Enlargement of monotone operators with applications to variational inequalities

Given a point-to-set operator T, we introduce the operator T-epsilon defined as T-epsilon(x) = {u: (u - v, x - y) greater than or equal to -epsilon for all y is an element of R-n,v is an element of T(y)}. When T is maximal monotone T-epsilon inherits most properties of the E-subdifferential, e.g. it is bounded on bounded sets, T-epsilon(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T-epsilon, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 is an element of H-epsilon(x), while the subproblems of the exact method are of the form 0 is an element of H(x). If epsilon(k) is the coefficient used in the kth iteration and the epsilon(k)'s are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.