# 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Convex Minimization over the Fixed Point Set of Demicontractive Mappings

, Volume 12 (2) – Jan 11, 2008
12 pages

/lp/springer_journal/convex-minimization-over-the-fixed-point-set-of-demicontractive-6cCpyPvxJJ
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-007-2066-x
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

PositivitySpringer Journals

Published: Jan 11, 2008

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