Nonsymmetric proximal point algorithm with moving proximal centers for variational inequalities: Convergence analysis

Abstract

The classical proximal point algorithm (PPA) requires a metric proximal parameter, which is positive definite and symmetric, because it plays the role of the measurement matrix of a norm in the convergence proof. In this paper, our main goal is to show that the metric proximal parameter can be nonsymmetric if the proximal center is shifted appropriately. The resulting nonsymmetric PPA with moving proximal centers maintains the same implementation difficulty and convergence properties as the original PPA, while the nonsymmetry of the metric proximal parameter allows us to design highly customized algorithms that can effectively take advantage of the structures of the model under consideration. We present both the exact and inexact versions of the nonsymmetric PPA with moving proximal centers, and analyze their convergence including the estimate of their worst-case convergence rates measured by the iteration complexity under mild assumptions and their asymptotically linear convergence rates under stronger assumptions.