Differential And Sensitivity Properties Of Gap Functions For Vector Variational Inequlities

It is well known that the variational inequalities plays an important role in modern nonlinear analysis. The theory of variational inequalities are widely applied to optimization , cybernetics, economic balance, etc. However, gap function is bridge between variational inequalities and optimization problems. The study of gap function become a hotspot gradually.In this paper, by usingΦ?contingent cone andΦ?contingent set, the first and second-order differential and sensitivity properties of a class of gap functions for (minty)vector variational-like equalities are obtained. These results generalize some known results in literature.The paper is divided into four chapters. In chapter 1, we review development and research issues related to this article. In chapter 2, we introduce some knowledge related this paper. In chapter 3, by usingΦ?contingent cone andΦ?contingent set, the first and second-oder contingent derivatives of set-valued maps G(x) involving vector variational-like inequatities are proved, and the differentinal and sensitivity properties of the gap function for vector variational-like inequatities are discussed, then some optimality conditions of solutions for vector variational-like inequatities are obtained. In chapter 4, we present the contingent derivatives of set-valued maps H(x) involving minty vector variational-like inequatities, and discuss the differentinal and sensitivity properties of the gap function for minty vector variational-like inequatities. Finally ,some optimality conditions of solutions for vector variational-like inequatities is given.