Abstract
In this paper, we introduce an iterative scheme with inertial effect using Mann iterative scheme and gradient-projection for solving the bilevel variational inequality problem over the intersection of the set of common fixed points of a finite number of nonexpansive mappings and the set of solution points of the constrained optimization problem. Under some mild conditions we obtain strong convergence of the proposed algorithm. Two examples of the proposed bilevel variational inequality problem are also shown through numerical results.
Highlights
Bilevel problem is defined as a mathematical program, where the problem contains another problem as a constraint
Denotes the bilevel variational inequality problem over the intersection of the set of common solution points of a finite number of equilibrium problems and the set of solution points of the constrained minimization problem given by find x ∈ Ω such that h F ( x ), x − xi ≥ 0, ∀ x ∈ Ω, where Ω is the solution-set of find x ∗ ∈ C such that f ( x ∗ ) = min f ( x ) and x ∗ ∈
Suppose that BVIPO-VM denotes the bilevel variational inequality problem over the intersection of the set of common solution points of finite number of variational inequality problems and the set of solution points of the constrained minimization problem given by find x ∈ Ω such that h F ( x ), x − xi ≥ 0, ∀ x ∈ Ω, where Ω is the solution-set of find x ∗ ∈ C such that f ( x ∗ ) = min f ( x ) and x ∗ ∈
Summary
, Anteneh Getachew Gebrie 2 and KMUTTFixed Point Research Laboratory, SCL 802 Fixed Point Laboratory & Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand Department of Mathematics, College of Computational and Natural Science, Debre Berhan University, P.O. Box 445, Debre Berhan, Ethiopia Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Received: 5 August 2019; Accepted: 2 September 2019; Published: 11 September 2019
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