Vector variational inequalities with cone-pseudomonotone bifunctions

In this article, two types of cone-pseudomonotone bifunctions have been introduced and the weaker form of pseudomonotonicity is used to establish an existence theorem for a Stampacchia-kind vector variational inequality problem given in terms of bifunctions. For a vector optimization problem, the necessary and sufficient optimality conditions in terms of an associated vector variational inequality problem have been established using a generalized form of cone pseudoconvexity of objective function.     

Ekeland’s principle for vector equilibrium problems

In this paper, the authors deal with bifunctions defined on complete metric spaces and with values in locally convex spaces ordered by closed convex cones. The aim is to provide a vector version of Ekeland’s theorem related to equilibrium problems. To prove this principle, a weak notion of continuity of a vector-valued function is considered, and some of its properties are presented. Via the vector Ekeland’s principle, existence results for vector equilibria are proved in both compact and noncompact domains.     

Generalized Vector Equilibrium Problems for Pseudomonotone Multivalued Bifunctions<Superscript><Emphasis Type="Bold">1</Emphasis></Superscript>

In this paper, we apply a new version of the Brézis, Nirenberg, and Stampacchia theorem; we use pseudomonotonicity and some coercivity conditions to establish some existence result for a solution of generalized vector equilibrium problems for multivalued bifunctions. The proper quasiconvexity of multivalued bifunctions is introduced and existence theorems for generalized vector equilibrium problems related to multivalued mappings with the KKM property are obtained. The new results extend and modify various existence theorems for similar problems.     

On polar generalized monotonicity in vector optimization

The aim of the paper is to put in evidence that scaianzation in vector optimization is not a mere technical tool but sometimes it is the only natural way to extend certain vectorial concepts like cone convexity and cone monotonicity. A deeper insight into the special properties of polarly quasimonotone bifunctions is given and it is shown that this con-cept is a natural extension of the (non polar) monotonicity concept     

VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle(see Qiu, 2014), we prove a new version of Ekeland variational principle(briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results(no matter scalar-valued case,or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.     

From Scalar to Vector Equilibrium Problems in the Quasimonotone Case

In a unified approach, existence results for quasimonotone vector equilibrium problems and quasimonotone (multivalued) vector variational inequality problems are derived from an existence result for a scalar equilibrium problem involving two (rather than one) quasimonotone bifunctions. The results in the vector case are not only obtained in a new way, but they are also stronger versions of earlier existence results.     

Equilibrium Problems in the Quasimonotone Case

Existence results for quasimonotone vector equilibrium problems and quasimonotone vector variational inequalities are obtained starting from an existence result for a scalar equilibrium problem involving two quasimonotone bifunctions. These results are established under weaker conditions than in previous works.     

Cyclically antimonotone vector equilibrium problems

ABSTRACT

In this paper, we extend the notion of cyclic antimonotonicity (known for scalar bifunctions) to the vector case, in order to obtain a vectorial equilibrium version of Ekeland's variational principle. We characterize the cyclic antimonotonicity in terms of a suitable approximation from below of the vector bifunction, which allows us to avoid the demanding triangle inequality property, usually required in the literature, when dealing with Ekeland's principle for bifunctions. Furthermore, a result for weak vector equilibria in the absence of convexity assumptions is given, without passing through the existence of approximate solutions.     

Existence results for Stampacchia and Minty type vector variational inequalities

In this article, we consider the general forms of Stampacchia and Minty type vector variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector variational inequalities for set-valued maps and prove the existence of their solutions in the setting of Banach spaces as well as topological vector spaces. We point out that our vector variational inequalities extend and generalize several vector variational inequalities that appeared in the literature. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector variational inequalities.     

Some Deterministic and Random Vector Equilibrium Problems

In this paper, some generalized concepts about semicontinuity and convexity for vector-valued bifunctions are introduced. A new existence theorem for vector equilibrium problems is proved. Further, a random version of this result is considered. Applications to cone saddle points, random vector optimization, random vector variational inequalities, and random vector best approximation problems are finally given.     

FC-空间中含伪单调集值映象的广义向量平衡问题

在FC-空间内引入和研究了几类含伪单调集值映象的广义向量平衡问题．利用第二作者的KKM定理,在FC空间内对这些广义向量平衡问题证明了平衡点的存在性定理,改进与推广了近期文献中的相应结果．     

Vector-valued perturbed equilibrium problems

The goal of this paper is to prove some general vector-valued perturbed equilibrium principles and some existence results of vector equilibrium points for bifunctions satisfying a new natural notion of lower semi-continuity. We obtain these results by going through a new concept of approximative equilibrium point.     

Equilibrium and least element problems for multivalued functions

The principal aim of this paper is to extend some recent results which concern problems involving bifunctions to similar generalized problems for multivalued bifunctions. To this end, by using the appropriate notions of strict pseudomonotonicity we establish the relationships between generalized vector equilibrium problems and generalized minimal element problems of feasible sets. Moreover relationships between generalized least element problems of feasible sets and generalized vector equilibrium problems are studied by employing the concept of Z-multibifunctions.     

A Generalized Variational Principle and Its Application to Equilibrium Problems

In this paper, we prove a generalized Ekeland-type variational principle for bifunctions, by showing the existence of solution for some generalized optimization problems. In a particular case, from this result, we obtain a Zhong-type variational principle for bifunctions, which may be important from algorithmic point of view, because the solution can be localized in a sphere. Contrary to the standard literature, we are able to guarantee the existence of solution without assuming the triangle property.     

New Type of Generalized Vector Quasiequilibrium Problem

In this paper, we introduce a new type of vector quasiequilibrium problem with set-valued mappings and moving cones. By using the scalarization method and fixed-point theorem, we obtain its existence theorem. As applications, we derive some existence theorems for vector variational inequalities and vector complementarity problems. This work was supported by the National Natural Science Foundation of China. The authors are grateful to Professor X.Q. Yang and the referees for valuable comments and suggestions improving the original draft.     

Local-global properties of bifunctions

Representations of composite systems, such as bilinear programming, models of consumer/producer behavior, and sensitivity problems involve bifunctions (functions of two vector arguments). Such bifunctions are typically convex, pseudoconvex, or quasiconvex in each of their arguments, but not jointly convex, pseudoconvex, or quasiconvex. These functions do not in general possess the strong local-global property, namely, that every stationary point is a global minimum. In this paper, we define conditions that ensure that a bifunction possesses only a global minimum. In exploring this question, we use P-convexity and pseudo P-convexity, which are classes of bifunctions that generalize quasiconvexity and pseudoconvexity.     

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.     

Generalized Equilibrium Problems for Quasimonotone and Pseudomonotone Bifunctions

By using quasimonotone and pseudomonotone bifunctions, we derive sufficient conditions which include weak coercivity conditions for existence of equilibrium points. As a consequence, we improve some recent results on the existence of such solutions.     

Refinements of existence results for relaxed quasimonotone equilibrium problems

We consider a general equilibrium problem in a normed vector space setting and we establish sufficient conditions for the existence of solutions in compact and non compact cases. Our approach is based on the concept of upper sign property for bifunctions, which turns out to be a very weak assumption for equilibrium problems. In the framework of variational inequalities, this notion coincides with the upper sign continuity for a set-valued operator introduced by Hadjisavvas. More in general, it allows to strengthen a number of existence results for the class of relaxed $\mu $ -quasimonotone equilibrium problems.     

Vector complementarity problems with a variable ordering relation

In this paper, we investigate vector complementarity problems with a variable ordering relation. We establish existence results of a solution of a vector complementarity problem under an inclusive type condition. We obtain equivalence results among a vector complementarity problem, a vector variational inequality problem and other related problems.