Abstract
Abstract In this paper, we introduce and analyze a relaxed iterative algorithm by virtue of Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, the regularization method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many variational inclusions, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others. MSC:49J30, 47H09, 47J20, 49M05.
Highlights
Let C be a nonempty closed convex subset of a real Hilbert space H and PC be the metric projection of H onto C
We introduce and study the following triple hierarchical variational inequality (THVI) with constraints of mixed equilibria, variational inequalities, and convex minimization problem
I(Bi, Ri) of the fixed point set of infinitely many nonexpansive mappings {Tn}∞ n=, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions and the set of minimizers of convex minimization problem (CMP)
Summary
Let C be a nonempty closed convex subset of a real Hilbert space H and PC be the metric projection of H onto C. It is well known that, if A is a strongly monotone and Lipschitz continuous mapping on C, VIP Let Tr(Θ,φ) : H → C be the solution set of the auxiliary mixed equilibrium problem, that is, for each x ∈ H, Tr(Θ,φ)(x) :=. Consider the convex minimization problem (CMP) of minimizing f over the constraint set C, minimize f (x) : x ∈ C It and its special cases were considered and studied in [ , , – ]. Let V : H → H be an l-Lipschitzian mapping with constant l
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