Subdifferential

analysis optimisation convex-analysis

Definition

Subdifferential

Let $f:X\to \mathbb{R}$ be a convex function on a convex set $X\subseteq {\mathbb{R}}^d$, and let $x_0\in X$. The subdifferential of $f$ at $x_0$ is the set of all subgradients of $f$ at $x_0$:

$$\partial f(x_0)=\{g\in {\mathbb{R}}^d\mid f(x)\ge f(x_0)+g^{\top }(x-x_0)\text{ for all }x\in X\}.$$

Each $g\in \partial f(x_0)$ defines an affine function that touches $f$ at $x_0$ and stays below $f$ everywhere on $X$.

Interpretation

The subdifferential collects every valid supporting slope at a point. At a smooth point of a differentiable convex function, it contains only the gradient. At a corner, it may contain a whole interval or region of slopes.

For convex minimisation, it gives the optimality condition

$$0\in \partial f(x_0)⟺x_0\text{ is a global minimiser of }f.$$

This generalises the smooth condition $\nabla f(x_0)=0$ to convex functions with corners.