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

Comments

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 $.

References

[a1]	R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 13–59 MR0568493 Zbl 0427.52003
[a2]	V. Barbu, Th. Precupanu, "Convexity and optimization in Banach spaces" , Reidel (1986) pp. 101ff MR0860772 Zbl 0594.49001