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An extended-real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b1109201.png" /> on a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b1109202.png" /> over the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b1109203.png" /> is said to be proper if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b1109204.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b1109205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b1109206.png" /> for at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b1109207.png" />. The epigraph of such a function is the subset of the product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b1109208.png" /> defined by
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<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/b/b110/b110920/b1109209.png" /></td> </tr></table>
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The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092010.png" /> is convex (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]) precisely when the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092011.png" /> is convex (cf. [[Convex set|Convex set]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092012.png" /> is lower semi-continuous (cf. [[Semi-continuous function|Semi-continuous function]]) precisely when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092013.png" /> is closed (cf. [[Closed set|Closed set]]). A continuous [[Linear functional|linear functional]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092015.png" /> (that is, a member of the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092016.png" />) is said to be a subgradient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092017.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092018.png" /> provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092020.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092021.png" />. The set of all subgradients to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092022.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092023.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092024.png" /> is finite) forms the subdifferential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092026.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092027.png" />. The Brøndsted–Rockafellar theorem [[#References|[a2]]] asserts that for a proper convex lower semi-continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092028.png" />, the set of points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092029.png" /> is non-empty is dense in the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092030.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092031.png" /> is finite (cf. [[Dense set|Dense set]]). This is related to the [[Bishop–Phelps theorem|Bishop–Phelps theorem]] [[#References|[a1]]] (and the proof uses techniques of the latter), since a subgradient at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092032.png" /> can be identified with a support functional (cf. [[Support function|Support function]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092033.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110920/b11092034.png" />. These techniques were again applied to obtain minimization results (the Ekeland variational principle) for non-convex lower semi-continuous functions [[#References|[a3]]]; see [[#References|[a4]]] for a survey.
+
An extended-real-valued function  $  f $
 +
on a [[Banach space|Banach space]]  $  E $
 +
over the real numbers  $  \mathbf R $
 +
is said to be proper if  $  - \infty < f ( x ) \leq  \infty $
 +
for all  $  x \in E $
 +
and  $  f ( x ) < \infty $
 +
for at least one point  $  x $.
 +
The [[epigraph]] of such a function is the subset of the product space  $  E \times \mathbf R $
 +
defined by
 +
 
 +
$$
 +
{ \mathop{\rm epi} } ( f ) = \left \{ {( x, r ) \in E \times \mathbf R } : {x \in E, r \geq  f ( x ) } \right \} .
 +
$$
 +
 
 +
The function  $  f $
 +
is convex (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]) precisely when the set $  { \mathop{\rm epi} } ( f ) $
 +
is convex (cf. [[Convex set|Convex set]]) and $  f $
 +
is lower semi-continuous (cf. [[Semi-continuous function|Semi-continuous function]]) precisely when $  { \mathop{\rm epi} } ( f ) $
 +
is closed (cf. [[Closed set|Closed set]]). A continuous [[Linear functional|linear functional]] $  x  ^ {*} $
 +
on $  E $(
 +
that is, a member of the dual space $  E  ^ {*} $)  
 +
is said to be a subgradient of $  f $
 +
at the point $  x $
 +
provided $  f ( x ) < \infty $
 +
and $  x  ^ {*} ( y - x ) \leq  f ( y ) - f ( x ) $
 +
for all $  y \in E $.  
 +
The set of all subgradients to $  f $
 +
at $  x $(
 +
where $  f ( x ) $
 +
is finite) forms the subdifferential $  \partial  f ( x ) $
 +
of $  f $
 +
at $  x $.  
 +
The Brøndsted–Rockafellar theorem [[#References|[a2]]] asserts that for a proper convex lower semi-continuous function $  f $,  
 +
the set of points where $  \partial  f ( x ) $
 +
is non-empty is dense in the set of $  x $
 +
where $  f $
 +
is finite (cf. [[Dense set|Dense set]]). This is related to the [[Bishop–Phelps theorem|Bishop–Phelps theorem]] [[#References|[a1]]] (and the proof uses techniques of the latter), since a subgradient at a point $  x $
 +
can be identified with a support functional (cf. [[Support function|Support function]]) of $  { \mathop{\rm epi} } ( f ) $
 +
at the point $  ( x, f ( x ) ) $.  
 +
These techniques were again applied to obtain minimization results (the Ekeland variational principle) for non-convex lower semi-continuous functions [[#References|[a3]]]; see [[#References|[a4]]] for a survey.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Bishop,  R.R. Phelps,  "The support functionals of a convex set"  P. Klee (ed.) , ''Convexity'' , ''Proc. Symp. Pure Math.'' , '''7''' , Amer. Math. Soc.  (1963)  pp. 27–35</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Brøndsted,  R.T. Rockafellar,  "On the subdifferentiability of convex functions"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 605–611</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Ekeland,  "On the variational principle"  ''J. Math. Anal. Appl.'' , '''47'''  (1974)  pp. 324–353</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Ekeland,  "Nonconvex minimization problems"  ''Bull. Amer. Math. Soc. (NS)'' , '''1'''  (1979)  pp. 443–474</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Bishop,  R.R. Phelps,  "The support functionals of a convex set"  P. Klee (ed.) , ''Convexity'' , ''Proc. Symp. Pure Math.'' , '''7''' , Amer. Math. Soc.  (1963)  pp. 27–35</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Brøndsted,  R.T. Rockafellar,  "On the subdifferentiability of convex functions"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 605–611</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Ekeland,  "On the variational principle"  ''J. Math. Anal. Appl.'' , '''47'''  (1974)  pp. 324–353</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Ekeland,  "Nonconvex minimization problems"  ''Bull. Amer. Math. Soc. (NS)'' , '''1'''  (1979)  pp. 443–474</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


An extended-real-valued function $ f $ on a Banach space $ E $ over the real numbers $ \mathbf R $ is said to be proper if $ - \infty < f ( x ) \leq \infty $ for all $ x \in E $ and $ f ( x ) < \infty $ for at least one point $ x $. The epigraph of such a function is the subset of the product space $ E \times \mathbf R $ defined by

$$ { \mathop{\rm epi} } ( f ) = \left \{ {( x, r ) \in E \times \mathbf R } : {x \in E, r \geq f ( x ) } \right \} . $$

The function $ f $ is convex (cf. Convex function (of a real variable)) precisely when the set $ { \mathop{\rm epi} } ( f ) $ is convex (cf. Convex set) and $ f $ is lower semi-continuous (cf. Semi-continuous function) precisely when $ { \mathop{\rm epi} } ( f ) $ is closed (cf. Closed set). A continuous linear functional $ x ^ {*} $ on $ E $( that is, a member of the dual space $ E ^ {*} $) is said to be a subgradient of $ f $ at the point $ x $ provided $ f ( x ) < \infty $ and $ x ^ {*} ( y - x ) \leq f ( y ) - f ( x ) $ for all $ y \in E $. The set of all subgradients to $ f $ at $ x $( where $ f ( x ) $ is finite) forms the subdifferential $ \partial f ( x ) $ of $ f $ at $ x $. The Brøndsted–Rockafellar theorem [a2] asserts that for a proper convex lower semi-continuous function $ f $, the set of points where $ \partial f ( x ) $ is non-empty is dense in the set of $ x $ where $ f $ is finite (cf. Dense set). This is related to the Bishop–Phelps theorem [a1] (and the proof uses techniques of the latter), since a subgradient at a point $ x $ can be identified with a support functional (cf. Support function) of $ { \mathop{\rm epi} } ( f ) $ at the point $ ( x, f ( x ) ) $. These techniques were again applied to obtain minimization results (the Ekeland variational principle) for non-convex lower semi-continuous functions [a3]; see [a4] for a survey.

References

[a1] E. Bishop, R.R. Phelps, "The support functionals of a convex set" P. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 27–35
[a2] A. Brøndsted, R.T. Rockafellar, "On the subdifferentiability of convex functions" Proc. Amer. Math. Soc. , 16 (1965) pp. 605–611
[a3] I. Ekeland, "On the variational principle" J. Math. Anal. Appl. , 47 (1974) pp. 324–353
[a4] I. Ekeland, "Nonconvex minimization problems" Bull. Amer. Math. Soc. (NS) , 1 (1979) pp. 443–474
How to Cite This Entry:
Brøndsted-Rockafellar theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Br%C3%B8ndsted-Rockafellar_theorem&oldid=11546
This article was adapted from an original article by R. Phelps (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article