View source for Subdifferential

''of a convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908201.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908202.png" />, defined on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908203.png" /> that is in duality with a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908204.png" />''

The set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908205.png" /> defined by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908206.png" /></td> </tr></table>

For example, the subdifferential of the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908207.png" /> in a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908208.png" /> with dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908209.png" /> takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082010.png" /></td> </tr></table>

The subdifferential of a convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082011.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082012.png" /> is a convex set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082013.png" /> is continuous at this point, then the subdifferential is non-empty and compact in the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082014.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082016.png" /> are convex functions and if, at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082017.png" />, at least one of the functions is continuous, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082018.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082019.png" /> (the Moreau–Rockafellar theorem).

At he origin, the subdifferential of the support function of a convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082021.png" /> that is compact in the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082022.png" /> coincides with the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082023.png" /> itself. This expresses the duality between convex compact sets and convex closed homogeneous functions (see also [[Support function|Support function]]; [[Supergraph|Supergraph]]; [[Convex analysis|Convex analysis]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.T. Rockafellar,   "Convex analysis" , Princeton Univ. Press  (1970)  {{MR|0274683}} {{ZBL|0193.18401}} </TD></TR></table>



====Comments====
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082025.png" />-topology is the [[Weak topology|weak topology]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082026.png" /> defined by the family of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082028.png" />; this is the weakest topology which makes all the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082029.png" /> continuous.

The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082030.png" /> are called subgradients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082031.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082032.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Schneider,   "Boundary structure and curvature of convex bodies"  J. Tölke (ed.)  J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser  (1979)  pp. 13–59  {{MR|0568493}} {{ZBL|0427.52003}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Barbu,   Th. Precupanu,   "Convexity and optimization in Banach spaces" , Reidel  (1986)  pp. 101ff  {{MR|0860772}} {{ZBL|0594.49001}} </TD></TR></table>