Quasi-equilibrium Problems Research Articles

Self-Adaptive Extragradient Algorithms for Quasi-Equilibrium Problems

Self-Adaptive Extragradient Algorithms for Quasi-Equilibrium Problems

Finite Convergence and Sharp Minima for Quasi-Equilibrium Problems

Finite Convergence and Sharp Minima for Quasi-Equilibrium Problems

A system of Generalized Operator Quasi-Equilibrium Problems and their solutions

A system of Generalized Operator Quasi-Equilibrium Problems and their solutions

Generalized Vector Quasi-Equilibrium Problems

The aim of this paper is to present new existence results for solutions to a generalized quasi-equilibrium problem with set-valued mappings and moving cones. The key to this approach is a new Browder-type fixed point theorem, which permits working in a new direction with the milder condition of transfer open-valued mapping and considering weaker assumptions on the coving cone. These results are applied to some generalized vector quasi-equilibrium problems with trifunctions and to a vector quasi-equilibrium problem with fuzzy mappings in a fuzzy environment.

Penalty method for set-valued quasi-equilibrium problems

Penalty method for set-valued quasi-equilibrium problems

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.

Equilibrium problems when the equilibrium condition is missing

Given a nonempty convex subset X of a topological vector space and a real bifunction f defined on X times X, the associated equilibrium problem consists in finding a point x_0 in X such that f(x_0, y) ge 0, for all y in X. A standard condition in equilibrium problems is that the values of f to be nonnegative on the diagonal of X times X. In this paper, we deal with equilibrium problems in which this condition is missing. For this purpose, we will need to consider, besides the function f, another one g: X times X rightarrow mathbb {R}, the two bifunctions being linked by a certain compatibility condition. Applications to variational inequality problems, quasiequilibrium problems and vector equilibrium problems are given.

Projected solutions for finite-dimensional quasiequilibrium problems

The concept of projected solution has been introduced in Aussel et al. (J Optim Theory Appl 170:818–837, 2016) for studying quasivariational problems where the constraint map may not be a self-map. Aim of this paper is to establish a new result on the existence of projected solutions for finite-dimensional quasiequilibrium problems without any monotonicity assumptions and without assuming the compactness of the feasible set. These two facts allow us to improve some recent results. Additionally, we deduce the existence of projected solutions for quasivariational inequalities, quasioptimization problems and generalized Nash equilibrium problems. Also, a comparison with similar results is provided.

New criteria for existence of solutions for equilibrium problems

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems, and noncooperative games as particular cases. In this paper sufficient conditions for the existence of solutions of an equilibrium problem are given by weakening the assumption of quasiconvexity of the involved equilibrium bifunction. The existence of solutions is established both in presence of compactness of the feasible set as well with a coercivity assumption. The results are obtained in an infinite dimensional setting, and they are based on the so called finite solvability property which is weaker than the recently introduced finite intersection property and in turn, weaker than most common cyclic and proper quasimonotonicity. Some examples are presented to illustrate the various cases in which other existence results for equilibrium problems do not apply. Finally, applications to the solution of quasiequilibrium problems, quasioptimization problems and generalized quasivariational inequalities are discussed.

On a threshold descent method for quasi-equilibria

We describe a special class of quasi-equilibrium problems in metric spaces and propose a novel simple threshold descent method for solving these problems. Due to the framework, the convergence of the method cannot be established with the usual convexity or generalized convexity assumptions. Under mild conditions, the iterative procedure gives solutions of the quasi-equilibrium problem. We apply this method to scalar and vector generalized quasi-equilibrium problems and to some classes of relative optimization problems.

Existence of solutions of a generalized bifunction-set optimization problem

<p style='text-indent:20px;'>An existence result is established for a generalized bifunction-set optimization problem, under some verifiable assumptions. Its proof is based on a Browder fixed point theorem and the KKMFan-Lemma. As applications of the main result of this paper, we obtain new existence results for some generalized versions of Kuroiwa set optimization problems and strong vector quasiequilibrium problems. Examples are provided. One of them is a non-cooperative generalized game with utility multifunctions.</p>

Existence Results for Generalized Vector Quasi-Equilibrium Problems in Hadamard Manifolds

The purpose of this article was to establish manifold versions of existence theorems for generalized vector quasi-equilibrium problems in locally compact and σ-compact spaces without any continuity assumption. The fixed-point theorem in a product Hadamard manifold is the key focus of our discussion. We further applied our theorems to saddle point and minimax problems.

Extragradient method with Bregman distances for solving vector quasi-equilibrium problems

Extragradient method with Bregman distances for solving vector quasi-equilibrium problems

Graphical Ekeland's variational principle with a generalized $w$-distance and a new approach to quasi-equilibrium problems

In this paper, we introduce the generalized Ekeland's variational principle in several forms. The general setting of our results includes a graphical metric structure and also employs a generalized $w$-distance. We then applied the proposed variational principles to obtain existence theorems for a class of quasi-equilibrium problems whose constraint maps are induced from the graphical structure. The conditions used in our existence results are based on a very general concept called a convergence class. Finally, we deduce the existence of a generalized Nash equilibrium via its quasi-equilibrium reformulation. A validating example is also presented.

Technical note on the existence of solutions for generalized symmetric set-valued quasi-equilibrium problems utilizing improvement set

In this paper, we establish some existence results for the solution of the generalized symmetric set-valued quasi-equilibrium problem (GSSQEP). The new forms of GSSQEP via improvement set and scalar generalized symmetric set-valued quasi-equilibrium problem (GSSQEP for short) are also introduced. By using Kakutani–Fan–Glicksberg fixed point method, maximal element principle and nonlinear scalarization technique, we develop two classes of sufficient conditions for the existence of solutions to GSSQEP. The drawback of some existing work for this problem is overcome. Moreover, some applications to the symmetric set-valued equilibrium problem (SSEP), symmetric vector quasi-equilibrium problem (SVQEP) and the set-valued equilibrium problem (SEP) are also given in this paper. Our results improve a few existing ones in Fakhar and Zafarani [Generalized symmetric vector quasiequilibrium problems. J Optim Theory Appl. 2008;136:397–409], Farajzadeh et al. [On the existence of solutions of symmetric vector equilibrium problems via nonlinear scalarization. Bull Iran Math Soc. 2019;45:35–58], Fu [Symmetric vector quasi-equilibrium problems. J Math Anal Appl. 2003;285:708–713], Han et al. [Existence of solutions for symmetric vector set-valued quasi-equilibrium problems with applications. Pacific J Optim. 2018;14:31–49].

Solving Equilibrium Problems with Bifunctions Having the Property of Negative Transitivity

In this article, we introduce the notion of bifunction with the property of negative transitivity and, using the Berge-Klee intersection theorem, we establish existence theorems of the solutions for the equilibrium problems, when the involved bifunction has this property. Then, using standard scalarization techniques, we obtain existence criteria of the solutions for vector equilibrium problems. The last section of the article is devoted to the scalar and vector quasi-equilibrium problems.

Solutions of system of generalized vector quasi-equilibrium problems in locally FC-uniform spaces

In this paper, we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally FC-uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally FC-uniform spaces. These theorems are new and generalize some known results in the literature.

Ekeland variational principle in complete weakly symmetric (1, q 2)-quasimetric spaces and applications

In this paper, Ekeland variational principle for bifunctions is given in complete weakly symmetric -quasimetric spaces and it is shown to be equivalent to Oettli–Théra theorem, Caristi–Kirk fixed point theorem and Takahashi's nonconvex minimization principle. Moreover, Ekeland variational principle for the system of bifunctions is obtained. Finally, by applying obtained results, the existence of solutions of quasi-equilibrium problems (QEP) and uncountable system of QEP is provided under the compactness condition and the generalized Caristi-like condition, respectively.

Path connectedness of the efficient solution set for generalized vector quasi-equilibrium problems

Path connectedness of the efficient solution set for generalized vector quasi-equilibrium problems

Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization

In this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.