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].

For the entire collection see [Zbl 0935.00054].

Reviewer: Stefan Nickel (Kaiserslautern)

##### 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 |