Abstract
Extending the Tikhonov regularization method to the fixed point problem for a nonexpansive self mapping P on a real Hilbert space H, generates a family of fixed points u r of strongly nonexpansive self mappings P r on H with positive parameter r tending to 0. If the fixed point set C of P is nonempty, then u r converges strongly to u * the unique solution to some monotone variational inequality on the (closed convex) subset C. The iteration method suitably combined with this regularization generates a sequence that converges strongly to u *. When C is a priori defined by finitely many convex inequality constraints, expressing C as the fixed point set of a suitable nonexpansive mapping and applying the above method lead to an iterative scheme in which each step is decomposed in finitely many successive or parallel projections or proximal computations.
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Lemaire, B., Ould Ahmed Salem, C. (2000). From Fixed Point Regularization to Constraint Decomposition in Variational Inequalities. In: Nguyen, V.H., Strodiot, JJ., Tossings, P. (eds) Optimization. Lecture Notes in Economics and Mathematical Systems, vol 481. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57014-8_17
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DOI: https://doi.org/10.1007/978-3-642-57014-8_17
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