Scalar Equilibrium Problem Research Articles

Reducing the complexity of equilibrium problems and applications to best approximation problems

"We consider the scalar equilibrium problems governed by a bifunction in a finite-dimensional framework and we characterize the solutions by means of extreme or exposed points."

Approximate Proper Solutions in Vector Equilibrium Problems

The paper concerns with properties of approximate Benson and Henig proper efficient solutions of vector equilibrium problems. Relationships between both kinds of solutions and the approximate weak efficient solutions of the problem are stated. Roughly speaking, it is proved that they coincide provided that the ordering cone of the problem can be separated from another closed cone by an approximating sequence of dilating cones. As a result, the limit behavior of approximate Benson and Henig proper efficient solutions when the error tends to zero is studied. It is shown that efficient and proper efficient solutions of the problem can be obtained as the limit of approximate Benson and Henig proper efficient solutions whenever a suitable domination set is considered. Finally, optimality conditions for approximate Benson and Henig proper efficient solutions are derived by approximate solutions of scalar equilibrium problems. They are formulated via linear scalarization and the necessary conditions hold true in problems satisfying a kind of nearly subconvexlikeness condition. The main results of the paper generalize some recent ones because they involve weaker assumptions.

Скалярная задача равновесия и предельное распределение нулей полиномов Эрмита-Паде II типа

С помощью скалярной задачи равновесия, поставленной на двулистной римановой поверхности, доказано существование предельного распределения нулей полиномов Эрмита-Паде II типа для пары функций, образующих систему Никишина. Обсуждается связь полученных результатов с результатами Г. Шталя 1988 г. Приводятся результаты численных экспериментов. Показано, что результаты, полученные в настоящей работе и в более ранних работах второго автора, вполне согласуются как с результатами Г. Шталя, так и с результатами численных экспериментов.

Scalar Equilibrium Problem and the Limit Distribution of Zeros of Hermite-Padé Polynomials of Type II

Using the scalar equilibrium problem posed on the two-sheeted Riemann surface, we prove the existence of a limit distribution of the zeros of Hermite-Pade polynomials of type II for a pair of functions forming a Nikishin system. We discuss the relation of the results obtained here to some results of H. Stahl (1988) and present results of numerical experiments. The results of the present paper and those obtained in earlier papers of the second author are shown to be in good accordance with both H. Stahl’s results and results of numerical experiments.

An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality

In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases.

Equivalence of a Scalar and a Vector Equilibrium Problem for a Pair of Functions Forming a Nikishin System

We prove the equivalence of a vector and a scalar equilibrium problem that naturally arise when studying the limit distribution of zeros of type I Hermite–Pade polynomials for a pair of functions forming a Nikishin system.

Extragradient Methods for Vector Equilibrium Problems in Banach Spaces

We consider the vector equilibrium problem, which extends the scalar equilibrium problem to vector valued bifunctions, in a Banach space setting. We propose an extragradient method for solving this problem. Under suitable assumptions on the bifunction, we prove that the generated sequence is weakly convergent to a solution of the problem. Then, we propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the problem.

On the Existence of Solutions of Symmetric Vector Equilibrium Problems via Nonlinear Scalarization

In this paper, by proposing a new type of generalized C-quasiconvexity for the set-valued mappings and using the nonlinear scalarization function $$\xi _q$$ and its properties, without assumption of monotonicity and boundedness, some existence results of the solutions for the symmetric vector equilibrium problems and symmetric scalar equilibrium problems are established. Moreover, the convexity of solution sets is also investigated. Finally, some examples to support our results are provided.

On a Sufficient Condition for Weak Sharp Efficiency in Multiobjective Optimization

In this paper, we provide sufficient conditions entailing the existence of weak sharp efficient points of a multiobjective optimization problem. The approach uses variational analysis techniques, like regularity and subregularity of the diagonal subdifferential map related to a suitable scalar equilibrium problem naturally associated to the multiobjective optimization problem.

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

A new approach to the problem of the zero distribution of type I Hermite—Pade polynomials for a pair of functions f1, f2 forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field which is posed on a two-sheeted Riemann surface.

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.

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.

Generalized scalar equilibrium problem with applications to best and coupled best approximations

In this paper, we present a new theorem on the existence of solutions for a three-function scalar equilibrium problem. Compactness is expressed in terms of the existence of feasibility sets of arbitrarily small measures of non-compactness, in addition to the completeness of the domain. Hereby we present some new results on the best approximations and coupled best approximations. We propose the generalized coupled coincidence problem and present a theorem on existence of its solution.

On Hölder Calmness and Hölder Well-Posedness of Vector Quasi-Equilibrium Problems

We study stability and well-posedness of parametric vector quasi-equilibrium problems. Weak continuity of orders not greater than one around a given point, in the sense of Holder calmness of such orders, of solution maps is under consideration. Namely, we consider stability in terms of Holder calmness of solution maps at the considered point of parameter. Sufficient conditions for such Holder calmness are established for weak and strong vector quasi-equilibrium problems. When applied to the particular case of scalar equilibrium problems, our results recover recent ones appearing online first in the literature. Then we propose a Holder well-posedness notion for parametric vector quasi-equilibrium problems, based on Holder calmness of approximate solution maps, and derive sufficient conditions for Holder well-posedness of both the mentioned weak and strong vector quasi-equilibrium problems.

LIPSCHITZ CONTINUITY OF AN APPROXIMATE SOLUTION MAPPING TO EQUILIBRIUM PROBLEMS

In this paper, with respect to Hausdorff metric, we establish the Lipschitz continuity of an approximate solution mapping for a parametric scalar equilibrium problem. Simultaneously, we give some applications to parametric optimization problems and parametric variational inequalities. Our main results are new and strengthen some results in the recent literature.

LIPSCHITZ CONTINUITY OF AN APPROXIMATE SOLUTION MAPPING TO EQUILIBRIUM PROBLEMS

In this paper, with respect to Hausdorff metric, we establish the Lipschitz continuity of an approximate solution mapping for a parametric scalar equilibrium problem. Simultaneously, we give some applications to parametric optimization problems and parametric variational inequalities. Our main results are new and strengthen some results in the recent literature.

Approximations of Equilibrium Problems

In this paper we study the scalar equilibrium problem (EP). We employ variational convergences of bifunctions (lopsided convergence in the maxinf framework, hypo-convergence, and continuous convergence) to study this problem by means of an approximation method. This method allows us to obtain not only existence but also stability results. We introduce a new notion of approximate solution of EPs and study its properties. Then, by coupling this notion of approximate solution and the above-mentioned variational convergences, we introduce new notions of well-posedness for EPs and characterize them. We identify various classes of problems that are well-posed. Finally, by employing the obtained results we prove convergence results of two numerical methods for pseudomonotone bifunctions.

Continuity of approximate solution mappings for parametric equilibrium problems

In this paper, we obtain sufficient conditions for Hausdorff continuity and Berge continuity of an approximate solution mapping for a parametric scalar equilibrium problem. By using a scalarization method, we also discuss the Berge lower semicontinuity and Berge continuity of a approximate solution mapping for a parametric vector equilibrium problem.

Vector equilibrium problems on unbounded sets

We consider a general vector equilibrium problem in a reflexive Banach space setting and propose a new coercivity condition for the case of unbounded sets. This condition enables us to obtain new existence results of solutions for vector equilibrium problems. We specialize these results for scalar equilibrium problems and vector variational inequalities.

Existence of solutions of Minty type scalar and vector variational inequalities

Scalar and vector variational inequalities involving bifunctions are considered in this article. Necessary and sufficient conditions for existence of solutions are proposed, based on the finite intersection property of a suitable set-valued function. In the case of a properly quasi-monotone bifunction the relation to the KKM property is established. The result of John (A note on Minty variational inequality and generalized monotonicity. Proc. Generalized Convexity and Monotonicity, 6 (2001), pp. 240–246) concerning the characterization of properly quasi-monotone functions in terms of existence of solutions of variational inequalities (scalar case) is extended to bifunctions both in the scalar and vector cases. Existence of solutions for scalar equilibrium problems with bifunctions are considered Bianchi and Pini (A note on equilibrium problems with properly quasimonotone bifunctions. J. Global Optim. 20 (2001), pp. 67–76) and their results admit a reformulation for variational inequalities as a special class of equilibrium problems. The present article also generalizes this result, even in the scalar case (here the bifunctions are not assumed quasi-convex). As for the vector case, it should be stressed that two types of variational inequalities are studied, and respectively, the quasi-monotonicity is understood in two different ways. Finally, as a particular case variational inequalities of differentiable type are discussed.