Subgradient

A vector that defines an affine global lower bound to a convex function at a point.

Subgradient

A subgradient of a convex function $f$ at a point $x\in\operatorname{dom} f$ (see domain ) is a vector $g\in\mathbb{R}^n$ such that

f(y)\ge f(x)+\langle g,\,y-x\rangle \quad \text{for all } y\in\mathbb{R}^n.

Subgradients generalize derivatives: when $f$ is differentiable at $x$ (see derivative ), the unique subgradient is $\nabla f(x)$ . The set of all subgradients at $x$ is the subdifferential $\partial f(x)$ , and each subgradient induces a supporting hyperplane to the epigraph of $f$ .

Examples:

For $f(x)=|x|$ on $\mathbb{R}$ , $\partial f(0)=[-1,1]$ , so any $g\in[-1,1]$ is a subgradient at $0$ .
For $f(x)=\max\{x,0\}$ on $\mathbb{R}$ , $\partial f(0)=[0,1]$ , while $\partial f(x)=\{1\}$ for $x>0$ and $\partial f(x)=\{0\}$ for $x<0$ .