Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems

We consider the following hierarchical equilibrium problem and variational inequality problem abbreviated as HEVP : find a point x∗ ∈ EP F, B such that 〈Ax∗, x − x∗〉 ≥ 0, for all x ∈ EP F, B , where A, B are two monotone operators and EP F, B is the solution of the equilibrium problem of finding z ∈ C such that F z, y 〈Bz, y − z〉 ≥ 0, for all y ∈ C. We note that the problem HEVP includes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solving HEVP , we propose a double-net algorithm which generates a net {xs,t}. We prove that the net {xs,t} hierarchically converges to the solution of HEVP ; that is, for each fixed t ∈ 0, 1 , the net {xs,t} converges in norm, as s → 0, to a solution xt ∈ EP F, B of the equilibrium problem, and as t → 0, the net {xt} converges in norm to the unique solution x∗ of HEVP .