Optimality conditions for vector equilibrium problems in normed spaces

By using the concepts of contingent epiderivative, radial epiderivative, Clarke tangent epiderivative and Y-epiderivative, we present necessary and sufficient conditions for the weakly efficient solution, the Henig efficient solution, and the globally proper efficient solution, respectively, to vector equilibrium problems with constraints.     

Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior

In this paper, we consider the set-valued vector optimization problems with constraint in locally convex spaces. We present the necessary and sufficient conditions for Henig efficient solution pair, globally proper efficient solution pair and super efficient solution pair without the ordering cones having the nonempty interior.     

Well-posedness for vector equilibrium problems

We introduce and study two notions of well-posedness for vector equilibrium problems in topological vector spaces; they arise from the well-posedness concepts previously introduced by the same authors in the scalar case, and provide an extension of similar definitions for vector optimization problems. The first notion is linked to the behaviour of suitable maximizing sequences, while the second one is defined in terms of Hausdorff convergence of the map of approximate solutions. In this paper we compare them, and, in a concave setting, we give sufficient conditions on the data in order to guarantee well-posedness. Our results extend similar results established for vector optimization problems known in the literature.      

Second-Order Optimality Conditions for Set-Valued Vector Equilibrium Problems

In this article, by using the generalized second-order contingent (adjacent) epiderivatives of set-valued maps, we obtain necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions, Henig efficient solutions to the set-valued vector equilibrium problems with constraints. Some results of this article improve the corresponding results in literatures by lessening the assumption of convexity.     

含参广义集值向量均衡问题有效解映射下半连续的最优条件

在实Hausdorff拓扑向量空间中研究一类含参广义集值向量均衡问题弱有效解与有效解映射的下半连续性. 在近似锥-次类凸的条件下, 运用标量化的方法得到弱有效解的标量化结果. 在适当条件下, 得到含参广义集值向量均衡问题弱有效解与有效解映射下半连续性定理.     

The connectedness of the solutions set for generalized vector equilibrium problems

In this paper, we establish the connectedness of the sets of Henig efficient solutions, globally efficient solutions, weak efficient solutions, superefficient solutions and efficient solutions for a class of generalized vector equilibrium problems without the assumptions of monotonicity and compactness.     

Existence results for strong vector equilibrium problems and their applications

New existence results for the strong vector equilibrium problem are presented, relying on a well-known separation theorem in infinite-dimensional spaces. The main results are applied to strong cone saddle-points and strong vector variational inequalities providing new existence results, and furthermore they allow recovery of an earlier result from the literature.     

Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces

In this paper, we introduce and discuss the notion of ε-solutions of vector variational inequalities. Using convex analysis and nonsmooth analysis, we provide some sufficient conditions and necessary conditions for a point to be an ε-solution of vector variational inequalities.      

Closedness of the solution map for parametric weak vector equilibrium problems

In this paper we introduced a new definition of vector topological pseudomonotonicity to study the parametric weak vector equilibrium problems. The main result gives sufficient conditions for closedness of the solution map defined on the set of parameters.     

Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints

This paper mainly intends to present some semicontinuity and convergence results for perturbed vector optimization problems with approximate equilibrium constraints. We establish the lower semicontinuity of the efficient solution mapping for the vector optimization problem with perturbations of both the objective function and the constraint set. The constraint set is the set of approximate weak efficient solutions of the vector equilibrium problem. Moreover, upper Painlevé–Kuratowski convergence results of the weak efficient solution mapping are showed. Finally, some applications to the optimization problems with approximate vector variational inequality constraints and the traffic network equilibrium problems are also given. Our main results are different from the ones in the literature.     

向量均衡问题的超有效性的对偶形式

在局部凸空间中引进了向量均衡问题的强超有效解、C-强超有效解、弱超有效解, C-弱超有效解、齐次超有效解、 C-齐次超有效解的概念,并在局部凸空间中用极理论为工具讨论了向量均衡问题的 C-弱超有效解, C-超有效解, C-齐次超有效解,以及C-强超有效解的对偶形式. 又在赋范线性空间中讨论了向量均衡问题的以上各种超有效解之间的等价性,并且在赋范线性空间具正规锥的条件下讨论了向量均衡问题的以上各种超有效解的对偶形式. 作为它的应用,给出了向量优化问题各种超有效解的对偶形式.     

Recession methods for generalized vector equilibrium problems

By using the recession method, we give some necessary and/or sufficient condition of solutions of generalized vector equilibrium problems.     

On implicit vector equilibrium problems

In this paper, we introduce and study a class of implicit vector equilibrium problems, which includes a number of (scalar) implicit equilibrium problems, implicit variational inequalities, and implicit complementarity problems as special cases. By using KKM-Fan theorem, we prove some new existence theorems of solutions for this kind of implicit vector equilibrium problems in Hausdorff topological vector spaces. Our results extend and unify some corresponding results of several authors.     

The system of generalized vector equilibrium problems with applications

In this paper, we introduce the system of generalized vector equilibrium problems which includes as special cases the system of generalized implicit vector variational inequality problems, the system of generalized vector variational and variational-like inequality problems and the system of vector equilibrium problems. By using a maximal element theorem, we establish existence results for a solution of these systems. As an application, we derive existence results for a solution of a more general Nash equilibrium problem for vector-valued functions.     

On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems

In this paper, we introduce the concepts of globally efficient solution and cone-Benson efficient solution for a vector equilibrium problem; we give some scalarization results for Henig efficient solution sets, globally efficient solution sets, weak efficient solution sets, and cone-Benson efficient solution sets in locally convex spaces. Using the scalarization results, we show the connectedness and path connectedness of weak efficient solution sets and various proper efficient solution sets of vector equilibrium problem. This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jinxing Province, China.     

Topological pseudomonotonicity and vector equilibrium problems

In this paper, we introduce the topological pseudomonotonicity to vector valued bifunctions, and derive some existence results for vector equilibrium problems with the corresponding bifunctions topologically pseudomonotone. Some applications to vector variational inequalities and existence of Pareto optima are given.     

Levitin-poyak well-posedness of generalized vector equilibrium problems with functional constraints

In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.     

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

The 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.