Subdifferential - Knowlpedia

A subdifferential of a convex function $f:\mathbb{R}^n\to(-\infty,+\infty]$ at a point $x$ in its effective domain is the set

\partial f(x) \;=\; \bigl\{g\in\mathbb{R}^n:\ f(z)\ge f(x)+\langle g,\,z-x\rangle \text{ for all } z\in\mathbb{R}^n\bigr\}.

Any $g\in\partial f(x)$ is called a subgradient of $f$ at $x$ .

Geometrically, $g\in\partial f(x)$ encodes a supporting hyperplane to the epigraph of $f$ at $(x,f(x))$ , with normal determined by $g$ (compare hyperplane). If $f$ is differentiable at $x$ (in the sense of the derivative / differentiable map), then $\partial f(x)=\{\nabla f(x)\}$ .

Examples

For $f(x)=|x|$ on $\mathbb{R}$ , one has $\partial f(0)=[-1,1]$ and $\partial f(x)=\{\mathrm{sign}(x)\}$ for $x\ne 0$ .
If $f=\delta_C$ is the indicator of a closed convex set $C$ , then $\partial f(x)$ consists of outward normal vectors to $C$ at $x$ (and is empty if $x\notin C$ ).