Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions

In this paper, by using the scalarization approach of Konnov, several kinds of strong and weak scalar variational inequalities (SVI and WVI) are introduced for studying strong and weak vector variational inequalities (SVVI and WVVI) with set-valued mappings, and their gap functions are suggested. The equivalence among SVVI, WVVI, SVI, WVI is then established under suitable conditions and the relations among their gap functions are analyzed. These results are finally applied to the error bounds for gap functions. Some existence theorems of global error bounds for gap functions are obtained under strong monotonicity and several characterizations of global (respectively local) error bounds for the gap functions are derived.     

Gap functions and error bounds for generalized vector variational inequalities

This paper deals with generalized vector variational inequalities. Without any scalarization approach, the gap functions and their regularized versions for generalized vector variational inequalities are first obtained. Then, in the absence of the projection operator method, some error bounds for generalized vector variational inequalities are established in terms of these regularized gap functions. Further, the results obtained in this paper are more simpler from the computational view.     

A Class of Differential Vector Variational Inequalities in Finite Dimensional Spaces

In this paper, we introduce and study a class of differential vector variational inequalities in finite dimensional Euclidean spaces. We establish a relationship between differential vector variational inequalities and differential scalar variational inequalities. Under various conditions, we obtain the existence and linear growth of solutions to the scalar variational inequalities. In particular we prove existence theorems for Carathéodory weak solutions of the differential vector variational inequalities. Furthermore, we give a convergence result on Euler time-dependent procedure for solving the initial-value differential vector variational inequalities.     

Vector variational inequalities on Hadamard manifolds involving strongly geodesic convex functions

This paper is intended to study the vector variational inequalities on Hadamard manifolds. Generalized Minty and Stampacchia vector variational inequalities are introduced involving generalized subdifferential. Under strongly geodesic convexity, relations between solutions of these inequalities and a nonsmooth vector optimization problem are established. To illustrate the relationship between a solution of generalized weak Stampacchia vector variational inequality and weak efficiency of a nonsmooth vector optimization problem, a non-trivial example is presented.     

On vector optimization problems and vector variational inequalities using convexificators

In this paper, we establish some results which exhibit an application of convexificators in vector optimization problems (VOPs) and vector variational inequaities involving locally Lipschitz functions. We formulate vector variational inequalities of Stampacchia and Minty type in terms of convexificators and use these vector variational inequalities as a tool to find out necessary and sufficient conditions for a point to be a vector minimal point of the VOP. We also consider the corresponding weak versions of the vector variational inequalities and establish several results to find out weak vector minimal points.     

On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities

In this paper, we consider a vector optimization problem involving approximately star-shaped functions. We formulate approximate vector variational inequalities in terms of Fréchet subdifferentials and solve the vector optimization problem. Under the assumptions of approximately straight functions, we establish necessary and sufficient conditions for a solution of approximate vector variational inequality to be an approximate efficient solution of the vector optimization problem. We also consider the corresponding weak versions of the approximate vector variational inequalities and establish various results for approximate weak efficient solutions.     

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     

混合向量变分不等式标量化及间隙函数误差界

利用Konnov对变分不等式问题的标量化方法,对一般的强变分不等式（SVI）和弱变分不等式（WVI）进行了进一步的推广.主要介绍了基于集值映射的强广义混合向量变分不等式（SGMVVI）和弱广义混合向量变分不等式（WGMVVI）,考虑了与它们相关的间隙函数,在合适的条件下讨论了强广义混合集值变分不等式（SGMVI）的间隙函数和SGMVVI的间隙函数之间的关系,以及WGMVVI和SGMVI的间隙函数之间的关系,最后讨论了它们的间隙函数的全局误差界.     

Scalarization and decomposition of vector variational inequalities governed by bifunctions

In this article we study the structure of solution sets within a special class of generalized Stampacchia-type vector variational inequalities, defined by means of a bifunction which takes values in a partially ordered Euclidean space. It is shown that, similar to multicriteria optimization problems, under appropriate convexity assumptions, the (weak) solutions of these vector variational inequalities can be recovered by solving a family of weighted scalar variational inequalities. Consequently, it is deduced that the set of weak solutions can be decomposed into the union of the sets of strong solutions of all variational inequalities obtained from the original one by selecting certain components of the bifunction which governs it.     

Weak minimizers,minimizers and variational inequalities for set-valued functions. A blooming wreath?

Recently, necessary and sufficient conditions in terms of variational inequalities have been introduced to characterize minimizers of convex set-valued functions. Similar results have been proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved using scalarization techniques that eventually provide original problems, not fully equivalent to the set-valued counterparts. Therefore, we try, in the course of this note, to close the network among the various notions proposed. More specifically, we prove that a minimizer is always a weak minimizer, and a solution to the stronger variational inequality always also a solution to the weak variational inequality of the same type. As a special case, we obtain a complete characterization of efficiency and weak efficiency in vector optimization by set-valued variational inequalities and their scalarizations. Indeed, this might eventually prove the usefulness of the set optimization approach to renew the study of vector optimization.     

弱向量变分不等式的解集及其连通性

建立Hausdorff拓扑向量空间的非空凸子集到其值域为连续线性映射空间L(X, Y) 内的C -单调映射的弱向量变分不等式和它的纯量型变分不等式问题解的存在性, 讨论该弱向量变分不等式与之相联系的纯量型变分不等式解集的关系, 利用映射的C -弱次连续和C -单调性及其集值映射的不动点定理,通过纯量型变分不等式解集所诱导的集值映射所具有的特性给出弱向量变分不等式解集的连通性.     

On F-Implicit Generalized Vector Variational Inequalities

In this paper, we study the F-implicit generalized (weak) case for vector variational inequalities in real topological vector spaces. Both weak and strong solutions are considered. These two sets of solutions coincide whenever the mapping T is single-valued, but not set-valued. We use the Ferro minimax theorem to discuss the existence of strong solutions for F-implicit generalized vector variational inequalities.     

Lower Semicontinuity for Parametric Weak Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, a key assumption similar to that of Li and Chen is introduced by virtue of a gap function for a class of parametric set-valued weak vector variational inequalities in Banach spaces. By using this key assumption, sufficient and necessary conditions of the continuity and Hausdorff continuity of the solution set mapping for such parametric set-valued weak vector variational inequalities are given in Banach spaces when the image space is infinite dimensional. The results presented in this paper generalize and improve some main results of Li and Chen.     

Vector equilibrium problems with elastic demands and capacity constraints

In this paper, a (weak) vector equilibrium principle for vector network problems with capacity constraints and elastic demands is introduced. A sufficient condition for a (weak) vector equilibrium flow to be a solution for a system of (weak) vector quasi-variational inequalities is obtained. By virtue of Gerstewitz’s nonconvex separation functional ξ, a (weak) ξ-equilibrium flow is introduced. Relations between a weak vector equilibrium flow and a (weak) ξ-equilibrium flow is investigated. Relations between weak vector equilibrium flows and two classes of variational inequalities are also studied.     

Relationships between interval-valued vector optimization problems and vector variational inequalities

In this paper, we study some relationships between interval-valued vector optimization problems and vector variational inequalities under the assumptions of LU-convex smooth and non-smooth objective functions. We identify the weakly efficient points of the interval-valued vector optimization problems and the solutions of the weak vector variational inequalities under smooth and non-smooth LU-convexity assumptions.     

Image Space Analysis for Constrained Inverse Vector Variational Inequalities via Multiobjective Optimization

In this paper, we employ the image space analysis to study constrained inverse vector variational inequalities. First, sufficient and necessary optimality conditions for constrained inverse vector variational inequalities are established by using multiobjective optimization. A continuous nonlinear function is also introduced based on the oriented distance function and projection operator. This function is proven to be a weak separation function and a regular weak separation function under different parameter sets. Then, two alternative theorems are established, which lead directly to sufficient and necessary optimality conditions of the inverse vector variational inequalities. This provides a partial answer to an open question posed in Chen et al. (J Optim Theory Appl 166:460–479, 2015).     

Global bounds for the distance to solutions of co-coercive variational inequalities

We establish two global bounds measuring the distance from any vector to the solution set of the co-coercive variational inequality. To prove our results, we use the fact that the co-coercivity condition is sufficient for the (strong) monotonicity of (perturbed) fixed point and normal maps associated with variational inequalities.     

On the Gap Functions of Prevariational Inequalities

Gap functions play a crucial role in transforming a variational inequality problem into an optimization problem. Then, methods solving an optimization problem can be exploited for finding a solution of a variational inequality problem. It is known that the so-called prevariational inequality is closely related to some generalized convex functions, such as linear fractional functions. In this paper, gap functions for several kinds of prevariational inequalities are investigated. More specifically, prevariational inequalities, extended prevariational inequalities, and extended weak vector prevariational inequalities are considered. Furthermore, a class of gap functions for inequality constrained prevariational inequalities is investigated via a nonlinear Lagrangian.     

Vector variational inequalities and vector optimization problems on Hadamard manifolds

In this paper, we introduce a Minty type vector variational inequality, a Stampacchia type vector variational inequality, and the weak forms of them, which are all defined by means of subdifferentials on Hadamard manifolds. We also study the equivalent relations between the vector variational inequalities and nonsmooth convex vector optimization problems. By using the equivalent relations and an analogous to KKM lemma, we give some existence theorems for weakly efficient solutions of convex vector optimization problems under relaxed compact assumptions.