Quasi-equilibrium Problems Research Articles

Well-posedness for general parametric quasi-variational inclusion problems

In this paper, we aim to suggest the new concept of well-posedness for the general parametric quasi-variational inclusion problems (QVIP). The corresponding concepts of well-posedness in the generalized sense are also introduced and investigated for QVIP. Some metric characterizations of well-posedness for QVIP are given. We prove that under suitable conditions, the well-posedness is equivalent to the existence of uniqueness of solutions. As applications, we obtain immediately some results of well-posedness for the parametric quasi-variational inclusion problems, parametric vector quasi-equilibrium problems and parametric quasi-equilibrium problems.

Existence and Connectedness of Solutions for Generalized Vector Quasi-Equilibrium Problems

In this paper, we establish three existence theorems for strongly efficient solutions, weakly efficient solutions and efficient solutions of generalized vector quasi-equilibrium problems by using the Fan-KKM theorem and the scalarization method. Moreover, we investigate the connectedness of the sets of weakly efficient solutions and efficient solutions for generalized vector quasi-equilibrium problems and make a new attempt to establish the path connectedness of the set of weakly efficient solutions for generalized vector quasi-equilibrium problems.

Gap functions for quasi-equilibria

An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimate of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm.

Existence of equilibrium for generalized games in choice form and applications

This paper has two central aims: first, to provide simple conditions under which the generalized games in choice form and, consequently, the abstract economies admit equilibrium; second, to study the solvability of several types of systems of vector quasi-equilibrium problems as an application. Our work outlines that there still is much to be gained from using the results concerning the existence of equilibrium of games as tools of research for other optimization problems.

Existence of solutions for generalized symmetric vector quasi-equilibrium problems in abstract convex spaces with applications

Existence of solutions for generalized symmetric vector quasi-equilibrium problems in abstract convex spaces with applications

Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems

Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems

Ekeland’s principle for cyclically antimonotone equilibrium problems

In this paper, we deal with equilibrium problems without convexity assumptions either for the domain or for the function involved. First we give an Ekeland’s variational principle for equilibrium problems on complete metric spaces which generalizes some recent results. The main improvement consists in widening the class of bifunctions for which the variational principle holds: instead of a triangle inequality property a suitable approximation from below of the bifunction f is required and we show that this condition is equivalent to the cyclic antimonotonicity of f. We furnish a simple and direct proof of the existence of equilibria for such equilibrium problems in presence of a compact feasible set without passing through the existence of approximate solutions. This fact allows us to remove the metric structure on the topological space and additional technical conditions. Subsequently, a weak coercivity condition is introduced to deal with the non weakly compact case in reflexive Banach space. Finally, applications both to uncountable systems of equilibrium problems and to the quasiequilibrium problems are also discussed.

Vector Quasi-Equilibrium Problems for the Sum of Two Multivalued Mappings

In this paper, we study vector quasi-equilibrium problems for the sum of two multivalued bifunctions. The assumptions are required separately on each of these bifunctions. Sufficient conditions for the existence of solutions of such problems are shown in the setting of topological vector spaces. The results in this paper unify, improve and extend some well-known existence theorems from the literature.

Semicontinuity and closedness of parametric generalized lexicographic quasiequilibrium problems

This paper is mainly concerned with the upper semicontinuity, closedness, and the lower semicontinuity of the set-valued solution mapping for a parametric lexicographic equilibrium problem where both two constraint maps and the objective bifunction depend on both the decision variable and the parameters. The sufficient conditions for the upper semicontinuity, closedness, and the lower semicontinuity of the solution map are established. Many examples are provided to ensure the essentialness of the imposed assumptions.

Existence of solutions of generalized quasi-variational relation problems by the inductive relations and some applications

By the inductive relations, this paper first obtains some existence theorems of solutions for generalized quasi-variational relation problems, which are different from other papers. As applications, some existence theorems of solutions for generalized quasi-equilibrium problems and NS-equilibria for noncooperative games under uncertainty are obtained.

Existence Theorems for a Variational Relation Problem

ABSTRACTThe aim of this article is to establish new existence results for a certain type of variational relation problem studied by Agarwal et al. in a very recent article. This problem is a general model for several quasivariational inclusions and quasiequilibrium problems investigated in many recent articles. The main tools in proving the existence theorems are two well-known fixed point theorems (Eilenberg-Montgomery and Kakutany-Fan Glicksberg, respectively). In the last section, we prove that, applied in special cases, our results are different or stronger than several earlier results.

Solutions and Approximate Solutions of Quasi-Equilibrium Problems in Banach Spaces

This paper deals with quasi-equilibrium problems in the setting of real Banach spaces. By a fixed point theory approach, we obtain existence results under mild conditions of continuity, improving some previous results in this area. By a selection theory approach, we make use of the Michael selection theorem to overcome the separability of the Banach spaces and generalize some results obtained recently in the literature. Finally, we deal with the existence of approximate solutions for quasi-equilibrium problems, and by arguments combining selection theory and fixed point theory, we obtain some qualitative results for quasi-equilibrium problems involving sub-lower semicontinuous set-valued mappings.

Approximate solutions of quasiequilibrium problems in Banach spaces

In this note we show that a recent existence result on quasiequilibrium problems, which seems to improve deeply some well-known results, is not correct. We exhibit a counterexample and we furnish a generalization of a lemma about continuous $$\varepsilon $$?-minimizers of quasiconvex functions depending on a parameter. This allows to establish an existence result of approximate solutions of quasiequilibrium problems.

A method to bypass the lack of solutions in MinSup problems under quasi-equilibrium constraints

MinSup problems with constraints described by quasi-equilibrium problems are considered in Banach spaces. The solutions set of such problems may be empty even in very good situations, so the aim of this paper is twofold. First, we determine appropriate regularizations (called inner regularizations) which allow to reach the value of the original problem. Then, among these regularizations we identify those which allow to bypass the lack of exact solutions to these problems by a suitable concept of “viscosity” solution whose existence is then proved under reasonable assumptions.

Well-posed symmetric vector quasi-equilibrium problems

In this paper, we consider well-posedness of symmetric vector quasi-equilibrium problems. Based on a nonlinear scalarization technique, we first establish the bounded rationality model M for symmetric vector quasi-equilibrium problems, and then introduce a well-posedness concept for symmetric vector quasi-equilibrium problems, which unifies its Hadamard and Tykhonov well-posedness. Finally, sufficient conditions on the well-posedness for symmetric vector quasi-equilibrium problems are given.

Topologically-based characterizations of the existence of solutions of optimization-related problems

Necessary and sufficient conditions for the existence of solutions of optimization-related problems, defined on general sets, are established by using topologically-based structures, without the usual linear structure. First, we prove that the existence of solutions to a variational relation problem is equivalent to the existence of either a KKM-structure or a connectedness structure, satisfying additionally some verifiable conditions. Then, applying these results, we obtain such full (two-way) characterizations for the existence of invariant points, solutions of quasiequilibrium problems of the Stampacchia-Minty type, saddle points, and Nash equilibria for noncooperative games. The main advantages of our scheme with the mentioned two structures are that full characterizations for existence are obtained in a unified way, with no convexity assumptions, and also that one has flexibility to choose a suitable structure in investigations. (This flexibility is decisive when the natural underlying structure of the given problem is scarcely employed.)

On Generalized Quasi-Vector Equilibrium Problems via Scalarization Method

In this paper, we consider the nonlinear scalarization function in the setting of topological vector spaces and present some properties of it. Moreover, using the nonlinear scalarization function and Fan---Glicksberg---Kakutani's fixed point theorem, we obtain an existence result of a solution for a generalized vector quasi-equilibrium problem without using any monotonicity and upper semi-continuity (or continuity) on the given maps. Our result can be considered as an improvement of the known corresponding result. After that, we introduce a system of generalized vector quasi-equilibrium problem which contains Nash equilibrium and Debreu-type equilibrium problem as well as the system of vector equilibrium problem posed previously. We provide two existence theorems for a solution of a system of generalized vector quasi-equilibrium problem. In the first one, our multi-valued maps have closed graphs and the maps are continuous, while in the second one, we do not use any continuity on the maps. Moreover, the method used for the existence theorem of a solution of a system of generalized vector quasi-equilibrium problem is not based upon a maximal element theorem. Finally, as an application, we apply the main results to study a system of vector optimization problem and vector variational inequality problem.

Well-posedness for lexicographic vector quasiequilibrium problems with lexicographic equilibrium constraints

We consider the well-posedness for lexicographic vector equilibrium problems and optimization problems with lexicographic equilibrium constraints in metric spaces. Sufficient conditions for a family of such problems to be (uniquely) well-posed at the reference point are established. Numerous examples are provided to explain that all the assumptions we impose are very relaxed and cannot be dropped.

Applications of order-theoretic fixed point theorems to discontinuous quasi-equilibrium problems

AbstractIn this paper, we apply order-theoretic fixed point theorems and isotone selection theorems to study quasi-equilibrium problems. Some existence theorems of solutions to quasi-equilibrium problems are obtained on Hilbert lattices, chain-complete lattices and chain-complete posets, respectively. In contrast to many papers on equilibrium problems, our approach is order-theoretic and all results obtained in this paper do not involve any topological continuity with respect to the considered mappings.

Existence of solutions for generalized vector quasi-equilibrium problems in abstract convex spaces with applications

In this paper, several kinds of generalized vector quasi-equilibrium problems are introduced and studied in abstract convex spaces. Using the properties of Γ-convex and \(\mathfrak{KC}\)-maps, some sufficient conditions are given to guarantee the existence of solutions in connection with these generalized vector quasi-equilibrium problems. As applications, some existence theorems of solutions for the generalized semi-infinite programs with vector quasi-equilibrium constraints are also given.