From Fixed Point Regularization to Constraint Decomposition in Variational Inequalities

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.Key wordsconstraint decompositionfixed pointiterationnonexpansiveregularizationvariational inequalities