# Subdifferential

of a convex function $f: X \rightarrow \mathbf R$ at a point $x _ {0}$, defined on a space $X$ that is in duality with a space $Y$

The set in $Y$ defined by:

$$\partial f( x _ {0} ) = \{ {y \in Y } : {f( x) - f( x _ {0} ) \geq \langle y, x- x _ {0} \rangle \ \ \textrm{ for } \textrm{ all } x \in X } \} .$$

For example, the subdifferential of the norm $f( x) = \| x \|$ in a normed space $X$ with dual space $X ^ \star$ takes the form

$$\partial f( x) = \left \{ \begin{array}{lttl} \{ {x ^ \star \in X ^ \star } : {\langle x ^ \star , x\rangle = \| x \| ,\ \| x ^ \star \| = 1 } \} &\{ {x ^ \star } : {\| x ^ \star \| = 1 } \} & \textrm{ if } x \neq 0, & \textrm{ if } x = 0. \\ \end{array} \right .$$

The subdifferential of a convex function $f$ at a point $x _ {0}$ is a convex set. If $f$ is continuous at this point, then the subdifferential is non-empty and compact in the topology $\sigma ( Y, X)$.

The role of the subdifferential of a convex function is similar to that of the derivative in classical analysis. Theorems for subdifferentials that are analogous to theorems for derivatives are valid. For example, if $f _ {1}$ and $f _ {2}$ are convex functions and if, at a point $\overline{x}\; \in ( \mathop{\rm Dom} f _ {1} ) \cap ( \mathop{\rm Dom} f _ {2} )$, at least one of the functions is continuous, then

$$\partial f _ {1} ( x) + \partial f _ {2} ( x) = \partial ( f _ {1} + f _ {2} )( x)$$

for all $x$( the Moreau–Rockafellar theorem).

At the origin, the subdifferential of the support function of a convex set $A$ in $X$ that is compact in the topology $\sigma ( Y, X)$ coincides with the set $A$ itself. This expresses the duality between convex compact sets and convex closed homogeneous functions (see also Support function; Supergraph; Convex analysis).

#### References

 [1] R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) MR0274683 Zbl 0193.18401

The $\sigma ( X, Y)$- topology is the weak topology on $X$ defined by the family of semi-norms $p _ {y} ( x) = | \langle x, y \rangle |$, $y \in Y$; this is the weakest topology which makes all the functionals $x \rightarrow \langle x, y \rangle$ continuous.
The elements $x ^ \star \in \partial f( x)$ are called subgradients of $f$ at $x$.