Subdifferential

of a convex function at a point , defined on a space that is in duality with a space

The set in defined by:

For example, the subdifferential of the norm in a normed space with dual space takes the form

The subdifferential of a convex function at a point is a convex set. If is continuous at this point, then the subdifferential is non-empty and compact in the topology .

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 and are convex functions and if, at a point , at least one of the functions is continuous, then

for all (the Moreau–Rockafellar theorem).

At he origin, the subdifferential of the support function of a convex set in that is compact in the topology coincides with the set 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 -topology is the weak topology on defined by the family of semi-norms , ; this is the weakest topology which makes all the functionals continuous.

The elements are called subgradients of at .

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