Subdifferential

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(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).


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