# Subdifferential

(of $LaTeX: f$ at $LaTeX: x$). $LaTeX: \textstyle \partial f(x) = \{y: x \in \argmax\{vy - f(v): v \in X\}\}$ Also see conjugate function. If $LaTeX: f$ is convex and differentiable with gradient, $LaTeX: \textstyle \nabla f, \partial f(x) = \{\nabla f(x)\}.$

$LaTeX: \mbox{Example: } f(x) = | x |. \mbox{ Then, } \partial f(0) = \begin{bmatrix} -1 & 1 \end{bmatrix}.$

The subdiffenential is built on the concept of supporting hyperplane, generally used in convex analysis. When $LaTeX: f$ is differentiable in a deleted neighborhood of $LaTeX: x$ (but not necessarily at $LaTeX: x$), the B-subdifferential is the set of limit points:

$LaTeX: \partial_B f(x) = {d: \mbox{ there exists } \{x^k\} \to x \mbox{ and } \{\nabla f(x^k)\} \to d\}.$

If $LaTeX: f$ is continuously differentiable in a neighborhood of $LaTeX: x$ (including $LaTeX: x$), $LaTeX: \textstyle \partial_B f(x) = \nabla f(x).$ Otherwise, $LaTeX: \textstyle \partial_B f(x)$ is generally not a convex set. For example, if $LaTeX: \textstyle f(x) = | x |, \partial_B f(0) = \{-1, 1\}.$

The Clarke subdifferential is the convex hull of $LaTeX: \textstyle \partial_B f(x).$