18.2 Subgradients
Definition 18.2.1 (Subdifferential).label Let $E$ be a locally convex space over $\real$, $f: E \to (-\infty, \infty]$, $x \in \bracs{f < \infty}$, and $\phi \in E^{*}$, then $\phi$ is a subgradient of $f$ if for any $h \in E$,
The set $\partial f(x)$ of all subgradients of $f$ at $x$ is the subdifferential of $f$ at $x$, and the mapping $\partial f$ is the subdifferential of $f$.
Proposition 18.2.2.label Let $E$ be a locally convex space over $\real$, $f, g: E \to (-\infty, \infty]$, and $x \in \bracs{f, g < \infty}$, then:
- (1)
$\partial f(x)$ is a weak*-closed convex set.
- (2)
$\partial(f + g)(x) \supset \partial f(x) + \partial g(x)$.
Proposition 18.2.3.label Let $E$ be a locally convex space over $\real$, $f: E \to (-\infty, \infty]$ be convex, and $x \in \bracs{f < \infty}$, then:
- (1)
For any $\phi \in \partial f(x)$ and $h \in E$,
\[\dpn{h, \phi}{E}\le |f(x + h) - f(x)|\] - (2)
If $f$ is continuous at $x$, then $\partial f(x)$ is non-empty, equicontinuous, and weak*-compact.
Proof, [Proposition 4.6, Cla13]. (1): By the subgradient inequality,
(2): Since $f$ is continuous at $x$, $\text{epi}(f)^{o} \ne \emptyset$. Since $(x, f(x)) \not\in \text{epi}(f)^{o}$, by the Hahn-Banach Theorem, there exists $\phi \in E^{*}$ and $\lambda \in \real$ such that for any $(y, \alpha) \in \text{epi}(f)^{o}$,
By continuity of $f$ at $x$, there exists $\alpha_{0} \in \real$ such that $\bracs{x}\times [\alpha_{0}, \infty) \subset \text{epi}(f)^{o}$. Thus $\lambda < 0$.
By Proposition 12.1.6, $\text{epi}(f) \subset \ol{\text{epi}(f)^o}$. Therefore for any $(y, \alpha) \in \text{epi}(f)$,
so $\phi \in \partial f(x) \ne \emptyset$.
(3): By Proposition 18.2.2 and the Banach-Alaoglu Theorem.$\square$
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