Convex Minimization over the Fixed Point Set of Demicontractive Mappings

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• 作者：Paul-Emile Maing¨¦
• 关键词：Demicontractive mapping ; Quasi ; nonexpansive mapping ; Nonexpansive mapping ; Fixed point problem ; Variational inequality problem ; Hybrid steepest descent method
• 刊名：Positivity
• 出版年：2008
• 出版时间：May, 2008
• 年：2008
• 卷：12
• 期：2
• 页码：269-280
• 全文大小：222.0 KB

文摘

This paper deals with a viscosity iteration method, in a real Hilbert space H{\mathcal H}, for minimizing a convex function Q:H ? \mathbbR\Theta:{\mathcal H} \rightarrow \mathbb{R} over the fixed point set of T:H ? HT:{\mathcal H} \rightarrow {\mathcal H}, a mapping in the class of demicontractive operators, including the classes of quasi-nonexpansive and strictly pseudocontractive operators. The considered algorithm is written as: x _n+1 := (1 − w) v _n + w T v _n , v _n := x _n − α _n Θ′(x _n ), where w ∈ (0,1) and (a_n) ¨¬ (0, 1)(\alpha_n) \subset (0, 1), Θ′ is the Gateaux derivative of Θ. Under classical conditions on T, Θ, Θ′ and the parameters, we prove that the sequence (x _n ) generated, with an arbitrary x₀ ? Hx_0 \in {\mathcal H}, by this scheme converges strongly to some element in Argmin _Fix(T) Θ.