Difference between revisions of "Minimizing sequence"
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| − | + | 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|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
$$ \rho ( x , y _ {n} ) \rightarrow \rho ( x , M ) = \inf \ \{ {\rho ( x , y ) } : {y \in M } \} . $$
How to Cite This Entry:
Minimizing sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimizing_sequence&oldid=47850
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