Vector Variational Inequalities and Vector Equilibria

Frontmatter

Vector Equilibrium Problems and Vector Variational Inequalities

Abstract

In this paper, we consider vector equilibrium problems and prove the existence of their solutions in the setting of Hausdorff topological vector spaces. We also derive some existence results for the scalar and vector variational inequalities.

Qamrul Hasan Ansari

Generalized Vector Variational-Like Inequalities and their Scalarizations

Abstract

In this paper, we consider a more general form of vector variational-like inequalities for multivalued maps and prove some results on the existence of solutions of our new class of vector variational-like inequalities in the setting of topological vector spaces. Several special cases were also discussed.

Abul Hasan Ansari, Abul Hasan Siddiqi, Jen-Chih Yao

Existence of Solutions for Generalized Vector Variational-Like Inequalities

Abstract

This paper aims to introduce a new kind of Variational Inequality, i.e. a ‘Generalized Vector Variational-Like Inequality’ which includes several classic and well-known Variational Inequalities as special cases. As an application of the Knaster-Kuratowski-Mazurkiewicz principle - in the extended form given by Fan in 1961 -, we prove that there exist solutions for our Generalized Vector Variational-Like Inequality under reasonable hypotheses. These results generalize corresponding results given by Chen et al. in (1992), Giannessi (1980), Harker and Pang (1990), Hartman and Stampacchia (1966), Isac (1990), Lee et al. (1993), Noor (1988), Saigal (1976), Siddiqi et al. (1995) and Yang (1993).

Shih-Sen Chang, Heven Bevan Thompson, George Xian-Zhi Yuan

On Gap Functions for Vector Variational Inequalities

Abstract

We extend the theory of gap functions for scalar variational inequality problems (see [1,8]) to the case of vector variational inequality. The gap functions for vector variational inequality are defined as set-valued mappings. The significance of the gap function is interpreted in terms of the inverse vector variational inequality. Convexity properties of these set-valued mappings are studied under different assumptions.

Guang-ya Chen, Chuen-Jin Goh, Xiao Qi Yang

Existence of Solutions for Vector Variational Inequalities

Abstract

The paper provides some of the most fundamental results on the existence of solutions for Vector Variational Inequalities. Vector Variational-Like Inequalities are also discussed.

Guang-ya Chen, Shui-Hung Hou

On the Existence of Solutions to Vector Complementarity Problems

Abstract

It is known that there is a close relation between vector complementarity problems and vector variational inequalities. There are several types of existence results for vector variational inequalities. This paper aims to review some existence results on vector complementarity problems; it includes existence of solutions for vector complementarity problems, vector implicit complementarity problems, and generalized vector complementarity problems.

Guang-ya Chen, Xiao Qi Yang

Vector Variational Inequalities and Modelling of a Continuum Traffic Equilibrium Problem

Abstract

The aim of this paper is to present a new model of traffic equilibrium problem in the continuous case using an user-optimization approach. The model leads to a Vector Variational Inequality problem defined in a new kind of convex set, for which existence theorems and a computational procedure are exhibited.

Patrizia Daniele, Antonino Maugeri

Generalized Vector Variational-Like Inequalities without Monotonicity

Abstract

In this paper, we introduce a class of generalized vector variational-like inequalities without monotonicity which generalizes and unifies generalized vector variational inequalities, vector variational inequalities as well as various extensions of the classic variational inequalities in the literature. Some existence theorems for the generalized vector variational-like inequality without monotonicity are obtained in noncompact setting of topological vector spaces.

Xie Ping Ding, Enayet Tarafdar

Generalized Vector Variational-Like Inequalities with C x - η-Pseudomonotone Set-Valued Mappings

Abstract

In this paper, we introduce and study a class of generalized vector variational-like inequalities involving C _x - η-pseudomonotone and weakly C _x - η-pseudomonotone set-valued mappings. The generalized vector variational-like inequality problem unifies and generalizes the generalized vector variational inequalities, vector variational inequalities and various extensions of the classic variational inequalities involving single-valued and set-valued mappings of various monotone types in the literature. Several existence theorems are established under noncompact setting in topological vector spaces. These new results unify and generalize many recent known results in the literature.

Xie Ping Ding, Enayet Tarafdar

A Vector Variational-Like Inequality for Compact Acyclic Multifunctions and its Applications

Abstract

A vector variational-like inequality for compact acyclic multifunctions is presented. This is used to introduce the generalized vector quasi-variational inequality and the generalized vector quasi-complementarity problem in ordered vector spaces. Some existence theorems for these problems are proved.

Junyi Fu

On the Theory of Vector Optimization and Variational Inequalities. Image Space Analysis and Separation

Abstract

By exploiting recent results, it is shown that the theories of Vector Optimization and of Vector Variational Inequalities can be based on the image space analysis and theorems of the alternative or separation theorems. It is shown that, starting from such a general scheme, several theoretical aspects can be developed - like optimality conditions, duality, penalization - as well as methods of solution - like scalarization.

Franco Giannessi, Giandomenico Mastroeni, Letizia Pellegrini

Scalarization Methods for Vector Variational Inequality

Abstract

Scalarization of vector problems is an important concept at least from the computational point of view. In this paper, scalarization is applied to Weak Vector Variational Inequalities, and results are established using a symmetric Jacobian condition. New relationships are presented for Vector Variational Inequalities and Vector Optimization problems, and sufficient and necessary conditions for reducing a Weak Vector Variational Inequality to a (scalar) Variational Inequality are discussed. An exact analytical method for solving a special case of the Weak Vector Variational Inequality involving only affine functions via scalarization is proposed.

Chuen-Jin Goh, Xiao Qi Yang

Super Efficiency for a Vector Equilibrium in Locally Convex Topological Vector Spaces

Abstract

This paper deals with the vector equilibrium problem. The concept of super efficiency for vector equilibrium is introduced. A scalar characterization of super efficient solution for vector equilibrium is given. By using of the scalarization result, we discuss the connectedness of super efficient solutions set to the vector equilibrium problems in locally convex topological vector spaces.

Xun Hua Gong, Wan Tao Fu, Wei Liu

The Existence of Essentially Connected Components of Solutions for Variational Inequalities

Abstract

The aim of this paper is to establish the existence of essentially connected components for Hartman-Stampacchia type variational inequalities for both set-valued and single-valued mappings in normed spaces. Our results show that each variational inequality problem has, at least, one connected component of its solutions which is stable though in general its solution set may not have a good behavior (i.e., not stable). Thus if a variational inequality problem has only one connected solution set, it must be stable. Here we don’t need to require the objective mapping to be either Lipschitz or differential.

George Isac, George Xian-Zhi Yuan

Existence of Solutions for Vector Saddle-Point Problems

Abstract

In this paper, we establish an existence theorem for weak saddle points of a vector valued function by making use for vector variational-like inequality and non-convex functions.

Kaleem Raza Kazmi

Vector Variational Inequality as a Tool for Studying Vector Optimization Problems

Abstract

The paper aims to show that a Vector Variational Inequality can be an useful tool for studying a Vector Optimization Problem.

Gue Myung Lee, Do Sang Kim, Byung Soo Lee, Nguyen Dong Yen

Vector Variational Inequalities in a Hausdorff Topological Vector Space

Abstract

Using Fan-Browder type fixed point theorem, we prove two theorems on the existence of solutions of Vector Variational Inequality in a Hausdorff topological vector space. Our results are fairly general enough to sharpen and cover earlier corresponding results of many authors. In particular, our results genaralize recent results of Lai and Yao, Yu and Yao. In addition, the equivalent relation between solutions of Generalized Minty Vector Variational Inequality and generalized vector-minimum points of Vector Optimization Problems is shown.

Gue Myung Lee, Sangho Kum

Vector Ekeland Variational Principle

Abstract

In this paper the well-known Ekeland variational principle is generalized to the case where vector-valued functions are involved. Namely, vector Ekeland variational principle is studied. In particular, the epsilon vector minimum point of a vector optimization problem is investigated via vector Ekeland variational principle.

Shen Jie Li, Xiao Qi Yang, Guang-ya Chen

Convergence of Approximate Solutions and Values in Parametric Vector Optimization

Abstract

New concepts of approximate values and solutions for vector optimization problems are introduced. Then, under conditions of minimal character, we present convergence results involving the above mentioned concepts.

Pierre Loridan, Jacqueline Morgan

On Minty Vector Variational Inequality

Abstract

The analysis of the connections among generalized systems, Vector Optimization Problems and Variational Inequalities allows to deepen the study of the properties of the Minty Vector Variational Inequality. In particular, under suitable regularity assumptions,the equivalence between Minty Vector Variational Inequality and Stampacchia Vector Variational Inequality is shown.

Giandomenico Mastroeni

Generalized Vector Variational-Like Inequalities

Abstract

In this paper, we obtain existence theorems for generalized vector variational-like inequalities both under compact and non-compact assumptions. The use of the concept of escaping sequences is crucial for this analysis.

Luo Qun

On Vector Complementarity Systems and Vector Variational Inequalities

Abstract

The weak ordering is generalized in Banach spaces; Vector Complementarity Systems and Vector Variational Inequalities are introduced based on this new ordering, and relations are discussed between them.

Tamás Rapcsák

Generalized Vector Variational Inequalities

Abstract

A vector and a set-valued generalization of Ky Fan minimax principle are proved. As application, several existence theorems for Generalized Vector Variational Inequalities with set-valued operators which have either lower (upper) semicontinuous or pseudomonotone properties in topological vector spaces are derived. The results obtained extend and unify a number of existence results for Vector Variational Inequalities. Moreover, we give the relationship between a kind of Generalized Vector Variational Inequality and a Vector Optimization Problem.

Wen Song

Vector Equilibrium Problems with Set-Valued Mappings

Abstract

An existence result for a generalized vector equilibrium problem is proved in a general topological vector space. As applications, existence results are derived for vector equilibrium problems and vector variational inequalities with set-valued mappings which have order-lower (upper) semicontinuous and pseudomonotone properties. These results extend and unify a number of existence results for Vector Variational Inequalities in the previous literature.

Wen Song

On Some Equivalent Conditions of Vector Variational Inequalities

Abstract

In this paper, we discuss some equivalent conditions for weak vector variational inequality and vector variational inequality problems, such as relations among vector variational inequalities, vector optimization problems and gap functions. These results are useful in the design of solution methods for vector variational inequalities.

Xiao Qi Yang

On Inverse Vector Variational Inequalities

Abstract

We study inverse (or dual) vector variational inequalities. Various relations among original vector variational inequalities and original vector variational inequalities are clarified. Examples are given to show relations of inverse vector variational inequalities with vector optimization problems.

Xiao Qi Yang, Guang-ya Chen

Vector Variational Inequalities, Vector Equilibrium Flow and Vector Optimization

Abstract

The study of vector variational inequalities and vector optimization problems has been advanced by extending the well-known Wardrop’s equilibrium principle to account for the situation where there are conflicting multicriteria weights. This paper sumarizes connections among vector variational inequalities, vector equilibrium flow problems and vector optimization problems. In particular, we consider the case where partial orderings are defined by a closed and convex cone. These relations allow one to solve a vector equilibrium flow problem or a vector variational inequality by using vector optimization methods.

Xiao Qi Yang, Chuen-Jin Goh

On Monotone and Strongly Monotone Vector Variational Inequalities

Abstract

By constructing an example we show that the solution sets of a strongly monotone vector variational inequality and of its relaxed inequality can be different from each other. A sufficient condition for the coincidence of these solution sets is given for general vector variational inequalities; connectedness and path-connectedness of the solution sets for some kinds of monotone problems in Hilbert spaces are studied in detail.

Nguyen Dong Yen, Gue Myung Lee

Connectedness and Stability of the Solution Sets in Linear Fractional Vector Optimization Problems

Abstract

As it was shown by C. Malivert (1996) and other Authors, in a Linear Fractional Vector Optimization Problem (for short, LFVOP) any point satisfying the first-order necessary optimality condition (a stationary point) is a solution. Therefore, solving such a problem is equivalent to solve a monotone affine vector variational inequality of a special type. This observation allows us to apply the existing results on monotone affine variational inequality to establish some facts about connectedness and stability of the solution sets in LFVOP. In particular, we are able to solve a question raised by E. U. Choo and D. R. Atkins (1983) by proving that the set of all the efficient points (Pareto solutions) of a LFVOP with a bounded constraint set is connected.

Nguyen Dong Yen, Ta Duy Phuong

Vector Variational Inequality and Implicit Vector Complementarity Problems

Abstract

In this paper, we consider the general vector variational inequality problem. The conception of new generalized monotone mappings is introduced. Existence theorems of the solution of this problem and topological properties of the solution set are presented. The results appear to be new even for the vector variational inequality considered by Chen in Ref.1 and the general variational inequality discussed by Isac in Ref.6. Meanwhile, implicit vector complementarity problems are proposed and the existence of the solution is studied.

Hongyou Yin, Chengxian Xu

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