Vector Variational Inequalities for Nondifferentiable Convex Vector Optimization Problems

In this paper, we consider a nondifferentiable convex vector optimization problem (VP), and formulate several kinds of vector variational inequalities with subdifferentials. Here we examine relations among solution sets of such vector variational inequalities and (VP). Mathematics Subject classification (2000). 90C25, 90C29, 65K10 This work was supported by the Brain Korea 21Project in 2003. The authors wish to express their appreciation to the anonymous referee for giving valuable comments.     

On Relations Between Vector Optimization Problems and Vector Variational Inequalities

We obtain equivalences between weak Pareto solutions of vector optimization problems and solutions of vector variational inequalities involving generalized directional derivatives.     

Vector Variational Inequality and Vector Pseudolinear Optimization

The study of a vector variational inequality has been advanced because it has many applications in vector optimization problems and vector equilibrium flows. In this paper, we discuss relations between a solution of a vector variational inequality and a Pareto solution or a properly efficient solution of a vector optimization problem. We show that a vector variational inequality is a necessary and sufficient optimality condition for an efficient solution of the vector pseudolinear optimization problem.     

Vector Variational Inequalities Involving Vector Maximal Points

This paper considers the existence of solutions and the equivalence of four kinds of vector variational inequalities (VVI). More precisely, a sufficient condition is provided under which the solution sets of these VVIs are nonempty and equal. An example is given, showing that such a sufficient condition is essential to ensure the results. Actually, the main theorems in this paper can be regarded as a suitable correction and a refinement of recent results due to Chang et al. (Ref. 1).     

Generalized Vector Variational Inequalities

In this paper, we introduce a generalized vector variational inequality problem (GVVIP) which extends and unifies vector variational inequalities as well as classical variational inequalities in the literature. The concepts of generalized C-pseudomonotone and generalized hemicontinuous operators are introduced. Some existence results for GVVIP are obtained with the assumptions of generalized C-pseudomonotonicity and generalized hemicontinuity. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained.     

Pointwise Well-Posedness of Perturbed Vector Optimization Problems in a Vector-Valued Variational Principle

In this note, we point out and correct some errors in Ref. 1. Another type of pointwise well-posedness and strong pointwise well-posedness of vector optimization problems is introduced. Sufficient conditions to guarantee this type of well-posedness are provided for perturbed vector optimization problems in connection with the vector-valued Ekeland variational principle.     

Generalized Vector Variational-Like Inequalities

Using a generalized Fan’s KKM theorem, some existence results for generalized vector variational-like inequalities in noncompact settings are established. Some applications to vector optimization problems are given. The results presented in this paper extend and unify corresponding results of other authors.     

Some Remarks on the Minty Vector Variational Inequality

In this paper, we establish some relations between a Minty vector variational inequality and a vector optimization problem under pseudoconvexity or pseudomonotonicity, respectively. Our results generalize those of Ref. 1.     

On Vector Implicit Variational Inequalities and Complementarity Problems

In this paper, two vector implicit variational inequalities and three vector implicit complementarity problems are introduced with a more general order in ordered Banach spaces and the equivalence between them is studied under certain assumptions. Furthermore, some existence theorems of the vector implicit variational inequalities are proved. We answered the open question proposed by Rapcsák [Nonconvex Optim., Appl., Vol. 38, Kluwer Acad. Publ., Dordrecht, 2000, pp. 371–380]. This work was supported by the National Natural Science Foundation of China and the Scientific Research Foundation for the Returned Overseas Chinese, Scholars, State Education Ministry.     

Weighted Variational Inequalities

In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities. The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees for comments and suggestions improving the final draft of this paper.     

Bound Constrained Smooth Optimization for Solving Variational Inequalities and Related Problems

Variational inequalities and related problems may be solved via smooth bound constrained optimization. A comprehensive discussion of the important features involved with this strategy is presented. Complementarity problems and mathematical programming problems with equilibrium constraints are included in this report. Numerical experiments are commented. Conclusions and directions of future research are indicated.     

Some Remarks On Vector Optimization Problems

We prove the existence of a weak minimum for vector optimization problems by means of a vector variational-like inequality and preinvex mappings.     

A Scalarization Approach for Vector Variational Inequalities with Applications

We consider an approach to convert vector variational inequalities into an equivalent scalar variational inequality problem with a set-valued cost mapping. Being based on this property, we give an equivalence result between weak and strong solutions of set-valued vector variational inequalities and suggest a new gap function for vector variational inequalities. Additional examples of applications in vector optimization, vector network equilibrium and vector migration equilibrium problems are also given Mathematics Subject Classification(2000). 49J40, 65K10, 90C29     

On Implicit Vector Variational Inequalities

In this paper, we study the existence of solutions of implicit vector variational inequalities for multifunctions. Generalized pseudomonotonicity concepts are introduced. Our results extend and unify corresponding earlier existence results of many authors for vector variational inequalities under the Hausdorff topological vector space setting.     

A Result on Vector Variational Inequalities with Polyhedral Constraint Sets

In this note, by using some well-known results on properly efficient solutions of vector optimization problems, we show that the Pareto solution set of a vector variational inequality with a polyhedral constraint set can be expressed as the union of the solution sets of a family of (scalar) variational inequalities.     

Duality Results for Generalized Vector Variational Inequalities with Set-Valued Maps

In this paper, we introduce new dual problems of generalized vector variational inequality problems with set-valued maps and we discuss a link between the solution sets of the primal and dual problems. The notion of solutions in each of these problems is introduced via the concepts of efficiency, weak efficiency or Benson proper efficiency in vector optimization. We provide also examples showing that some earlier duality results for vector variational inequality may not be true. This work was supported by the Brain Korea 21 Project in 2006.     

Existence of Solutions to Implicit Vector Variational Inequalities

In this paper, we study a class of implicit vector variational inequalities which contain implicit variational inequalities and generalized quasivariational inequalities as special cases. By employing the Fan–Kakutani fixed-point theorem and the Oettli scalarization procedure, respectively, we establish several existence results for implicit vector variational inequalities.     

随机向量值变分不等式

本文先讨论了一个KyFan截口定理的随机化形式，然后利用它讨论了一个隐式随机向量值变分不等式和一个显式随机向量值变分不等式的随机解的存在性．     

Vector Variational Inequalities with Semi-monotone Operators

In this paper, the concept of a vector semi-monotone operator is introduced. This concept is applied to establish several existence results of vector variational inequalities. These extend some results of others. The work is supported by the Natural Science Foundation of China (Grant No. 19671042) and the Doctoral Program Foundation of the Ministry of Education of China.     

Solving Fuzzy Variational Inequalities

This paper studies the variational inequality problem over a fuzzy domain and variational inequalities for fuzzy mappings over a fuzzy domain. It is shown that such problems can be reduced to bilevel programming problems. A penalty function algorithm is introduced with a convergence proof. Numerical examples are also included to illustrate the solution procedure.