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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 \[ \langle F(u^{\ast}), v - u^{\ast}\rangle \geq 0,\qquad \text{for } v \in C. \] Applications to constrained pseudoinverses are included.

MSC:
49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
90C30 Nonlinear programming
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