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 Abstract and Applied Analysis , 2013, DOI: 10.1155/2013/492103 Abstract: This paper is concerned with gap functions of generalized vector variational inequalities (GVVI). By using scalarization approach, scalar-valued variational inequalities of (GVVI) are introduced. Some relationships between the solutions of (GVVI) and its scalarized versions are established. Then, by using these relationships and some mild conditions, scalar-valued gap functions for (GVVI) are established. 1. Introduction The concept of vector variational inequalities was firstly introduced by Giannessi [1] in a finite-dimensional space. Since then, extensive study of vector variational inequalities has been done by many authors in finite- or infinite-dimensional spaces under generalized monotonicity and convexity assumptions. See [2–10] and the references therein. Among solution approaches for vector variational inequalities, scalarization is one of the most analyzed topics at least from the computational point of view; see [8–10]. Gap functions are very useful for solving vector variational inequalities. One advantage of the introduction of gap functions in vector variational inequalities is that vector variational inequalities can be transformed into optimization problems. Then, powerful optimization solution methods and algorithms can be applied for finding solutions of vector variational inequalities. Recently, some authors have investigated the gap functions for vector variational inequalities. Yang and Yao [11] introduced gap functions and established necessary and sufficient conditions for the existence of a solution of vector variational inequalities. Chen et al. [12] extended the theory of gap function for scalar variational inequalities to the case of vector variational inequalities. They also obtained the set-valued gap functions for vector variational inequalities. Li and Chen [13] introduced set-valued gap functions for a vector variational inequality and obtained some related properties. Li et al. [14] investigated differential and sensitivity properties of set-valued gap functions for vector variational inequalities and weak vector variational inequalities. Meng and Li？？[15] also investigated the differential and sensitivity properties of set-valued gap functions for Minty vector variational inequalities and Minty weak vector variational inequalities. The purpose of this paper is to define a single variable gap function for generalized vector variational inequalities by using the scalarization approach. To this end, we first transform the generalized vector variational inequality into an equivalent scalar variational inequality by using
 Applied Mathematics (AM) , 2017, DOI: 10.4236/am.2017.812135 Abstract: One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of gap function. The gap functions are useful tools in deriving the error bounds which provide an estimated distance between a specific point and the exact solution of variational inequality problem. In this paper, we follow a similar approach for set-valued vector quasi variational inequality problems and define the gap functions based on scalarization scheme as well as the one with no scalar parameter. The error bounds results are obtained under fixed point symmetric and locally α-Holder assumptions on the set-valued map describing the domain of solution space of a set-valued vector quasi variational inequality problem.
 - , 2017, DOI: 10.6040/j.issn.1671-9352.0.2016.469 Abstract: 摘要： 利用线性标量化方法构造广义向量变分不等式的间隙函数,并利用广义f-投影算子的性质验证了正则间隙函数。在广义强伪单调的条件下得到了误差界结论。Abstract: Gap functions for generalized vector variational inequalities were established via linear scalarization approaches. By using some properties of generalized f-projection operators, the regularized gap function was verified. With the condition of the generalized strong pseudomonotonicity, error bounds were obtained
 - , 2017, DOI: 10.13718/j.cnki.xdzk.2017.08.015 Abstract: 鉴于间隙函数与误差界在优化方法中有重要的作用，特别地，误差界能刻画可行点和变分不等式解集之间的有效估计距离.利用像空间分析法，构造了带锥约束变分不等式的间隙函数.然后，利用此间隙函数，得到了带锥约束变分不等式的误差界.The gap function and the error bound play an important role in optimization methods and the error bound, especially, can characterize the effective estimated distance between a feasible point and the solution set of variational inequalities. In this article, by using the image space analysis, gap functions for a class of variational inequalities with cone constraints are proposed. Moreover, error bounds, which provide an effective estimated distance between a feasible point and the solution set, for the variational inequalities are established via the gap functions
 Applied Mathematics (AM) , 2012, DOI: 10.4236/am.2012.31010 Abstract: In this paper, we define the concepts of (η,h)-quasi pseudo-monotone operators on compact set in locally convex Hausdorff topological vector spaces and prove the existence results of solutions for a class of generalized quasi variational type inequalities in locally convex Hausdorff topological vector spaces.
 Journal of Inequalities and Applications , 2006, Abstract: This paper is devoted to the study of behaviour and sensitivity analysis of the solution for a class of parametric problem of completely generalized quasi-variational inequalities.
 Journal of Inequalities and Applications , 2006, Abstract: This paper is devoted to the study of behaviour and sensitivity analysis of the solution for a class of parametric problem of completely generalized quasi-variational inequalities.
 Mathematics , 2008, DOI: 10.1214/09-AOP496 Abstract: We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence, this suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs.
 Giandomenico Mastroeni Le Matematiche , 1994, Abstract: After revisiting the well-known relationship with the minimax theory, some duality results for constrained extremum problems are related to variational inequalities. In particular, the connection with saddle point conditions and gap functions associated to the variational inequality are analysed.
 American Journal of Operations Research (AJOR) , 2013, DOI: 10.4236/ajor.2013.33029 Abstract: In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in Hausdorff topological vector spaces, which include the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector quasi-variational inequalities, and several other systems as special cases. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the g-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for the system of generalized vector quasi-variational-like inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.
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