An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions

We present a new Bregman-function-based algorithm which is a modification o f the generalized proximal point method for solving the variational inequal ity problem with a maximal monotone operator. The principal advantage of th e presented algorithm is that it allows a more constructive error tolerance criterion in solving the proximal point subproblems. Furthermore, we elimi nate the assumption of pseudomonotonicity which was, until now, standard in proving convergence for paramonotone operators. Thus we obtain a convergen ce result which is new even for exact generalized proximal point methods. F inally, we present some new results on the theory of Bregman functions. For example, we show that the standard assumption of convergence consistency i s a consequence of the other properties of Bregman functions, and is theref ore superfluous.