Convergence analysis of an extended Auxiliary Problem Principle for solving variational inequalities

We study the Extended Proximal Auxiliary Problem Principle-method (EPAPP) by Kaplan and Tichatschke [17, 20] for solving variational inequalities whose operator is the sum of a maximal monotone and a continuous operator. As in comparable methods using Bregman distances the authors required that the operator of the considered variational inequality (here called main operator) is paramonotone (see [11] for definition and properties of paramonotone operators). The main purpose of this paper is to establish the convergence of the EPAPP-method without use of paramonotonicity. A sort of error summability criterion is used to allow inexact solutions of the auxiliary problems, and we also admit an outer approximation of the set-valued component of the operator. Due to the use of Bregman-like functions to construct the symmetric components of the auxiliary operators an interior point effect is provided, that is, – with a certain precaution – the auxiliary problems can be treated as unconstrained ones.