Existence of solution and algorithms for a class of bilevel variational inequalities with hierarchical nesting structure

*Correspondence: [email protected] 1School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China 2Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China Full list of author information is available at the end of the article Abstract In this paper we consider a class of bilevel variational inequalities with hierarchical nesting structure. We first of all get the existence of a solution for this problem by using the Himmelberg fixed point theorem. Then the uniqueness of the solution for an upper-level variational inequality is given under some mild conditions. By using gap functions of the upper-level and lower-level variational inequalities, we transform bilevel variational inequalities into a one-level variational inequality. Moreover, we propose two iterative algorithms to find the solutions of the bilevel variational inequalities. Finally, the convergence of the proposed algorithm is derived under some mild conditions.