Existence of a Solution and Variational Principles for Vector Equilibrium Problems

In this paper, we prove an existence result for a solution to the vector equilibrium problems. Then, we establish variational principles, that is, vector optimization formulations of set-valued maps for vector equilibrium problems. A perturbation function     

Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities

We consider five different types of systems of generalized vector variational inequalities and derive relationships among them. We introduce the concept of pseudomonotonicity for a family of multivalued maps and prove the existence of weak solutions of these problems under these pseudomonotonicity assumptions in the setting of Hausdorff topological vector spaces as well as real Banach spaces. We also establish the existence of a strong solution of our problems under lower semicontinuity for a family of multivalued maps involved in the formulation of the problems. By using a nonlinear scalar function, we introduce gap functions for our problems by which we can solve systems of generalized vector variational inequalities using optimization techniques. The first two authors were supported by SABIC and Fast Track Research Grants SAB-2006-05. They are grateful to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities.     

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 Results for Set-Valued Vector Quasiequilibrium Problems

This paper deals with the set-valued vector quasiequilibrium problem of finding a point (z ₀,x ₀) of a set E×K such that (z ₀,x ₀)∈B(z ₀,x ₀)×A(z ₀,x ₀), and, for all η∈A(z ₀,x ₀),

Generalized Variational Principle and Vector Optimization

In this note, we present a generalization of the celebrated Ekeland variational principle and its equivalent forms. The results presented in this paper unify, improve, and extend the corresponding result in Refs. 1–7.     

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.     

Systems of Simultaneous Generalized Vector Quasiequilibrium Problems and their Applications

In this paper, we introduce systems of simultaneous generalized vector equilibrium problems and prove the existence of their solutions. As application of our results, we derive the existence theorems for solutions of systems of vector saddle–point problems. Consequently, we prove the existence of a solution of systems of generalized minimax inequalities. Further application of our results is also given to establish the existence of a solution of a Debreu-type equilibrium problem for vector-valued functions. The first author thanks the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The second and third authors were supported by the National Science Council of the Republic of China. The authors are grateful to the referees for valuable suggestions and comments.     

Characterizations of Solutions for Vector Equilibrium Problems

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.     

Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.     

Gap Functions and Existence of Solutions for a System of Vector Equilibrium Problems

In this paper, a gap function for a system of vector equilibrium problems is introduced and studied. Some necessary and sufficient conditions for the system of vector equilibrium problems are established. Characterizations of the solutions set for the system of vector equilibrium problems are also derived. Furthermore, some existence results of solutions for the system of vector equilibrium problems are proved. This work was supported by the National Natural Science Foundation of China, the Youth Foundation, Sichuan Education Department of China, the National Natural Science Foundation, Sichuan Education Department of China (2004C018), and a grant from the National Science Council of ROC.     

Partial Minimization for Vector Variational Problems

The basic idea of performing a partial minimization with respect to some components in a vector variational problem, while keeping the other fixed, is explored and implemented in various examples. In one-dimensional problems, it leads sometimes to nonstandard variational problems. We also include a situation for a genuine vector variational problem coming from the reformulation of some optimal design problems in conductivity.     

Existence of Solutions for Generalized Vector Variational-like Inequalities

In this paper, we consider a generalized vector variational-like inequality problem (for short, GVVLIP), which includes generalized vector variational inequalities, vector variational inequalities and classical variational inequalities as special cases. The concepts of generalized C-pseudomonotone-like and generalized H-hemicontinuous-like operators are introduced. Some existence results for GVVLIP are obtained under the assumptions of generalized C-pseudomonotone-like property and generalized H-hemicontinuous-like property. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained. In this research, the first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai. The third author was partially supported by Grant NSC 94-2213-E-110-035.     

Existence Theorems for Systems of Generalized Vector Quasiequilibrium Problems and Optimization Problems

In this paper, we use existence theorems for the equilibria of generalized abstract economies proved recently to establish existence theorems for systems of generalized vector quasiequilibrium problems. Then, these existence theorems for equilibrium problems are used to derive existence theorems for systems of generalized vector quasivariational-like inequality problems, and vector quasioptimization problems.This research was supported by the National Science Council of the Republic of China. The authors express their gratitude to the referees for valuable suggestions.     

Generalized Vector Variational Inequalities over Countable Product of Sets

In this paper, we consider vector variational inequalities with set-valued mappings over countable product sets in a real Banach space setting. By employing concepts of relative pseudomonotonicity, we establish several existence results for generalized vector variational inequalities and for systems of generalized vector variational inequalities. These results strengthen previous existence results which were based on the usual monotonicity type assumptions     

Derivative-Free Methods for Monotone Variational Inequality and Complementarity Problems

Monotone variational inequality problems with box constraints and complementarity problems are reformulated as simple-bound optimization problems. Some derivative-free methods for these problems are proposed. It is shown that, for these new methods, the updated point sequence remains feasible with respect to its simple constraints if the initial point is feasible. Under certain conditions, these methods are globally convergent.     

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.     

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     

Variational characterization of the contingent epiderivative

In this paper the existence of the contingent epiderivative of a set-valued map is studied from a variational perspective. We give a variational characterization of the ideal minimal of a weakly compact set. As a consequence we characterize the existence of the contingent epiderivative in terms of an associated family of variational systems. When a set-valued map takes values in Rn we show that these systems can be formulated in terms of the contingent epiderivatives of scalar set-valued maps. By applying these results we extend some existing theorems.     

Gap Function for Set-Valued Vector Variational-Like Inequalities

Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational-like inequalities (VVLI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVLI. We investigate the existence of a solution for the generalized VVLI with a set-valued mapping by exploiting the existence of a solution of the VVLI with a single-valued function and a continuous selection theorem. The research of first author was partially supported by the Council of Scientific and Industrial Research, New Delhi, Ministry of Human Resources Development, Government of India Grant 25(0132)/ER-II/2004.