Differential And Sensitivity Properties Of Gap Functions For Mixed Vector Variational Inequlities

It is well known that the variational inequalities plays an important role in modernnonlinear analysis. The theory of variational inequalities are widely applied tooptimization, cybernetics, economic balance, etc. However gap function is bridgebetween variational inequalities and optimization problems.In this paper, by suitable conditions, differential and sensitivity properties of a classof gap functions for mixed vector variational equalities are obtained. Relationshipbetween their Contingent derivatives are discussed. A formula computing contingentderivative of the gap functions is established. then some optimality conditions ofsolutions for mixed vector variational inequalities are obtained.By usingΦ contingent cone,we present the contingent derivatives of set-valued maps H(x)involving minty mixed vector variational inequalities, and discuss the differentinal andsensitivity properties of the gap function for minty mixed vector variationalinequalities. Relationship between their Contingent derivatives are discussed.Anexplicit expression of the contingent derivative for the class of set-valued maps isestablished.Finally, some optimality conditions of solutions for mixed vectorvariational inequalities is given.These results generalize some known results inliterature.The paper is divided into four chapters.In chapter1, we review development and research issues related to this article.In chapter2, we introduce some knowledge related this paper.In chapter3,By suitable conditions,contingent derivatives of set-valued maps G(x)involving mixed vector variational inequalities are proved, and the differential and sensitivity properties of the gap function for mixed vector variational inequalities arediscussed, Relationship between their Contingent derivatives are discussed. then someoptimality conditions of solutions for mixed vector variational inequalities areobtained.In chapter4, by usingΦ contingent cone,we present the contingent derivativesof set-valued maps H(x) involving minty mixed vector variational inequalities, anddiscuss the differentinal and sensitivity properties of the gap function for minty mixedvector variational inequalities. Relationship between their Contingent derivatives arediscussed.An explicit expression of the contingent derivative for the class of set-valuedmaps is established. Finally, some optimality conditions of solutions for mixed vectorvariational inequalities is given.