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Bregman functions and auxiliary problem principle

Abstract An extension of the auxiliary problem principle for solving variational inequalities with maximal monotone operators is studied. The main idea of this approach consists in an application of Bregman functions for constructing symmetric (regularizing) components of the auxiliary operators. This provides an interior point effect, i.e., auxiliary problems can be treated as unconstrained ones. However, up to now, such Bregman functions were proposed only for linearly constrained variational inequalities or in the case where the constraint set K is a ball. In this article, considering a slightly modified concept of a Bregman function, we introduce appropriate functions for a wide class of sets K including, in particular, convex sets described by a system of inequalities with affine and strictly convex functions. The convergence analysis allows that the auxiliary problems are solved inexactly by using a sort of error summability criterion.