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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 η-strongly monotone and κ-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 H. The sequence (x n ) is shown to converge in norm to the unique solution u * of the variational inequality

F(u * ),v-u * 0,forvC·

Applications to constrained pseudoinverses are included.


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