An Implicit Extragradient Method for Hierarchical Variational Inequalities. (28th November 2010) - Yao, Yonghong; Liou, Yeong Cheng
Abstract : As a well-known numerical method, the extragradient method solves numerically the variational inequalityV I ( C, A ) of findingu ∈ C such that〈 A u, v − u 〉 ≥ 0, for allv ∈ C . In this paper, we devote to solve the following hierarchical variational inequalityH V I ( C, A, f ) Findx ˜ ∈ V I ( C, A ) such that〈 ( I − f ) x ˜, x − x ˜ 〉 ≥ 0, for allx ∈ V I ( C, A ) . We first suggest and analyze an implicit extragradient method for solving the hierarchical variational inequalityH V I ( C, A, f ) . It is shown that the net defined by the suggested implicit extragradient method converges strongly to the unique solution ofH V I ( C, A, f ) in Hilbert spaces. As a special case, we obtain the minimum norm solution of the variational inequalityV I ( C, A ) .