Auxiliary problem principle extended to variational inequalities

The auxiliary problem principle has been proposed by the author as a framework to describe and analyze iterative optimization algorithms such as gradient or subgradient as well as decomposition/coordination algorithms (Refs. 1–3). In this paper, we extend this approach to the computation of solutions to variational inequalities. In the case of single-valued operators, this may as well be considered as an extension of ideas already found in the literature (Ref. 4) to the case of nonlinear (but still strongly monotone) operators. The case of multivalued operators is also investigated.This research has been supported by the Centre National de la Recherche Scientifique (ATP Complex Technological Systems) and by the Centre National d'Études des Télécommunications (Contract No. 83.5B.034.PAA). It has been conducted partly while the author was visiting the Department of Electrical Engineering of the Pontificia Catholic University of Rio de Janeiro in July–August 1984 under the CAPES/COFECUB scientific exchange program.