A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem

Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C -> H be a rho-contraction. Let S : C -> C be a nonexpansive mapping. Let B, B : H -> H be two strongly positive bounded linear operators. Consider the triple-hierarchical constrained optimization problem of finding a point x* such that x* is an element of Omega, <((B) over tilde - gamma f)x* - (I - B)Sx*, x - x*> >= 0, for all x is an element of Omega, where Omega is the set of the solutions of the following variational inequality: x* is an element of EP(F, A), <((B) over tilde - S)x*, x - x*> >= 0, for all x is an element of EP(F, A), where EP(F, A) is the set of the solutions of the equilibrium problem of finding z is an element of C such that F (z, y) + < Az, y - z > >= 0, for all y is an element of C. Assume Omega not equal circle divide. The purpose of this paper is the solving of the above triple-hierarchical constrained optimization problem. For this purpose, we first introduce an implicit double-net algorithm. Consequently, we prove that our algorithm converges hierarchically to some element in EP(F, A) which solves the above triple-hierarchical constrained optimization problem. As a special case, we can find the minimum norm x* is an element of EP( F, A) which solves the monotone variational inequality <(I - S)x*, x - x*> >= 0, for all x is an element of EP(F, A). (C) 2011 Elsevier Ltd. All rights reserved.