proximaloperator
The proximal operator, or proximal mapping, of a proper lower semicontinuous convex function f on a Euclidean space is defined at a point v by prox_f(v) = argmin_x { f(x) + (1/2) ||x - v||^2 }. A scaled version is prox_{t f}(v) = argmin_x { f(x) + (1/(2t)) ||x - v||^2 } for t > 0. Equivalently, prox_{t f}(v) can be written as (I + t ∂f)^{-1}(v), linking it to the subdifferential of f.
Proximal operators have several key properties. If f is proper, convex, and l.s.c., prox_f is well-defined and
Relation to Moreau decomposition and subgradients is central to their theory. For any v, prox_f(v) satisfies
Common examples illustrate their utility. The proximal operator for the L1 norm, prox_{λ||·||1}(v), is the soft-thresholding
Applications include proximal gradient methods (Forward-Backward), ADMM, and other operator-splitting algorithms used in signal processing, statistics,