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

Fixexd point theory and applications

Jia-wei Chen¹  Xiaoke Zhao²  Gaoxi Li³  Zhongping Wan³ 

[1] Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, China;School of Mathematics and Statistics, Southwest University, Chongqing, China;School of Mathematics and Statistics, Wuhan University, Wuhan, China

关键词: bilevel variational inequalities;    Himmelberg fixed point theorem;    existence;    gap function;    iterative algorithm;    convergence;    90C33;    49J40;    90C30;   

DOI  :  10.1186/s13663-016-0524-5

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

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.

【 授权许可】

CC BY