Abstract:

A class of bilevel variational inequalities (shortly (BVI)) with hierarchical
nesting structure is firstly introduced and investigated. The relationship between
(BVI) and some existing bilevel problems are presented. Subsequently, the existence
of solution and the behavior of solution sets to (BVI) and the lower level variational
inequality are discussed without coercivity. By using the penalty method, we transform
(BVI) into one-level variational inequality, and establish the equivalence between
(BVI) and the one-level variational inequality. A new iterative algorithm to
compute the approximate solutions of (BVI) is also suggested and analyzed. The
convergence of the iterative sequence generated by the proposed algorithm is derived
under some mild conditions. Finally, some relationships among (BVI), system
of variational inequalities and vector variational inequalities are also given.

Key words: Bilevel variational inequalities , Bilevel programs , System of
variational inequality , Vector variational inequality , Penalty method