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Zbl 1045.49018
Xu, H.K.; Kim, T.H.
Convergence of hybrid steepest-descent methods for variational inequalities.
(English)
[J] J. Optimization Theory Appl. 119, No. 1, 185-201 (2003). ISSN 0022-3239; ISSN 1573-2878/e

Summary: Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is $\eta$-strongly monotone and $\kappa$-Lipschitzian on a nonempty closed convex subset $C$ of $H$. Assume also that $C$ is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on $H$. We devise an iterative algorithm which generates a sequence ($x_n$) from an arbitrary initial point $x_0 \in H$. The sequence ($x_n$) is shown to converge in norm to the unique solution $u^{\ast}$ of the variational inequality $$\langle F(u^{\ast}), v - u^{\ast}\rangle \ge 0,\qquad \text{for } v \in C.$$ Applications to constrained pseudoinverses are included.
MSC 2000:
*49J40 Variational methods including variational inequalities
47J20 Inequalities involving nonlinear operators
90C30 Nonlinear programming

Keywords: iterative algorithms; hybrid steepest-descent methods; convergence; nonexpansive mappings; Hilbert space; constrained pseudoinverses

Cited in: Zbl 1158.65047 Zbl 1092.49013

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