Self-dual Regularization of Monotone Operators via the Resolvent Average

Abstract

We give two self-dual regularizations of maximal monotone operators on Hilbert spaces. These regularizations and their set-valued inverses are strongly monotone, single-valued, and Lipschitz with full domain. Moreover, these regularizations graphically converge to the original monotone operator. If a maximal monotone operator has nonempty zeros, these self-dual regularizations can be used to find its least norm solution. When the maximal monotone operator is the subdifferential of a proper lower semicontinuous convex function with nonempty minimizers, this translates to finding the least norm minimizer.