Proper closed convex extended-real functions on ${\mathbb{R}}^n$

Let $n\in \mathbb{N}$ and let $g:{\mathbb{R}}^n\to (-\mathrm{\infty },+\mathrm{\infty }]$. Its effective domain is $\mathrm{dom}g=\{x:g(x)<+\mathrm{\infty }\}$. The function is proper if $\mathrm{dom}g\mathrm{\ne }\mathrm{\varnothing }$; it is convex if $g((1-\theta )x+\theta y)\le (1-\theta )g(x)+\theta g(y)(x,y\in {\mathbb{R}}^n,0\le \theta \le 1),$ using the endpoint convention $0\cdot (+\mathrm{\infty })=0$ of Convex and strictly convex functionals on a convex subset of a real vector space. It is closed (equivalently, lower semicontinuous in the extended-real convex-analysis convention used here) if its epigraph $\mathrm{epi}g=\{(x,s)\in {\mathbb{R}}^n\times \mathbb{R}:g(x)\le s\}$ is closed in ${\mathbb{R}}^{n+1}$. The proximal-gradient items on this page assume $g$ is proper, closed and convex.

This page's closed-epigraph convention is explicit: it does not silently use an equivalence between sequential lower semicontinuity and topological closedness that would require an additional choice principle in the library's generic sequential convention. In finite dimensions this is the standard closed proper convex input to proximal algorithms.