Let $C \subset \mathbb{R}^n$ be a convex set, let $f: C \to \mathbb{R}$ be a [convex function](/page/Convex%20Function), and let $x_0 \in C$. Suppose the subdifferential $\partial f(x_0)$ is understood relative to the domain $C$, meaning
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\begin{align*}
\partial f(x_0)=\{g \in \mathbb{R}^n : f(x) \ge f(x_0)+g\cdot (x-x_0) \text{ for every } x \in C\}.
\end{align*}
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Then $x_0$ is a global minimizer of $f$ on $C$ if and only if $0 \in \partial f(x_0)$.