Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions
This paper applies a scalarization approach of Konnov for studying strong and weak vector variational inequalities with set valued mappings. More precisely, let \(X\) be a real Banach space, \((Y,P)\) an ordered Banach space induced by a closed convex and pointed cone \(P\). Let \(L(X,Y)\) be the set of continuous linear mappings from \(X\) to \(Y\), and let \(K\subset X\) be nonempty, closed and convex. Given \(T:X\to 2^{L(X,Y)}\) a set-valued mapping, the authors consider two kinds of vector variational inequalities as follows. The \textit{Strong Vector Variational Inequality}: Find \(x^*\in K\) such that \[ \exists\,\,t\in T(x^*) : \langle t^*,y-x^* \rangle \not< 0,\;\;\forall\;y\in K. \] The \textit{Weak Vector Variational Inequality}: Find \(x^*\in K\) such that \[ \forall\;y\in K,\,\exists\,\,t\in T(x^*) : \langle t^*,y-x^* \rangle \not< 0. \] The authors consider several kinds of strong and weak variational inequalities and their gap functions. Under certain conditions, they prove the equivalence between the scalar and vector version of these problems and analyze the relations among their corresponding gap functions. They apply their results to derive error bounds for gap functions. Finally, they obtain some existence results of global error bounds for gap functions and derive characterizations of global and local error bounds fo these gap functions.
scientific article; zbMATH DE number 5770313
scientific article