Subdifferential of a Real-Valued Function on a Convex Subset of $\mathbb{R}^n$

Let $n$ be a natural number with $1\le n$ and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points and the scalar multiple, and write $y-x$ and $q\cdot z$ for the difference and dot product of points of $\mathbb{R}^{n}$.

Let $C\subseteq\mathbb{R}^{n}$ be convex, let $f:C\to\mathbb{R}$, and let $x\in C$.

The subdifferential of $f$ at $x$ relative to $C$ (i) is the subset of $\mathbb{R}^{n}$

$$\partial_{C}f(x)=\bigl\{\,q\in\mathbb{R}^{n}\ :\ f(y)\ge f(x)+q\cdot(y-x)\ \text{ for every }y\in C\,\bigr\}.$$

An element $q$ of $\partial_{C}f(x)$ is called a subgradient of $f$ at $x$ relative to $C$. (ii)

When the set $C$ is clear from the context the subdifferential is also written $\partial f(x)$.