Generalized Equilibrium Problems Research Articles

Hybrid method for equilibrium problems and variational inclusions

By providing a new iterative method our aim is finding a common element of the set of fixed points of two nonexpansive mappings, the set of solutions to a variational inclusion and the set of solutions of a generalized equilibrium problem in a real Hilbert space. We review the strong convergence of the new iterative method in the framework of Hilbert spaces. Finally, we show that our main result is a generalization for some known theorems in this field.

An extragradient algorithm for split generalized equilibrium problem and the set of fixed points of quasi-φ-nonexpansive mappings in Banach spaces

In this paper, we study the problem of finding a common solution to split generalized mixed equilibrium problem and fixed point problem for quasi-$% \phi $-nonexpansive mappings in 2-uniformly convex and uniformly smooth Banach space $E_1$ and a smooth, strictly convex, and reflexive Banach space $% E_2$. An iterative algorithm with Armijo linesearch rule for solving the problem is presented and its strong convergence theorem is established. The convergence result is obtained without using the hybrid method which is mostly used when strong convergence is desired. Finally, numerical experiments are presented to demonstrate the practicability, efficiency, and performance of our algorithm in comparison with other existing algorithms in the literature. Our results extend and improve many recent results in this direction.

On the resolution of misspecified convex optimization and monotone variational inequality problems

We consider a misspecified optimization problem, requiring the minimization of a function $$f(\cdot;\theta ^*)$$ over a closed and convex set X where $$\theta ^*$$ is an unknown vector of parameters that may be learnt by a parallel learning process. Here, we develop coupled schemes that generate iterates $$(x_k,\theta _k)$$ as $$k \rightarrow \infty$$ , then $$x_k \rightarrow x^*$$ , a minimizer of $$f(\cdot;\theta ^*)$$ over X and $$\theta _k \rightarrow \theta ^*$$ . In the first part of the paper, we consider the solution of problems where f is either smooth or nonsmooth. In smooth strongly convex regimes, we demonstrate that such schemes still display a linear rate of convergence, albeit with larger constants. When strong convexity assumptions are weakened, it can be shown that the convergence in function value sees a modification in the canonical convergence rate of $${{{\mathcal {O}}}}(1/K)$$ by an additive factor proportional to $$\Vert \theta _0-\theta ^*\Vert$$ where $$\Vert \theta _0-\theta ^*\Vert$$ represents the initial misspecification in $$\theta ^*$$ . In addition, when the learning problem is assumed to be merely convex but admits a suitable weak-sharpness property, then the convergence rate deteriorates to $${\mathcal {O}}(1/\sqrt{K})$$ . In both convex and strongly convex regimes, diminishing steplength schemes are also provided and are less reliant on the knowledge of problem parameters. Finally, we present an averaging-based subgradient scheme that displays a rate of $${\mathcal {O}}(1/\sqrt{K})+ \mathcal{O}(\|\theta_0-\theta^*\|(1/K))$$ , implying no effect on the canonical rate of $${{{\mathcal {O}}}}(1/\sqrt{K})$$ . In the second part of the paper, we consider the solution of misspecified monotone variational inequality problems, motivated by the need to contend with more general equilibrium problems as well as the possibility of misspecification in the constraints. In this context, we first present a constant steplength misspecified extragradient scheme and prove its asymptotic convergence. This scheme is reliant on problem parameters (such as Lipschitz constants) and leads to a misspecified variant of iterative Tikhonov regularization, an avenue that does not necessitate the knowledge of such constants.

Convergence analysis for combination of equilibrium problems and k-nonspreading set-valued mappings

We find a common solution of generalized equilibrium problems and the set of fixed points of a k-nonspreading setvalued mapping by using shrinking projection hybrid method. Finally, we compare the shrinking solution set after randomization by giving numerical example which justifies our main result.

New two types of semi-implicit viscosity iterations for approximating the fixed points of nonexpansive operators associated with contraction operators and applications

Motivated and inspired by the growing contribution with respect to iterative approximations from some researchers in the literature, we design and investigate two types of brand-new semi-implicit viscosity iterative approximation methods for finding the fixed points of nonexpansive operators associated with contraction operators in complete {operatorname{CAT}(0)} spaces and for solving related variational inequality problems. Under some suitable assumptions, strong convergence theorems of the sequences generated by the approximation iterative methods are devised, and a numerical example and some applications to related variational inequality problems are included to verify the effectiveness and practical utility of the convergence theorems. Our main results presented in this paper do not only improve, extend and refine some corresponding consequences in the literature, but also show that the additional variational inequalities, general variational inequality systems and equilibrium problems can be solved via approximation of the iterative sequences. Finally, we provide an open question for future research.

Stochastic variational formulation for a general random time-dependent economic equilibrium problem

In the paper, in a Hilbert space setting, a random time-dependent oligopolistic market equilibrium problem in presence of both production and demand excesses is studied and the random time-dependent Cournot–Nash equilibrium principle by means of a stochastic variational inequality is characterized. Then, some existence results to such problem are established and the stochastic continuity of the equilibrium solution is proved. Moreover a simple numerical example illustrates the theoretical results.

Strong convergence of an inertial iterative algorithm for variational inequality problem, generalized equilibrium problem, and fixed point problem in a Banach space

We propose and analyze an inertial iterative algorithm to approximate a common solution of generalized equilibrium problem, variational inequality problem, and fixed point problem in the framework of a 2-uniformly convex and uniformly smooth real Banach space. Further, we study the convergence analysis of our proposed iterative method. Finally, we give application and a numerical example to illustrate the applicability of the main algorithm.

Iterative method for split generalized equilibrium and fixed point problems for multivalued demicontractive-type

In [4], [Fixed Point Theory and Applications (2018) 2018:6], W. Phuengrattana and K. Lerkchaiyaphum obtained some results on solving split generalized equilibrium problem and fixed point of multi-valued nonexpansive mappings in real Hilbert spaces. We observed a gap in the proof of their main result, Theorem 3.1. Motivated by their result, we first correct the observed error and study in this article, approximation of solution of generalized split equilibrium problem and common fixed point problem for a finite family of multi-valued demicontractive-type mappings in real Hilbert spaces.

A Viscosity Iterative Algorithm Technique for Solving a General Equilibrium Problem System

In the recent decade, a considerable number of Equilibrium problems havebeen solved successfully based on the iteration methods. In this paper, we suggest a viscosity iterative algorithm for nonexpansive semigroup in the framework of Hilbert space. We prove that, the sequence generated by this algorithm under the certain conditions imposed on parameters strongly convergence to a common solution of general equilibrium problem system. Results presented in this paper extend and unify the previously known results announced by many other authors. Further, we give some numerical examples to justify our main results.

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.

Approximation of common solutions for a fixed point problem of asymptotically nonexpansive mapping and a generalized equilibrium problem in Hilbert space

In this paper, we introduce an iterative algorithm to approximate a common solution of a generalized equilibrium problem and a fixed point problem for an asymptotically nonexpansive mapping in a real Hilbert space. We prove that the sequences generated by the iterative algorithm converge strongly to a common solution of the generalized equilibrium problem and the fixed point problem for an asymptotically nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area. Some applications of main results are also provided.

Viscosity method for generalized equilibrium problems with perturbation and nonexpansive mappings in Hilbert spaces

In this paper, we introduce a new iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem with perturbation and the fixed points for a new kind of combination of two nonexpansive mapping in Hilber space, we prove that the sequence converges strongly to a common element of two sets. Our results generalize and improve some results in the litretures.

Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward-backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

A new mapping for finding a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem

In this paper, we introduce and study a general iterative algorithm to approximate a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. Further, we prove a strong convergence theorem for the sequences generated by the proposed iterative scheme. Finally, we derive some consequences from our main result. The results presented in this paper extended and unify many of the previously known results in this area.

Parallel hybrid viscosity method for fixed point problems, variational inequality problems and split generalized equilibrium problems

In this paper, we first propose a new parallel hybrid viscosity iterative method for finding a common element of three solution sets: (i) finite split generalized equilibrium problems; (ii) finite variational inequality problems; and (iii) fixed point problem of a finite collection of demicontractive operators. And we prove that the sequence generated by the iterative scheme strongly converges to a common solution of the above-mentioned problems. Also, we present numerical examples to demonstrate the effectiveness of our algorithm. Our results presented in this paper improve and extend many recent results in the literature.

On ill-posedness and stability of tensor variational inequalities: application to an economic equilibrium

The general tensor variational inequalities, recently introduced in Barbagallo et al. (J Nonconvex Anal 19:711–729, 2018), are very useful in order to analyze economic equilibrium models. For this reason, the study of existence and regularity results for such inequalities has an important rule to the light of applications. To this aim, we start to consider some existence and uniqueness theorems for tensor variational inequalities. Then, we investigate on the approximation of solutions to tensor variational inequalities by using suitable perturbed tensor variational inequalities. We establish the convergence of solutions to the regularized tensor variational inequalities to a solution of the original tensor variational inequality making use of the set convergence in Kuratowski’s sense. After that, we focus our attention on some stability results. At last, we apply the theoretical results to examine a general oligopolistic market equilibrium problem.

Constrained stability of conservative static equilibrium

Modelling of static structural stability problems is considered. Focus is set on problems where passive physical constraints affect the response to applied forces, and where more than one free parameter describes the setting. The existence of vibration frequencies at equilibrium states is investigated, as an indication of stability. The relevant Jacobian matrix is developed, with an emphasis on the necessity to formulate the constraint equations from an energy form in a conservative problem. The corresponding mass matrix is introduced, with zero mass contribution from constraint equations. Three different forms of the relevant Jacobians are considered, and alternative methods for the eigenvalue extraction given. Stability is discussed in a context of generalized equilibrium problems, where auxiliary parameters and equations can be included in a continuation setting. Examples show the formulation, implementation and interpretation of stability.

Benchmarking of State of Formal Theoretical Modeling in Realm of Finance

It is a stylized fact of formal mathematical modeling that a 'space' cannot be described by attempts at parameterization of distances or areas (under or above curves) that feasibly could occur within the space. Tools for parameterization of mathematical spaces, such as Set Theory, Connectedness, or 'Reducio Ad Absurdum' are tools of 'pure mathematics' (quali tools). By the same token, research objectives that attempt to model changes within a system require parameterization of either of distances, areas, or points for arrival at formal mathematical solutions. Tools for parameterization of distances, areas, or points, such as ODEs, PDEs, Utility Functions, or Integrands are tools of 'quantitative mathematics' (quant tools). Let 'problems of Finance' refer to 'finance research questions' that require workouts of formal theoretical models for arrival at resolution. This literature review achieves three main objectives. In no particular order, first, it demonstrates that application of quali tools to resolving of problems of Finance that can be reformulated for application of quant tools produce more rigorous model outcomes. Second, whereas quali tools are formally shown to produce workouts of problems of Finance that are more robust, there exists a severe drought of sole applications of such tools to resolving of either of General Equilibrium or game theoretic problems. Third, formal theoretical predictions and supporting evidence show models that attempt to combine quant tools and quali tools for resolving of problems of Finance inherently are non-robust. This prediction is a general equilibrium outcome, that is, is normative, is not restricted to problems of Finance. Using in addition, reviews of 12 studies that are published in top tier Finance journals, non-robustness of mixtures of quant and quali tools for resolving of problems of Finance is unequivocally validated. In presence of validation of the general equilibrium prediction, a precondition for robustness of workouts of formal theoretical models is a pre-determination as to whether problems of Finance are best worked out with sole application of quant tools, or sole application of quali tools. Using this criterion, and `inferential' transformations of this criterion, this review of the literature reveals paucity of robust workouts of problems of Finance in top tier Finance journals.

Equilibrium formulations of relative optimization problems

We consider relative or subjective optimization problems where the goal function and feasible set are dependent of the current state of the system under consideration. We propose equilibrium formulations of the corresponding problems that lead to general (quasi-)equilibrium problems. We propose to apply a regularized version of the penalty method for the general quasi-equilibrium problem, which enables us to establish existence results under weak coercivity conditions and replace the quasi-equilibrium problem with a sequence of the usual equilibrium problems. We describe several examples of applications and show that the subjective approach can be extended to non-cooperative game problems.

OPERATOR SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEMS IN HAUSDORFF TOPOLOGICAL VECTOR SPACES

In this paper, we consider the generalized equilibrium problems with operator solutions and derive some existence results for generalized operator equilibrium problems in Hausdorff topological vector spaces by employing KKM theorem. The results presented of this paper generalize and unify the corresponding results of the several authors.