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Convergence of hybrid steepest-descent methods for variational inequalities. (English) Zbl 1045.49018
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.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
90C30Nonlinear programming
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