Coupling the Auxiliary Problem Principle and Epiconvergence Theory to Solve General Variational Inequalities

Many algorithms for solving variational inequality problems can be derived from the auxiliary problem principle introduced several years ago by Cohen. In recent years, the convergence of these algorithms has been established under weaker and weaker monotonicity assumptions: strong (pseudo) monotonicity has been replaced by the (pseudo) Dunn property. Moreover, well-suited assumptions have given rise to local versions of these results.In this paper, we combine the auxiliary problem principle with epiconvergence theory to present and study a basic family of perturbed methods for solving general variational inequalities. For example, this framework allows us to consider barrier functions and interior approximations of feasible domains. Our aim is to emphasize the global or local assumptions to be satisfied by the perturbed functions in order to derive convergence results similar to those without perturbations. In particular, we generalize previous results obtained by Makler-Scheimberg et al.

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