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=1 n xi log xi We do not use in our analysis the assumption of paramonotonicity, which is standard in proving convergence of Bregman-based proximal methods. © by SIAM.