ProX

Prox, short for proximal operator, is a concept in convex analysis and optimization. It is widely used in iterative methods to minimize functions that are convex but may be non-smooth.

For a convex function f: R^n -> R ∪ {+∞}, the proximal operator is defined by prox_f(x) = argmin_y { f(y) + (1/2) || y - x ||^2 }. Scaled versions prox_{λ f} use (1/(2λ)) in place of (1/2). The operator is firmly nonexpansive and 1-Lipschitz; Moreau decomposition relates prox_f to the convex conjugate f* via x = prox_f(x) + prox_{f*}(x).

Proximal operators enable splitting algorithms such as the proximal gradient method and ADMM, which minimize composite objective functions g(x) + h(x) by alternating gradient steps on the smooth part g and proximal updates on the non-smooth part h.

Common closed-form proximals include prox_{λ ||·||_1}(v) = soft-thresholding, prox_{δ_C}(v) as the projection onto a convex set C (where δ_C is the indicator function of C), and prox_{λ ||·||_*}(V) as singular value thresholding for the nuclear norm. For many functions, the proximal step requires an auxiliary optimization, but many standard regularizers have efficient forms.

Applications of proximal operators span sparse recovery, compressed sensing, image denoising, and regularized learning. In practice, the proximal operator is a fundamental building block in modern convex optimization and variational problems, enabling efficient handling of non-smooth terms in large-scale problems. See also proximal mapping, Moreau envelope, proximal gradient method, and ADMM.