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''of a function''
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s0912201.png" /> be a function defined on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s0912202.png" /> with values in the extended real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s0912203.png" />. Then the supergraph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s0912204.png" /> is the subset of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s0912205.png" /> consisting of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s0912206.png" /> lying  "above"  the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s0912207.png" />. The supergraph is sometimes called the epigraph and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s0912208.png" />:
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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/s/s091/s091220/s0912209.png" /></td> </tr></table>
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''of a function''
  
The projection of the supergraph onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122010.png" /> is known as the effective domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122011.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122012.png" />:
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Let  $  f:  X \rightarrow \overline{\mathbf R}\; $
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be a function defined on some set  $  X $
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with values in the extended real line  $  \overline{\mathbf R}\; = \mathbf R \cup \{ - \infty \} \cup \{ + \infty \} $.  
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Then the supergraph of  $  f $
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is the subset of the Cartesian product  $  X \times \mathbf R $
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consisting of all points  $  ( x, \alpha ) $
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lying  "above" the graph of  $  f $.  
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The supergraph is sometimes called the epigraph and is denoted by $  \mathop{\rm epi}  f $:
  
<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/s091/s091220/s09122013.png" /></td> </tr></table>
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$$
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\mathop{\rm epi}  f  = \
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\{ {( x, \alpha ) \in X \times \mathbf R } : {\alpha \geq  f ( x) } \}
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.
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$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122014.png" /> is said to be proper if
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The projection of the supergraph onto  $  X $
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is known as the effective domain of  $  f $,
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denoted by  $  \mathop{\rm dom}  f $:
  
<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/s091/s091220/s09122015.png" /></td> </tr></table>
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$$
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\mathop{\rm dom}  f  = \{ {x \in X } : {f ( x) < + \infty } \}
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.
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$$
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122016.png" /> defined on a real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122017.png" /> is convex if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122018.png" /> is a convex subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122019.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122020.png" /> defined on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122021.png" /> is upper semi-continuous if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091220/s09122022.png" /> is a closed set.
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The function $  f $
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is said to be proper if
  
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$$
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f ( x)  >  - \infty , \forall x,\  \textrm{ and } \ \
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\mathop{\rm dom}  f  \neq  \emptyset .
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$$
  
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A function  $  f:  X \rightarrow \mathbf R $
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defined on a real vector space  $  X $
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is convex if and only if  $  \mathop{\rm epi}  f $
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is a convex subset of  $  X \times \mathbf R $.
 +
A function  $  f:  X \rightarrow \overline{\mathbf R}\; $
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defined on a topological space  $  X $
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is upper semi-continuous if and only if  $  \mathop{\rm epi}  f $
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is a closed set.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)  pp. 23; 307</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)  pp. 23; 307</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


of a function

Let $ f: X \rightarrow \overline{\mathbf R}\; $ be a function defined on some set $ X $ with values in the extended real line $ \overline{\mathbf R}\; = \mathbf R \cup \{ - \infty \} \cup \{ + \infty \} $. Then the supergraph of $ f $ is the subset of the Cartesian product $ X \times \mathbf R $ consisting of all points $ ( x, \alpha ) $ lying "above" the graph of $ f $. The supergraph is sometimes called the epigraph and is denoted by $ \mathop{\rm epi} f $:

$$ \mathop{\rm epi} f = \ \{ {( x, \alpha ) \in X \times \mathbf R } : {\alpha \geq f ( x) } \} . $$

The projection of the supergraph onto $ X $ is known as the effective domain of $ f $, denoted by $ \mathop{\rm dom} f $:

$$ \mathop{\rm dom} f = \{ {x \in X } : {f ( x) < + \infty } \} . $$

The function $ f $ is said to be proper if

$$ f ( x) > - \infty , \forall x,\ \textrm{ and } \ \ \mathop{\rm dom} f \neq \emptyset . $$

A function $ f: X \rightarrow \mathbf R $ defined on a real vector space $ X $ is convex if and only if $ \mathop{\rm epi} f $ is a convex subset of $ X \times \mathbf R $. A function $ f: X \rightarrow \overline{\mathbf R}\; $ defined on a topological space $ X $ is upper semi-continuous if and only if $ \mathop{\rm epi} f $ is a closed set.

Comments

References

[a1] R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) pp. 23; 307
How to Cite This Entry:
Supergraph. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Supergraph&oldid=17155
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article