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

Abstract

Let C be a nonempty closed convex subset of a real Hilbert space H . Let f : C → H be a ρ -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 ∗ ∈ Ω , 〈 ( B ˜ − γ f ) x ∗ − ( I − B ) S x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ Ω , where Ω is the set of the solutions of the following variational inequality: x ∗ ∈ E P ( F , A ) , 〈 ( B ˜ − S ) x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ E P ( F , A ) , where E P ( F , A ) is the set of the solutions of the equilibrium problem of finding z ∈ C such that F ( z , y ) + 〈 A z , y − z 〉 ≥ 0 , ∀ y ∈ C . Assume Ω ≠ 0̸ . 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 E P ( F , A ) which solves the above triple-hierarchical constrained optimization problem. As a special case, we can find the minimum norm x ∗ ∈ E P ( F , A ) which solves the monotone variational inequality 〈 ( I − S ) x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ E P ( F , A ) .