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Difference between revisions of "Minimizing sequence"

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A sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m0640101.png" /> from a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m0640102.png" /> for which the corresponding sequence of function values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m0640103.png" /> tends to the greatest lower bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m0640104.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m0640105.png" />, that is,
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The compactness of a minimizing sequence, that is, the existence of a subsequence converging to an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m0640107.png" />, in combination with the lower semi-continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m0640108.png" />, guarantees the existence of an optimal element
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A sequence of elements  $  y _ {n} $
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from a set  $  M $
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for which the corresponding sequence of function values  $  \phi ( y _ {n} ) $
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tends to the greatest lower bound of $  \phi $
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on  $  M $,
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that is,
  
<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/m/m064/m064010/m0640109.png" /></td> </tr></table>
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$$
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\lim\limits _ {n \rightarrow \infty }  \phi ( y _ {n} )  = \
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\inf _ {y \in M }  \phi ( y) .
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$$
  
In approximation theory, a minimizing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m06401010.png" /> for a given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m06401011.png" /> of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064010/m06401012.png" /> is a sequence for which
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The compactness of a minimizing sequence, that is, the existence of a subsequence converging to an element of $  M $,
 +
in combination with the lower semi-continuity of  $  \phi $,
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guarantees the existence of an optimal element
  
<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/m/m064/m064010/m06401013.png" /></td> </tr></table>
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$$
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y  ^ {n}  \in  M ,\  \phi ( y  ^ {n} )  = \min _ {
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y \in M }  \phi ( y) .
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$$
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In approximation theory, a minimizing sequence  $  \{ y _ {n} \} \in M $
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for a given element  $  x $
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of a metric space  $  X = ( X , \rho ) $
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is a sequence for which
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$$
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\rho ( x , y _ {n} )  \rightarrow  \rho ( x , M )  = \inf \
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\{ {\rho ( x , y ) } : {y \in M } \}
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.
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$$
  
 
See [[Approximately-compact set|Approximately-compact set]].
 
See [[Approximately-compact set|Approximately-compact set]].

Latest revision as of 08:00, 6 June 2020


A sequence of elements $ y _ {n} $ from a set $ M $ for which the corresponding sequence of function values $ \phi ( y _ {n} ) $ tends to the greatest lower bound of $ \phi $ on $ M $, that is,

$$ \lim\limits _ {n \rightarrow \infty } \phi ( y _ {n} ) = \ \inf _ {y \in M } \phi ( y) . $$

The compactness of a minimizing sequence, that is, the existence of a subsequence converging to an element of $ M $, in combination with the lower semi-continuity of $ \phi $, guarantees the existence of an optimal element

$$ y ^ {n} \in M ,\ \phi ( y ^ {n} ) = \min _ { y \in M } \phi ( y) . $$

In approximation theory, a minimizing sequence $ \{ y _ {n} \} \in M $ for a given element $ x $ of a metric space $ X = ( X , \rho ) $ is a sequence for which

$$ \rho ( x , y _ {n} ) \rightarrow \rho ( x , M ) = \inf \ \{ {\rho ( x , y ) } : {y \in M } \} . $$

See Approximately-compact set.

How to Cite This Entry:
Minimizing sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimizing_sequence&oldid=47850
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article