From Fixed Point Regularization to Constraint Decomposition in Variational Inequalities

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.