- André F. T. Martins, Noah A. Smith, E. Xing, P. Aguiar, Mário A. T. Figueiredo. 2011. A PROXIMITY OPERATORS AND MOREAU PROJECTIONS. Abstract: the function Mφ : R→ R̄ is called the Moreau envelope of φ, and the map proxφ : R→ R is the proximity operator of φ (Combettes and Wajs, 2006; Moreau, 1962). Proximity operators generalize Euclidean projectors: consider the case φ = ιC , where C ⊆ R is a convex set and ιC denotes its indicator (i.e., φ(x) = 0 if x ∈ C and +∞ otherwise). Then, proxφ is the Euclidean projector onto C and Mφ is the residual. Two other important examples of proximity operators follow:
