Enlargement of Monotone Operators with Applications to Variational Inequalities

Abstract

Given a point-to-set operator T, we introduce the operator Te defined as Te(x)= {u: 〈 u − v, x − y 〉 ≥ −e for all y ɛ Rn, v ɛ T(y)}. When T is maximal monotone Te inherits most properties of the e-subdifferential, e.g. it is bounded on bounded sets, Te(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of Te, 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 ɛ He(x), while the subproblems of the exact method are of the form 0 ɛ H(x). If ek is the coefficient used in the kth iteration and the ek'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.