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 single-valued, and it is firmly nonexpansive (and hence 1-Lipschitz). These operators serve as resolvents of subgradient maps and are central to many splitting algorithms that handle non-smooth terms.

Relation to Moreau decomposition and subgradients is central to their theory. For any v, prox_f(v) satisfies v − prox_f(v) ∈ ∂f(prox_f(v)). The Moreau decomposition states that v = prox_f(v) + prox_{f}(v), where f is the convex conjugate of f. Proximal calculus provides rules for compositions with linear operators and scaling, enabling efficient computation in many settings.

Common examples illustrate their utility. The proximal operator for the L1 norm, prox_{λ||·||1}(v), is the soft-thresholding operator. The proximal operator of the indicator function of a convex set C is the projection onto C. For (λ/2)||x||^2, prox_{f}(v) = v / (1 + λ).

Applications include proximal gradient methods (Forward-Backward), ADMM, and other operator-splitting algorithms used in signal processing, statistics, and machine learning.