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A perturbed and inexact version of the auxiliary problem method for solving general variational inequalities with a multivalued operator. (English) Zbl 1017.90078
Nguyen, Van Hien (ed.) et al., Optimization. Proceedings of the 9th Belgian-French-German conference, Namur, Belgium, September 7-11, 1998. Berlin: Springer. Lect. Notes Econ. Math. Syst. 481, 396-418 (2000).
In 1988 Cohen developed the auxiliary problem method for solving general variational inequalities with a multivalued maximal monotone operator in a Hilbert space. In the literature this method was combined with perturbation techniques to get a basic family of perturbation methods. However, the influence of a variational perturbation on the convergence of the auxiliary problem method has only been studied in the case of a single-valued operator. In this paper the authors present convergence results for the perturbed auxiliary problem scheme when the operator is multivalued. Especially, the concept of $$\varepsilon$$-enlargement of a maximal monotone operator is looked at. Under classical assumptions, the sequence generated by this scheme is shown to be bounded and weakly convergent to a solution of the problem. Also additional conditions are stated under which even strong convergence can be obtained. For nondifferentiable convex optimization the $$\varepsilon$$-enlargement is replaced by the $$\varepsilon$$-subdifferential. In the nonperturbed situation the presented scheme reduces to the projected inexact subgradient procedure.
For the entire collection see [Zbl 0935.00054].

##### MSC:
 90C25 Convex programming 49J40 Variational inequalities 47J20 Variational and other types of inequalities involving nonlinear operators (general) 58E35 Variational inequalities (global problems) in infinite-dimensional spaces