Slope and G-Value characterization of Set-valued Functions and Applications to Non-differentiable Optimization Problems

In this paper we derive a generalizing concept of G-norms, which we call G-values, which is used to characterize minimizers of nondifferentiable regularization functionals. Moreover, the concept is closely related to the definition of slopes as published in a recent book by Ambrosio, Gigli, Savaré. A paradigm of regularization models fitting in this framework is robust bounded variation regularization. Two essential properties of this regularization technique are documented in the literature and it is shown that these properties can also be achieved with metric regularization techniques, which also have the advantage that they attain a unique minimizers.