Quasi-equilibrium Problems Research Articles

An alternative extragradient projection method for quasi-equilibrium problems

For the quasi-equilibrium problem where the players’ costs and their strategies both depend on the rival’s decisions, an alternative extragradient projection method for solving it is designed. Different from the classical extragradient projection method whose generated sequence has the contraction property with respect to the solution set, the newly designed method possesses an expansion property with respect to a given initial point. The global convergence of the method is established under the assumptions of pseudomonotonicity of the equilibrium function and of continuity of the underlying multi-valued mapping. Furthermore, we show that the generated sequence converges to the nearest point in the solution set to the initial point. Numerical experiments show the efficiency of the method.

Levitin-Polyak well-posedness for parametric quasivariational inclusion and disclusion problems

In this paper, we aim to suggest the new concept of Levitin-Polyak (for short, LP) well-posedness for the parametric quasivariational inclusion and disclusion problems (for short, (QVIP) (resp. (QVDP))). Necessary and sufficient conditions for LP well-posedness of these problems are proved. As applications, we obtained immediately some results of LP well-posedness for the quasiequilibrium problems and for a scalar equilibrium problem.

On system of vector quasi-equilibrium problems for multivalued mappings

On system of vector quasi-equilibrium problems for multivalued mappings

A Stackelberg quasi-equilibrium problem via quasi-variational inequalities

In this paper, we consider a class of Stackelberg quasi-equilibrium problem with two players in finite dimensional spaces. Existence and location of the Stackelberg quasi-equilibrium is discussed by employing the quasivariational inequality techniques and the fixed point arguments. The results presented in this paper generalize some corresponding ones due to Nagy [Nagy, S., Stackelberg equilibia via variational inequalities and projections, J. Global Optim., 57 (2013), 821–828].

Painleve-Kuratowski convergences of the approximate solution sets for vector quasiequilibrium problems

In this paper, we study vector quasiequilibrium problems. After that, the Painlev´e-Kuratowski upper convergence, lower convergence and convergence of the approximate solution sets for these problems are investigated by using a sequence of mappings ΓC -converging. As applications, we also consider the Painlev´e-Kuratowski upper convergence of the approximate solution sets in the special cases of variational inequality problems of the Minty type and Stampacchia type. The results presented in this paper extend and improve some main results in the literature.

An extragradient-type method for solving nonmonotone quasi-equilibrium problems

In this paper, a quasi-equilibrium problem with a nonmonotone bifunction is considered in a finite-dimensional space. The primary difficulty with this problem is related to the fact that one must simultaneously solve a nonmonotone equilibrium problem and calculate a fixed point of a multivalued mapping. An extragradient-type method is presented and analysed for its solution. The convergence of the method is proved under the assumption that the solution set of an associated dual equilibrium problem is nonempty. Finally, some numerical experiments are reported.

A note on quasi-equilibrium problems

The purpose of this paper is to prove the existence of solutions of quasi-equilibrium problems without any generalized monotonicity assumption. Additionally, we give some applications.

Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems

In this paper, we consider vector quasi-equilibrium problems under perturbation in terms of suitable asymptotically solving sequences, not embedding given problems into a parameterized family. By employing some types of convergences for mapping and set sequences, we obtain the Painleve–Kuratowski upper convergence of solution sets for the reference problems. Then, using nonlinear scalarization functions, we propose gap functions for such problems, and later employing these functions, we study necessary and sufficient conditions for Painleve–Kuratowski lower convergence and Painleve–Kuratowski convergence. As an application, we discuss the special case of vector quasi-variational inequality.

Quasi-Equilibrium Problems and Fixed Point Theorems of Separately l.s.c and u.s.c Mappings

ABSTRACTIn this paper, we apply our new results on quasi-variational inequality problems to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of solutions are shown. In particular, we establish several results on the existence of solutions to fixed points of separately lower and upper semicontinuous (lower and lower, upper and upper) mappings.These generalize some well-known fixed point theorems obtained by previous authors as Browder and Ky Fan, Wu, et al., etc.

Refinements on Gap Functions and Optimality Conditions for Vector Quasi-Equilibrium Problems via Image Space Analysis

By means of some new results on generalized systems, vector quasi-equilibrium problems with a variable ordering relation are investigated from the image perspective. Lagrangian-type optimality conditions and gap functions are obtained under mild generalized convexity assumptions on the given problem. Applications to the analysis of error bounds for the solution set of a vector quasi-equilibrium problem are also provided. These results are refinements of several authors’ works in recent years and also extend some corresponding results in the literature.

Lower semicontinuity of solution mappings for parametric fixed point problems with applications

In this paper, we establish the lower semicontinuity of the solution mapping and the approximate solution mapping for parametric fixed point problems under some suitable conditions. As applications, the lower semicontinuity result applies to the parametric vector quasi-equilibrium problem, and allows to prove the existence of solutions for generalized Stackelberg games.

Existence Results and Optimal Control for a Class of Quasi Mixed Equilibrium Problems Involving the (f,\xa0g,\xa0h)-Quasimonotonicity

In this paper, by introducing a new concept of the (f, g, h)-quasimonotonicity and applying the maximal monotonicity of bifunctions and KKM technique, we show the existence results of solutions for quasi mixed equilibrium problems when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. Next, we also obtain a result of optimal control to a minimization problem. Our main results can be applied to the problems of evolution equations, differential inclusions and hemivariational inequalities.

Abstract generalized vector quasi-equilibrium problems in noncompact Hadamard manifolds

This paper deals with the abstract generalized vector quasi-equilibrium problem in noncompact Hadamard manifolds. We prove the existence of solutions to the abstract generalized vector quasi-equilibrium problem under suitable conditions and provide applications to an abstract vector quasi-equilibrium problem, a generalized scalar equilibrium problem, a scalar equilibrium problem, and a perturbed saddle point problem. Finally, as an application of the existence of solutions to the generalized scalar equilibrium problem, we obtain a weakly mixed variational inequality and two mixed variational inequalities. The results presented in this paper unify and generalize many known results in the literature.

Characterization of duality for a generalized quasi-equilibrium problem

In this paper, the duality theory of a generalized quasi-equilibrium problem (also called generalized Ky Fan quasi-inequality) is investigated by using the image space approach. Generalized quasi-equilibrium problem is transformed into a minimization problem. The minimization problem is further reformulated as an image problem by virtue of linear/nonlinear separation function. The dual problem of the image problem is constructed in the image space, then zero duality gap between the image problem and its dual problem is derived under saddle point condition as well as the equivalent regular linear/nonlinear separation condition. Finally, some more sufficient conditions guaranteeing zero duality gap are also proposed.

Quasi-equilibrium Problems and Fixed Point Theorems of the Product Mapping of Lower and Upper Semicontinuous Mappings

Quasi-equilibrium Problems and Fixed Point Theorems of the Product Mapping of Lower and Upper Semicontinuous Mappings

Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems

In this paper, we consider parametric strong vector quasiequilibrium problems in Hausdorff topological vector spaces. Firstly, we introduce parametric gap functions for these problems, and study the continuity property of these functions. Next, we present two key hypotheses related to the gap functions for the considered problems and also study characterizations of these hypotheses. Then, afterwards, we prove that these hypotheses are not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of solution mappings to these problems. Finally, as applications, we derive several results on Hausdorff (lower) continuity properties of the solution mappings in the special cases of variational inequalities of the Minty type and the Stampacchia type.

Existence and Hadamard well-posedness of a system of simultaneous generalized vector quasi-equilibrium problems

An existence result for the solution set of a system of simultaneous generalized vector quasi-equilibrium problems (for short, (SSGVQEP)) is obtained, which improves Theorem 3.1 of the work of Ansari et al. (J. Optim. Theory Appl. 127:27-44, 2005). Moreover, a definition of Hadamard-type well-posedness for (SSGVQEP) is introduced and sufficient conditions for Hadamard well-posedness of (SSGVQEP) are established.

New existence criteria of the solutions for a variational relation problem

Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are special cases of a variational relation problem proposed in a recent paper by Agarwal et al. (J. Optim. Theory Appl. 2012;155:417–429). The purpose of this paper is to establish new existence results for the solutions of this problem. The main ingredients in the proofs are some continuous selection and fixed point theorems, and an interesting section result. In the last section, we prove that, applied for some concrete relations, our results are different from those obtained by other authors.

Iterative Schemes for Nonconvex Quasi-Variational Problems with V-Prox-Regular Data in Banach Spaces

In this paper, we propose an extension of quasi-equilibrium problems from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. The proposed problem is called quasi-variational problem. We study the convergence of some algorithms to solutions of the proposed nonconvex problems in Banach spaces.

On well-posedness for parametric vector quasiequilibrium problems with moving cones

In this paper we consider weak and strong quasiequilibrium problems with moving cones in Hausdorff topological vector spaces. Sufficient conditions for well-posedness of these problems are established under relaxed continuity assumptions. All kinds of wellposedness are studied: (generalized) Hadamard well-posedness, (unique) well-posedness under perturbations. Many examples are provided to illustrate the essentialness of the imposed assumptions. As applications of the main results, sufficient conditions for lower and upper bounded equilibrium problems and elastic traffic network problems to be well-posed are derived.