Communications in Mathematical Sciences

Volume 3 (2005)

Number 4

Slope and $G$-set characterization of set-valued functions and applications to non-differentiable optimization problems

Pages: 479 – 492



Stanley Osher

Otmar Scherzer

Wotao Yin


In this paper we derive a generalizing concept of $G$-norms, which we call $G$-sets, which is used to characterize minimizers of non-differentiable regularization functionals. Moreover, the concept is closely related to the defnition 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.

2010 Mathematics Subject Classification

Primary 49J40. Secondary 65J20.

Published 1 January 2005