In this paper, we introduce a new iteration method, that is a combination of the hybrid steepestdescent method with two-step iteration method of linearization, for finding a minimizer of a convex differentiable functional over the fixed point set of a nonexpansive mapping in Hilbert spaces. Key words: Metric projection, Fixed point, Nonexpansive Mappings. | Nguyễn Bường và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 99(11): 161 - 164 AN ITERATION METHOD FOR A CONVEX OPTIMIZATION OVER THE FIXED POINT SET OF A NONEXPANSIVE MAPPING Nguyen Buong1*, Nguyen Duong Nguyen2 1 Vietnamese Academy of Science and Technology 2 Vietnam Foreign Trade University SUMMARY In this paper, we introduce a new iteration method, that is a combination of the hybrid steepestdescent method with two-step iteration method of linearization, for finding a minimizer of a convex differentiable functional over the fixed point set of a nonexpansive mapping in Hilbert spaces. Key words: Metric projection, Fixed point, Nonexpansive Mappings. AMS 2000 Mathematics Subject Classification (MSC): 41A65, 47H17, 47H20. INTRODUCTION* Let H be a real Hilbert space with the scalar product and norm denoted by the symbols and ||.||, respectively, and let C be a nonempty, closed and convex subset of H . Denote by PC ( x) the metric projection from x ∈ H onto C . Let T be a nonexpansive mapping on H , ., T : H → H and Tx − Ty ≤ x − y for all x, y ∈ H . We use Fix(T ) to denote the set of fixed points of T , ., Fix(T ) = {x ∈ H : x = Tx} . The convex optimization problem is to find a minimizer p*∈ C such that φ( p*) = inf φ( p) , p∈C () where φ is a convex functional on H . It is well known that when φ , in addition, is smooth with gradient ∇φ( x) , () is equivalent to the variational inequality problem: find a point p*∈ C such that ≤ 0, ∀p ∈ C . () It is also well known that, if ∇φ is L Lipschitz continuous and η -strongly monotone, ., ∇φ satisfies, respectively, the following conditions || ∇φ( x) − ∇φ( y ) || ≤ L || x − y || and ≥ η || x − y ||2 , where L and η are fixed positive numbers, then () has a unique solution and is equivalent to the fixed-point equation p* = PC ( p * − µ∇φ( p*)) , () * Email: nbuong@ where µ is a fixed positive constant. Usually, the metric projection PC in () is not easy to comput, due to the .
