Error bounds for proximal point subproblems and associated inexact proximal point algorithms

We study various error measures for approximate solution of proximal point regularizations of the variational inequality problem, and of the closely related problem of finding a zero of a maximal monotone operator. A new merit function is proposed for proximal point subproblems associated with the latter. This merit function is based on Burachik–Iusem–Svaiter's concept of ϵ-enlargement of a maximal monotone operator. For variational inequalities, we establish a precise relationship between the regularized gap function, which is a natural error measure in this context, and our new merit function. Some error bounds are derived using both merit functions for the corresponding formulations of the proximal subproblem. We further use the regularized gap function to devise a new inexact proximal point algorithm for solving monotone variational inequalities. This inexact proximal point method preserves all the desirable global and local convergence properties of the classical exact/inexact method, while providing a constructive error tolerance criterion, suitable for further practical applications. The use of other tolerance rules is also discussed.