Class Of Equilibrium Problems Research Articles

Determination of material properties of a linearly elastic peridynamic material

We investigate the properties of an isotropic linear elastic peridynamic material in the context of a three-dimensional state-based peridynamic theory, which considers both length and relative angle changes, and is based on a free energy function proposed in previous work that contains four material constants. To this end, we consider a class of equilibrium problems in mechanics to show that, in interior points of the body where deformations are smooth, the corresponding solutions in classical linear elasticity are also equilibrium solutions in peridynamics. More generally, we show that the equations of equilibrium are satisfied even when two of the four peridynamic constants are arbitrary. Pure torsion of a cylindrical shaft and pure bending of a cylindrical beam are particular cases of this class of problems and are used together with a correspondence argument proposed elsewhere to determine these two constants in terms of the elasticity constants of an isotropic material from the classical linear elasticity. One of the constants has a singularity in the Poisson ratio, which needs further investigation. Two additional experiments concerning bending of cylindrical beam by terminal load and anti-plane shear of a hollow cylinder, which do not belong to the previous class of problems, are used to validate these results.

A parallel hybrid accelerated extragradient algorithm for pseudomonotone equilibrium, fixed point, and split null point problems

This paper provides iterative construction of a common solution associated with the classes of equilibrium problems (EP) and split convex feasibility problems. In particular, we are interested in the EP defined with respect to the pseudomonotone bifunction, the fixed point problem (FPP) for a finite family of -demicontractive operators, and the split null point problem. From the numerical standpoint, combining various classical iterative algorithms to study two or more abstract problems is a fascinating field of research. We, therefore, propose an iterative algorithm that combines the parallel hybrid extragradient algorithm with the inertial extrapolation technique. The analysis of the proposed algorithm comprises theoretical results concerning strong convergence under a suitable set of constraints and numerical results.

Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature.

Two strongly convergent methods governed by pseudo-monotone bi-function in a real Hilbert space with applications

Many iterative schemes have already been developed to solve the equilibrium problems, one of which is the most efficient two-step extragradient method. The objective of this research is to propose two new iterative methods with inertial effect to solve equilibrium problems. These iterative methods are based on an extra-gradient method and a Mann-type iterative method. Two strong convergence theorems have been proved in the setting of real Hilbert space, with mild assumptions that the underlying bi-function is Lipschitz-type continuous and pseudo-monotone. The primary advantage of the second method is that it does not require the information of the Lipschitz-type bi-functional constants. We have also studied the applications of our research results to solve particular classes of equilibrium problems. Numerical studies are carried out to show the behaviour of proposed methods and to compare them with the existing ones in the 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.

Iterative Schemes for Triequilibrium-Like Problems

In this work, we present a new class of equilibrium problems, termed as triequilibrium-like problems with trifunction in invexity settings. Classical varilunational-like inequalities and equilibrium-like problems can be obtained as specific variants of triequilibrium-like problems. Certain new iterative methods are proposed and examined for the solution of triequilibrium-like problems by using auxiliary principle technique. Convergence analysis of these proposed methods is examined under some mild conditions.

Higher Order Strongly Biconvex Functions and Biequilibrium Problems

In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and k-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the k-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.

Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models

In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods.

The Ishikawa Subgradient Extragradient Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

In this paper, we introduce a novel algorithm for finding a common point of the solution set of a class of equilibrium problems involving pseudo-monotone bifunctions and satisfying the Lipschitz-type condition and the set of fixed points of a quasi-nonexpansive in a real Hilbert space. This algorithm can be considered as a combination of the subgradient extragradient method for equilibrium problems and the Ishikawa method for fixed point problems. The strong convergence of the iterates generated by the proposed method is obtained under the main assumptions that the fixed-point mapping is demiclosed at 0 and Lipschitz-type constant of the cost bifunction is unknown. Some numerical examples are implemented to show the computational efficiency of the proposed algorithm.

General regularized nonconvex mixed equilibrium problems: iterative methods by auxiliary principle technique

This paper introduces a new class of equilibrium problems named general regularized nonconvex mixed equilibrium problem. By using the auxiliary principle technique, some predictor–corrector methods for solving such class of general regularized nonconvex mixed equilibrium problems are suggested and analyzed. The study of convergence analysis of the proposed iterative algorithms requires either jointly pseudomonotonicity or partially mixed relaxed and strong monotonicity property of the operator and bifunction involved in the considered problem. As a consequence of our main results, we provide the correct versions of the algorithms and results presented in the literature.

New inertial algorithm for a class of equilibrium problems

The article introduces a new algorithm for solving a class ofequilibrium problems involving strongly pseudomonotone bifunctions with Lipschitz-type condition. We describe how to incorporate the proximal-like regularized technique with inertial effects. The main novelty of the algorithm is that it can be done without previously knowing the information on the strongly pseudomonotone and Lipschitz-type constants of cost bifunction. A reasonable explain for this is that the algorithm uses a sequence of stepsizes which is diminishing and non-summable. Theorem of strong convergence is proved. In the case, when the information on the modulus of strong pseudomonotonicity and Lispchitz-type constant is known,the rate of linear convergence of the algorithm has been established. Several of numerical experiments are performed to illustrate the convergence of the algorithm and also compare it with other algorithms.

Generalized -D-gap functions and error bounds for a class of equilibrium problems

In this paper, we introduce and study the generalized -D-gap function for an equilibrium problem. We give sufficient conditions under which a critical point of the generalized -D-gap function is a solution of the equilibrium problem, and develop a descent scheme for locating these critical points. Using the generalized -D-gap function, we construct an error bound for the equilibrium problem. We support our approach with concrete examples.

Solving normalized stationary points of a class of equilibrium problem with equilibrium constraints

This paper focuses on solving normalized stationary points of a class of equilibrium problem with equilibrium constraints (EPEC). We show that, under some kind of separability assumption, normalized C-/M-/S-stationary points of EPEC are actually C-/M-/S-stationary points of an associated mathematical program with equilibrium constraints (MPEC), which implies that we can solve MPEC to obtain normalized stationary points of EPEC. In addition, we demonstrate the proposed approach on competition of manufacturers for similar products in the same city.

New extragradient method for a class of equilibrium problems in Hilbert spaces

The paper proposes a new extragradient algorithm for solving strongly pseudomonotone equilibrium problems which satisfy a Lipschitz-type condition recently introduced by Mastroeni in auxiliary problem principle. The main novelty of the paper is that the algorithm generates the strongly convergent sequences in Hilbert spaces without the prior knowledge of Lipschitz-type constants and any hybrid method. Several numerical experiments on a test problem are also presented to illustrate the convergence of the algorithm.

Generalized multivalued equilibrium-like problems: auxiliary principle technique and predictor-corrector methods

This paper is dedicated to the introduction a new class of equilibrium problems named generalized multivalued equilibrium-like problems which includes the classes of hemiequilibrium problems, equilibrium-like problems, equilibrium problems, hemivariational inequalities, and variational inequalities as special cases. By utilizing the auxiliary principle technique, some new predictor-corrector iterative algorithms for solving them are suggested and analyzed. The convergence analysis of the proposed iterative methods requires either partially relaxed monotonicity or jointly pseudomonotonicity of the bifunctions involved in generalized multivalued equilibrium-like problem. Results obtained in this paper include several new and known results as special cases.

Interior proximal extragradient method for equilibrium problems

The Bregman function-based Proximal Point Algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additional feature consists in an application of zone coercive regularizations. The latter allows to treat the generated subproblems as unconstrained ones, albeit with a certain precaution in numerical experiments. However, compared to the (classical) Proximal Point Algorithm for equilibrium problems, convergence results require additional assumptions which may be seen as the price to pay for unconstrained subproblems. Unfortunately, they are quite demanding – for instance, as they imply a sort of unique solvability of the given problem. The main purpose of this paper is to develop a modification of the BPPA, involving an additional extragradient step with adaptive (and explicitly given) stepsize. We prove that this extragradient step allows to leave out any of the additional assumptions mentioned above. Hence, though still of interior proximal type, the suggested method is applicable to an essentially larger class of equilibrium problems, especially including non-uniquely solvable ones.

Some Iterative Methods for Solving Nonconvex Bifunction Equilibrium Variational Inequalities

We introduce and consider a new class of equilibrium problems and variational inequalities involving bifunction, which is called the nonconvex bifunction equilibrium variational inequality. We suggest and analyze some iterative methods for solving the nonconvex bifunction equilibrium variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only monotonicity. Some special cases are also considered. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.

The existence and the stability of solutions for equilibrium problems with lower and upper bounds

In this paper, we study a class of equilibrium problems with lower and upper bounds. We obtain some existence results of solutions for equilibrium problems with lower and upper bounds by employing some classical fixed-point theorems. We investigate the stability of the solution sets for the problems, and establish sufficient conditions for the upper semicontinuity, lower semicontinuity and continuity of the solution set mapping $S:\Lambda_1\times\Lambda_2\to2^{X}$ in a Hausdorff topological vector space, in the case where a set $K$ and a mapping $f$ are perturbed respectively by parameters $\lambda$ and $\mu.$

Mixed equilibrium variational inequalities

In this paper, we introduce and consider a new class of equilibrium problems and variational inequalities, which is called the mixed equilibrium-variational inequality. We use the auxiliary principle technique to suggest and analyze some explicit and proximal-point iterative methods for solving the mixed equilibrium-variational inequalities. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper can be viewed as refinement and improvement of the previously known and new results. The ideas and techniques of this paper may stimulate further research in this dynamic area of pure and applied sciences. Key words: Mixed variational inequality, equilibrium problems, convergence, auxiliary principle technique.

Some iterative algorithms for trifunction equilibrium variational inequalities

In this paper, we use the auxiliary principle technique to suggest and analyze some iterative methods for solving a new class of equilibrium problems and variational inequalities, which is called the trifunction equilibrium variational inequality. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper continue to hold for these known and new classes of equilibrium and variational inequalities. Key words: Mixed variational inequality, equilibrium problems, convergence, auxiliary principle technique.