convergence results, we examine two well-known Nash–Cournot equilibrium models for the numerical experiment and show that the competitive advantage of our proposed methods is based on time of execution and number of iterations.

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

A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions of split generalized equilibrium problems in Hilbert spaces

ABSTRACT The purpose of this article is to present a new iterative technique for approximating solutions of split generalized equilibrium problem and common fixed points of multivalued demicontractive mappings satisfying the gate conditions in real Hilbert spaces. Unlike the earlier results in this direction, we obtain a strong convergence result using an Armijo line search rule for determining the best appropriate step size for the next iteration. Also, our algorithm is designed in such a way that it does not require a projection onto the feasible set. We further give an analysis of the convergence rate of our method which is shown to be . Finally, the performances and comparisons with some existing methods are presented through numerical experiments. This result extends, generalizes and improves many of the existing related results in a unified way.

Two-Step Proximal Method for Equilibrium Problems in Hadamard spaces

We propose a novel two-step proximal method for solving equilibrium problems in Hadamard spaces. The equilibrium problem is very general in the sense that it includes as special cases many applied mathematical models such as: variational inequalities, optimization problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithm is the analog of the two-step algorithm for solving the equilibrium problem in Hilbert spaces explored earlier. We prove the weak convergence of the sequence generated by the algorithm for pseudo-monotone bifunctions. Our results extend some known results in the literature for pseudo-monotone equilibrium problems.

A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.

An inertial based forward\u2013backward algorithm for monotone inclusion problems and split mixed equilibrium problems in Hilbert spaces

Iterative algorithms are widely applied to solve convex optimization problems under a suitable set of constraints. 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 monotone inclusion problem and the split mixed equilibrium problem in Hilbert spaces. Moreover, numerical experiments compare favorably the efficiency of the proposed algorithm with the existing algorithms. As a consequence, our results improve various existing results in the current literature.

Fixed Point Theorems for a Demicontractive Mapping and Equilibrium Problems in Hilbert Spaces

In this research, we introduce some properties of demicontractive mapping and the combination of equilibrium problem. Then, we prove a strong convergence for the iterative sequence converging to a common element of fixed point set of demicontractive mapping and a common solution of equilibrium problems. Finally, we give a numerical example for the main theorem to support our results.

Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators

The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding a common element of the fixed point of pseudocontractive operators and the pseudomonotone equilibrium problem in Hilbert spaces. The suggested iterative algorithm is based on the projected method and subgradient method with a linearsearch technique. We show the strong convergence result for the iterative sequence generated by this algorithm. Some applications are also included. Our result improves and extends some existing results in the literature.