Averaging Schemes for Variational Inequalities and Systems of Equations

We study averaging methods for solving variational inequalities whose underlying maps are nonexpansive and for solving systems of (asymmetric) equations. Our goal is to establish global convergence results using weaker assumptions than are traditional in the literature. We examine averaging schemes for relaxation algorithms and for their specialization as projection and linearization methods and as Cohen's auxiliary framework. For solving systems of equations, we consider averaging for a general class of methods that includes, as a special case, a generalized steepest descent method. We also develop a new interpretation of a norm condition typically used for establishing convergence of relaxation schemes, by associating it with a strong-f-monotonicity condition.