Convex Minimization over the Fixed Point Set of Demicontractive Mappings

Convex Minimization over the Fixed Point Set of Demicontractive Mappings This paper deals with a viscosity iteration method, in a real Hilbert space $${\mathcal H}$$ , for minimizing a convex function $$\Theta:{\mathcal H} \rightarrow \mathbb{R}$$ over the fixed point set of $$T:{\mathcal H} \rightarrow {\mathcal H}$$ , 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 $$(\alpha_n) \subset (0, 1)$$ , Θ′ is the Gâteaux derivative of Θ. Under classical conditions on T, Θ, Θ′ and the parameters, we prove that the sequence (x n ) generated, with an arbitrary $$x_0 \in {\mathcal H}$$ , by this scheme converges strongly to some element in Argmin Fix(T) Θ. Positivity Springer Journals

Convex Minimization over the Fixed Point Set of Demicontractive Mappings

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SP Birkhäuser Verlag Basel
Copyright © 2008 by Springer Science + Business Media B.V.
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
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