On the proximity operator of the sum of two convex functions
Samir Adly, Loïc Bourdin, Fabien Caubet · HAL (Le Centre pour la Communication Scientifique Directe) · 2017
The main result of this paper provides an explicit decomposition of the proximity operator of the sum of two proper, lower semicontinuous and convex functions. For this purpose, we introduce a new operator, called f-proximity operator, generalizing the classical notion. After providing some properties and characterizations, we discuss the relations between the f-proximity operator and the classical Douglas-Rachford operator. In particular we provide a one-loop algorithm allowing to compute numerically this new operator, and thus the proximity operator of the sum of two proper, lower semicontinuous and convex functions. Finally we illustrate the usefulness of our main result in the context of sensitivity analysis of linear variational inequalities of second kind in a Hilbert space.