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''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" />''
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The set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s0908205.png" /> defined by:
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{{TEX|auto}}
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{{TEX|done}}
  
<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>
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''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 $''
  
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
+
The set in $  Y $
 +
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/s09082010.png" /></td> </tr></table>
+
$$
 +
\partial  f( x _ {0} )  = \{ {y \in Y } : {f( x) - f( x _ {0} ) \geq  \langle  y, x- x _ {0} \rangle \ \
 +
\textrm{ for }  \textrm{ all }  x \in X } \}
 +
.
 +
$$
  
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" />.
+
For example, the subdifferential of the norm  $  f( x) = \| x \| $
 +
in a normed space  $  X $
 +
with dual space  $  X  ^  \star  $
 +
takes the form
  
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
+
$$
 +
\partial  f( x)  = \left \{
  
<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>
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\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}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090820/s09082019.png" /> (the Moreau–Rockafellar theorem).
+
\right .$$
  
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]]).
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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) $.
  
====References====
+
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} $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.T. Rockafellar,   "Convex analysis" , Princeton Univ. Press (1970)</TD></TR></table>
+
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|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====
 
====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 $  \sigma ( X, Y) $-
 +
topology is the [[Weak topology|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 <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" />.
+
The elements $  x  ^  \star  \in \partial  f( x) $
 +
are called subgradients of $  f $
 +
at $  x $.
  
 
====References====
 
====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</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</TD></TR></table>
+
<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>

Latest revision as of 14:55, 7 June 2020


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
How to Cite This Entry:
Subdifferential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subdifferential&oldid=14652
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article