Inexact Proximal Point Methods for Variational Inequality Problems

We present a new family of proximal point methods for solving monotone variational inequalities. Our algorithm has a relative error tolerance criterion in solving the proximal subproblems. Our convergence analysis covers a wide family of regularization functions, including double regularizations recently introduced by Silva, Eckstein, and Humes, Jr. [SIAM J. Optim., 12 (2001), pp. 238–261] and the Bregman distance induced by h(x)=∑ᵢ₌₁ⁿxᵢlog xᵢ. We do not use in our analysis the assumption of paramonotonicity, which is standard in proving convergence of Bregman-based proximal methods.