Solutions Of Vector Equilibrium Problems Research Articles

Optimality and scalarization of approximate solutions for vector equilibrium problems via Michel-Penot subdifferential

Optimality and scalarization of approximate solutions for vector equilibrium problems via Michel-Penot subdifferential

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.

Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

Higher-order efficiency conditions for constrained vector equilibrium problems

ABSTRACT In this paper, we investigate some optimality conditions of higher-order for nonsmooth nonconvex vector equilibrium problems with constraints (or without constraints) in terms of the higher-order upper and lower Studniarski derivatives in Banach spaces. The calculus rule of all the data involved in the problem is taken into account. Using the notion of higher-order upper Studniarski derivative with the class of m-stable/m-steady functions, we first provide the higher-order necessary optimality conditions for the local weak efficient solution of vector equilibrium problem without constraints and then we present the higher-order necessary and sufficient optimality conditions for the local strict minimum of order m to such problem. We second obtain the necessary and sufficient optimality conditions for those efficient solutions of vector equilibrium problem with cone and equality constraints through Lagrange multiplier rules in finite-dimensional spaces. Final, the higher-order necessary and sufficient optimality conditions in terms of the higher-order upper and lower Studniarski derivatives for the local weak efficient solution of vector equilibrium problem with set, cone and equality constraints in Banach spaces are also established. Some examples are proposed to demonstrate our findings.

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.

Vectorial Ekeland variational principle for cyclically antimonotone vector equilibrium problems

ABSTRACTIn this paper, we first obtain some results for the vectorial Ekeland variational principle with a W-distance, in which the final space is a real linear space not necessarily endowed with a topology. We introduce a cyclically antimonotone bifunction taking values in a real linear space, show its properties, and then apply the results to deduce a general vector equilibrium version of Ekeland variational principle. Moreover, we apply the properties of the cyclically antimonotone bifunction to study the existence of solutions for vector equilibrium problems defined on compact or noncompact sets. We also prove the existence of solutions for uncountable systems of vector equilibrium problems. Finally, our main results are applied to establish two new general versions of Caristi-Kirk fixed point theorem for set-valued mappings and some new existence theorems for vector variational inequalities under mild conditions.

Optimally Local Dense Conditions for the Existence of Solutions for Vector Equilibrium Problems

In this paper, by using C-sequentially sign property for bifunctions, we provide sufficient conditions that ensure the existence of solutions of some vector equilibrium problems in Hausdorff topological vector spaces which ordered by a cone. The conditions which we consider are not imposed on the whole domain of the operators involved, but just on a locally segment-dense subset of the domain.

Necessary and sufficient optimality conditions for constrained vector equilibrium problems using contingent hypoderivatives

In this paper, we study the Fritz John necessary and sufficient optimality conditions for weak efficient solutions of vector equilibrium problem with constraints via contingent hypoderivatives in finite-dimensional spaces. Using the stability of objective functions at a given optimal point and assumming, in addition, that the regularity condition (RC) holds, some primal and dual necessary optimality conditions for weak efficient solutions are derived. Furthermore, a dual necessary optimality condition is also established for the case of Frechet differentiable functions. Making use of the concept of a support function on the feasible set of vector equilibrium problems with constraints, some primal and dual sufficient optimality conditions are given for the class of stable functions and Frechet differentiable functions at a given feasible point. As an application, several necessary and sufficient optimality conditions for weak efficient solution are also obtained with the class of Hadamard differentiable functions. Examples to illustrate our results are provided as well.

New Second-Order Optimality Conditions for Vector Equilibrium Problems with Constraints in Terms of Contingent Derivatives

In this paper, we establish the primal and dual second-order necessary and sufficient optimality conditions for (local) weakly efficient solutions of vector equilibrium problems with constraints in terms of contingent derivatives with 2-steady functions in real finite spaces. Using the 2-steadiness of objective and constraint functions at a given optimal point, the second-order necessary conditions for weakly efficient solutions of constrained vector equilibrium problems are derived. Under the suitable assumptions on 2-steadiness of objective and constraint functions, the dual second-order necessary optimality conditions will become the dual second-order sufficient optimality conditions. We also give several examples that illustrate our results.

Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems

In this paper, we provide variants of the Ekeland variational principle for a type of approximate proper solutions of a vector equilibrium problem, whose final space is finite dimensional and partially ordered by a polyhedral cone. Depending on the choice of an approximation set that defines these solutions, we prove that they approximate suitably exact weak efficient/proper efficient/efficient solutions of the problem. The variants of the Ekeland variational principle are obtained for an unconstrained and also for a cone-constrained vector equilibrium problem, through a nonlinear scalarization, and expressed by means of the matrix that defines the ordering cone, which makes them easier to handle. At the end, the results are applied to multiobjective optimization problems, for which a related vector variational inequality problem is defined.

Vectorial form of Ekeland variational principle with applications to vector equilibrium problems

ABSTRACTWe first derive an Ekeland type variational principle for vector-valued bifunctions without assuming triangular inequality for bifunctions. Then, by using such variational principle, we derive some existence results for solutions of vector equilibrium problems and vector quasi-equilibrium problems.

Optimality Conditions for the Efficient Solutions of Vector Equilibrium Problems with Constraints in Terms of Directional Derivatives and Applications

The aim of this paper is to establish the Kuhn–Tucker optimality conditions for the efficient solution types of constrained vector equilibrium problems in terms of directional derivatives in Banach spaces. Under the suitable assumptions on generalized convexity of objective and constraint functions, the Kuhn–Tucker necessary and sufficient optimality conditions for efficient solution, weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem with set and cone constraints are established. Some applications to the constrained vector variational inequality problem and the constrained vector optimization problem are also given. Besides, the Karush–Kuhn–Tucker necessary and sufficient optimality conditions for weakly efficient solutions to the model of transportation–production and Nash–Cournot equilibria problems are obtained. We also provide several examples to illustrate our results.

On Optimality Conditions for Henig Efficiency and Superefficiency in Vector Equilibrium Problems

Necessary optimality conditions for local Henig efficient and superefficient solutions of vector equilibrium problems involving equality, inequality, and set constraints in Banach space with locally Lipschitz functions are established under a suitable constraint qualification via the Michel–Penot subdifferentials. With assumptions on generalized convexity, necessary conditions for Henig efficiency and superefficiency become sufficient ones. Some applications to vector variational inequalities and vector optimization problems are given as well.

Contingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraints

We establish Fritz John necessary conditions for local weak efficient solutions of vector equilibrium problems with constraints in terms of contingent derivatives. Under suitable constraint qualifications, Karush–Kuhn–Tucker necessary conditions for those solutions are investigated.

A note on existence of weak efficient solutions for vector equilibrium problems

This work concerns with vector equilibrium problems where the image space of the bifunction is not endowed with any topology. To be precise, a kind of “semi-algebraic” upper semicontinuity notion is introduced and, by means of a recent algebraic version of the so-called Gerstewitz’s functional, a new existence result of weak efficient solutions is obtained that significantly improves some previous ones stated in the topological setting since it requires weaker assumptions.

Optimality conditions for Henig efficient and superefficient solutions of vector equilibrium problems

Optimality conditions for Henig efficient and superefficient solutions of vector equilibrium problems

New optimality conditions for unconstrained vector equilibrium problem in terms of contingent derivatives in Banach spaces

This article presents necessary and sufficient optimality conditions for weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem without constraints in terms of contingent derivatives in Banach spaces with stable functions. Using the steadiness and stability on a neighborhood of optimal point, necessary optimality conditions for efficient solutions are derived. Under suitable assumptions on generalized convexity, sufficient optimality conditions are established. Without assumptions on generalized convexity, a necessary and sufficient optimality condition for efficient solutions of unconstrained vector equilibrium problem is also given. Many examples to illustrate for the obtained results in the paper are derived as well.

Existence results for globally efficient solutions of vector equilibrium problems via a generalized kkm principle

AbstractThe aim of this article is to present new existence results for globally efficient solutions of a strong vector equilibrium problem given by a sum of two functions via a generalized KKM principle, and to establish the connectedness of the solutions set.

Existence of Solutions and Algorithms for Bilevel Vector Equilibrium Problems: An Auxiliary Principle Technique

The main goal of this paper is to introduce and study bilevel vector equilibrium problems. We first establish some existence results for solutions of vector equilibrium problems and mixed vector equilibrium problems. We study the existence of solutions of bilevel vector equilibrium problems by considering a vector Thikhonov-type regularization procedure. By using this regularization procedure and existence results for mixed vector equilibrium problems, we establish some existence results for solutions of bilevel vector equilibrium problems. By using the auxiliary principle, we propose an algorithm for finding the approximate solutions of bilevel vector equilibrium problems. The strong convergence of the proposed algorithm is also studied.

Ekeland Variational Principles in Vector Equilibrium Problems

In this paper, we focus on vector equilibrium problems whose final space is a real linear space not necessarily endowed with a topology. In this framework, some exact and approximate Ekeland variational principles are obtained. The main mathematical tool is a kind of strict fixed point theorem for set-valued mappings, from which the Ekeland variational principles are derived by means of algebraic notions and a concept of approximate solution for vector equilibrium problems based on free-disposal sets. The obtained results improve other recent ones of the literature, and clarify the role of some assumptions usually required on the vector bifunction to state Ekeland variational principles in vector equilibrium problems.