Difference between revisions of "Brøndsted-Rockafellar theorem"
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| − | The function | + | 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=22191
Brøndsted-Rockafellar theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Br%C3%B8ndsted-Rockafellar_theorem&oldid=22191
This article was adapted from an original article by R. Phelps (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article