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Hybrid methods for a class of monotone variational inequalities. (English) Zbl 1176.90462
The paper deals with the study of certain hybrid methods for a special class of ill-posed monotone variational inequality problems in a Hilbert space setting, where the underlying operator is the complement of a nonexpansive mapping and the constraint set equals the set of fixed points of another nonexpansive mapping. Problems of this type include in particular monotone inclusions and convex optimization problems with a constraint set of the latter type. It is shown that both implicit and explicit iterative schemes are strongly convergent, where the employed regularization technique uses contractions of the nonexpansive operator in the variational inequality and allows a proof under considerably less restrictive conditions than they were needed for a related earlier approach by other authors. The paper is completed with an application to the class of hierarchical optimization problems where a proper lower semi-continuous convex function on a Hilbert space is minimized over the set of minimizers of another function of this type.

MSC:
90C25Convex programming
47H05Monotone operators (with respect to duality) and generalizations
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)