Set-valued Equilibrium Problems Research Articles

Higher-Order Radial Hadamard Directional Derivatives and Applications to Set-Valued Equilibrium Problems

In this paper, we first introduce the higher-order lower and upper radial Hadamard directional derivatives for set-valued maps, which can capture the global information of concerning maps, and discuss the relationships with other existed higher-order derivatives. Second, based on the introduced derivatives, we establish optimality conditions for Henig efficient solutions of a set-valued equilibrium problem with constraints. Particularly, the optimality conditions hold in the case where the derivatives of objective and constraint functions are separated. Finally, we give some duality theorems for a mixed type of primal-dual set-valued equilibrium problem. The main results of this paper are illustrated by several concrete examples.

Approximate Benson properly efficient solutions for set-valued equilibrium problems

Approximate Benson properly efficient solutions for set-valued equilibrium problems

Semi-continuity of the solution maps of set-valued equilibrium problems with equilibrium constraints

In this paper, we study the parametric set-valued equilibrium problems with equilibrium constraints based on the comparison of objective values of set-valued maps by set order relations. We introduce new notions of generalized concavity for set-valued maps and study their properties as well as their relationship with other existing well-known notions. By using the generalized concavity and semi-continuity of set-valued maps, we study the existence of solutions for set-valued equilibrium problems when the equilibrium condition is missing. We further establish sufficient conditions for lower/upper semi-continuity of the solution maps of the set-valued equilibrium problems involving set order relations. Several examples are provided to illustrate the derived results.

Lower semicontinuity of the 解 maps of unified parametric generalized set-valued equilibrium problems with the improvement sets

Lower semicontinuity of the 解 maps of unified parametric generalized set-valued equilibrium problems with the improvement sets

Set-valued equilibrium problems based on the concept of null set with applications

In this paper, by means of the null set defined by Wu (J Math Anal Appl. 2019;472:1741–1761), we introduce a set-valued equilibrium problem based on the null set, where the objective mapping takes values in a hyperspace equipped with a convex cone. Moreover, we obtain new existence results for set-valued equilibrium problems defined on compact or noncompact sets. Some applications are given to set optimization problems, to a set-valued variational inequality, to saddle point theorems for set-valued mappings, and to a generalized noncooperative game.

Well-posedness for set-valued equilibrium problems

"In this paper we extend a concept of well-posedness for vector equilibrium problems to the more general framework of set-valued equilibrium problems in topological vector spaces using an appropriate reformulation of the concept of minimality for sets. Su cient conditions for well-posedness are given in the generalized convex settings and we are able to single out classes of well-posed set-valued equilibrium problems. On the other hand, in order to relax some conditions, we introduce a concept of minimizing sequences for a set-valued problem, in the set criterion sense, and further we will have a concept of well-posedness for the set-valued equilibrium problem we are interested in. Suficient results are also given for this well-posedness concept."

Scalarization and Optimality Conditions of E-Globally Proper Efficient Solution for Set-Valued Equilibrium Problems

In this paper, our purpose is to use the improvement set to investigate the scalarization and optimality conditions of E-globally proper efficient solution for the set-valued equilibrium problems with constraints. First, the notion of E-globally proper efficient solution for set-valued equilibrium problems with constraints is introduced in locally convex Hausdorff topological spaces. Second, the linear scalarization theorems of E-globally proper efficient solution are derived. Finally, under the assumption of nearly E-subconvexlikeness, the Kuhn–Tucker and Lagrange optimality conditions for set-valued equilibrium problems with constraints are obtained in the sense of E-globally proper efficiency. Meanwhile, we give some examples to illustrate our results. The results obtained in this paper improve and generalize some known results in the literature.

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].

On optimality conditions for set-valued equilibrium problems

On optimality conditions for set-valued equilibrium problems

Karush–Kuhn–Tucker Multiplier Rules for Efficient Solutions of Set-Valued Equilibrium Problem with Constraints

Karush–Kuhn–Tucker Multiplier Rules for Efficient Solutions of Set-Valued Equilibrium Problem with Constraints

Nonlinear Evolutionary Systems Driven by Set-Valued Mixed Equilibrium Problems

This paper is concerned with the system (NEESSVMEP) obtained by mixing a nonlinear evolutionary equation and a strong set-valued mixed equilibrium problem (SSVMEP). In our setting, the set of constraints is not necessarily compact and the problem is driven by a not necessarily monotone set-valued mapping. We show that the solution set for (SSVMEP) is nonempty, closed, convex and bounded. We then establish the upper semicontinuity and the measurability properties of the involved functions in the nonlinear evolutionary equation. Utilizing these results, we prove that the solution set for (NEESSVMEP) is nonempty and compact.

On Lipschitz continuity of approximate solutions to set-valued equilibrium problems via nonlinear scalarization

In this paper, we consider parametric set-valued equilibrium problems in normed spaces. By virtue of the Gerstewitz nonlinear scalarization function along with relaxed concavity assumptions, we obtain the Lipschitz continuity property of solution maps to such problems. The treatment and obtained results for these problems are new and different from the existing ones in the literature. We apply the main results to the Browder variational inclusion to illustrate for their applicability.

Second-Order characterizations for set-valued equilibrium problems with variable ordering structures

<p style='text-indent:20px;'>The concepts of weakly efficient solutions and globally efficient solutions are introduced for constrained set-valued equilibrium problems with variable ordering structures. By applying the second-order tangent epiderivative and a nonlinear functional, necessary optimality conditions for weakly efficient solutions and globally efficient solutions are established without any convexity assumption. Under the cone-convexity of the objective and constraint functions, sufficient optimality conditions are given. In addition, the tangent derivatives of objective and constraint functions are separated. Simultaneously, a unified necessary and sufficient optimality conditions for weakly efficient solutions is derived, and the same goes for globally efficient solutions. In particular, we give specific examples to illustrate the optimality conditions, respectively.</p>

Stability analysis for set-valued equilibrium problems with applications to Browder variational inclusions

Stability analysis for set-valued equilibrium problems with applications to Browder variational inclusions

Local dense solutions for equilibrium problems with applications to noncooperative games

In this work, we establish several results concerning the existence of solutions for set-valued and single-valued equilibrium problems in real Hausdorff topological vector spaces. Firstly we introduce some generalizations of convexity and continuity conditions to set-valued mappings and then apply them to special dense subsets of the domain to obtain the existence of local dense solutions of equilibrium problems. Then the existence of the global solutions follows from a condition that is weaker than semistrict quasiconvexity. Specifically, we give an existence theorem for noncooperative n-person games, under assumptions imposed on a locally segment-dense subset of the strategy set of each player.

Approximate Benson efficient solutions for set-valued equilibrium problems

In locally convex Hausdorff topological vector spaces, the approximate Benson efficient solution is proposed for set-valued equilibrium problems and its relationship to the Benson efficient solution is discussed. Under the assumption of generalized convexity, by using a separation theorem for convex sets, Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued equilibrium problems are established, respectively.

Sensitivity analysis for set-valued equilibrium problems

In the paper, we mention a parametrized vector equilibrium problem via sum of two given set-valued maps, considered as a unified model of some problems in variational analysis and optimization. Then, we study sensitivity analysis for this problem in terms of the second-order contingent derivatives. Finally, corresponding results for special cases are implied.

OPTIMALITY CONDITIONS FOR SOLVING NONCONVEX SET-VALUED EQUILIBRIUM PROBLEMS

In this paper, sufficient conditions ensuring the existence of solutions for set-valued equilibrium problems are obtained. The convexity assumption on the whole domain is not necessary and just the closure of a quasi-self-segment-dense subset of the domain is convex. Using a KKM theorem and a notion of Q-selected preserving $R_{-}^{*}$-intersection($R_{-}^{*}$-inclusion) for set-valued mapping, the existence results are proved in real Hausdorff topological vector spaces.

Existence of solutions of set-valued equilibrium problems in topological vector spaces with applications

In this paper, we first introduce the new notions of $$\overline{{\mathbb {R}}}_+$$-local-intersection property, $$\overline{{\mathbb {R}}}_+$$-local-inclusion property, $$\overline{{\mathbb {R}}}_+$$-strongly transfer lower semicontinuity, $${\mathbb {R}}_-$$-weakly transfer lower semicontinuity and $${\mathbb {R}}_-$$-strongly transfer lower semicontinuity for a set-valued mapping $$\Phi $$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.

On the stability of approximate solutions to set-valued equilibrium problems

In this paper, we consider set-valued equilibrium problems in Hausdorff locally convex topological vector spaces. Based on linear scalarization techniques for sets, we study sufficient conditions for the stability of approximate solutions to such problems. Variational inequalities with equilibrium constraints and weak traffic network equilibrium problems are also discussed as applications of the main results.