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A minimal value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s0905901.png" /> taken by a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s0905902.png" /> at a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s0905903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s0905904.png" />, such that
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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/s090/s090590/s0905905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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for all comparison curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s0905906.png" /> satisfying the condition of being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s0905908.png" />-near of zero order, i.e.
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A minimal value  $  J ( \widetilde{y}  ) $
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taken by a functional  $  J ( y) $
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at a curve  $  \widetilde{y}  ( x) $,  
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$  x _ {1} \leq  x \leq  x _ {2} $,
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such that
  
<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/s090/s090590/s0905909.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{1 }
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J ( \widetilde{y}  ) \leq  J ( y)
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$$
  
on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059010.png" />. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059012.png" /> satisfy given boundary conditions.
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for all comparison curves  $  y ( x) $
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satisfying the condition of being  $  \epsilon $-
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near of zero order, i.e.
  
If, along with condition (2), which requires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059014.png" />-nearness of ordinates, one adds the condition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059015.png" />-nearness of the derivatives:
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$$ \tag{2 }
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| y ( x) - \widetilde{y}  ( x) |  \leq  \epsilon
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$$
  
<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/s090/s090590/s09059016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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on the whole interval  $  [ x _ {1} , x _ {2} ] $.  
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It is assumed that  $  \widetilde{y}  ( x) $,
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$  y ( x) $
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satisfy given boundary conditions.
  
on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059017.png" />, then one speaks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059018.png" />-nearness of first order.
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If, along with condition (2), which requires  $  \epsilon $-
 +
nearness of ordinates, one adds the condition of $  \epsilon $-
 +
nearness of the derivatives:
  
The value taken by the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059019.png" /> at a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059020.png" /> for which (1) is satisfied for all comparison curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059021.png" /> which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059022.png" />-near of first order, is called a weak relative minimum.
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$$ \tag{3 }
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| y  ^  \prime  ( x) - \widetilde{y}  {}  ^  \prime  ( x) |  \leq  \epsilon
 +
$$
  
Since the condition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059023.png" />-nearness of zero order selects a broader class of curves than the condition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059024.png" />-nearness of first order, every strong minimum is simultaneously a weak minimum (cf. also [[Weak relative minimum|Weak relative minimum]]); but not every weak minimum is strong. In this connection the necessary, and also sufficient, conditions of optimality for strong and weak relative minima do not have the same form.
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on the whole interval  $  [ x _ {1} , x _ {2} ] $,
 +
then one speaks of $  \epsilon $-
 +
nearness of first order.
  
Alongside the idea of a strong relative minimum the idea of an absolute minimum can be introduced. An absolute minimum is the minimal value taken by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090590/s09059025.png" /> on the whole set of curves on which it has a meaning. An absolute minimum is global, whereas strong and weak relative minima are local.
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The value taken by the functional  $  J ( y) $
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at a curve  $  \widetilde{y}  ( x) $
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for which (1) is satisfied for all comparison curves  $  y ( x) $
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which are  $  \epsilon $-
 +
near of first order, is called a weak relative minimum.
 +
 
 +
Since the condition of  $  \epsilon $-
 +
nearness of zero order selects a broader class of curves than the condition of  $  \epsilon $-
 +
nearness of first order, every strong minimum is simultaneously a weak minimum (cf. also [[Weak relative minimum|Weak relative minimum]]); but not every weak minimum is strong. In this connection the necessary, and also sufficient, conditions of optimality for strong and weak relative minima do not have the same form.
 +
 
 +
Alongside the idea of a strong relative minimum the idea of an absolute minimum can be introduced. An absolute minimum is the minimal value taken by $  J ( y) $
 +
on the whole set of curves on which it has a meaning. An absolute minimum is global, whereas strong and weak relative minima are local.
  
 
An absolute minimum is also a strong relative minimum, but not every strong relative minimum is an absolute minimum.
 
An absolute minimum is also a strong relative minimum, but not every strong relative minimum is an absolute minimum.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  L.A. Lyusternik,  "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  L.A. Lyusternik,  "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.E. [L.E. El'sgol'ts] Elsgolc,  "Calculus of variations" , Pergamon  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Cesari,  "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.E. [L.E. El'sgol'ts] Elsgolc,  "Calculus of variations" , Pergamon  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Cesari,  "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


A minimal value $ J ( \widetilde{y} ) $ taken by a functional $ J ( y) $ at a curve $ \widetilde{y} ( x) $, $ x _ {1} \leq x \leq x _ {2} $, such that

$$ \tag{1 } J ( \widetilde{y} ) \leq J ( y) $$

for all comparison curves $ y ( x) $ satisfying the condition of being $ \epsilon $- near of zero order, i.e.

$$ \tag{2 } | y ( x) - \widetilde{y} ( x) | \leq \epsilon $$

on the whole interval $ [ x _ {1} , x _ {2} ] $. It is assumed that $ \widetilde{y} ( x) $, $ y ( x) $ satisfy given boundary conditions.

If, along with condition (2), which requires $ \epsilon $- nearness of ordinates, one adds the condition of $ \epsilon $- nearness of the derivatives:

$$ \tag{3 } | y ^ \prime ( x) - \widetilde{y} {} ^ \prime ( x) | \leq \epsilon $$

on the whole interval $ [ x _ {1} , x _ {2} ] $, then one speaks of $ \epsilon $- nearness of first order.

The value taken by the functional $ J ( y) $ at a curve $ \widetilde{y} ( x) $ for which (1) is satisfied for all comparison curves $ y ( x) $ which are $ \epsilon $- near of first order, is called a weak relative minimum.

Since the condition of $ \epsilon $- nearness of zero order selects a broader class of curves than the condition of $ \epsilon $- nearness of first order, every strong minimum is simultaneously a weak minimum (cf. also Weak relative minimum); but not every weak minimum is strong. In this connection the necessary, and also sufficient, conditions of optimality for strong and weak relative minima do not have the same form.

Alongside the idea of a strong relative minimum the idea of an absolute minimum can be introduced. An absolute minimum is the minimal value taken by $ J ( y) $ on the whole set of curves on which it has a meaning. An absolute minimum is global, whereas strong and weak relative minima are local.

An absolute minimum is also a strong relative minimum, but not every strong relative minimum is an absolute minimum.

A variational problem having more than one strong relative minimum is called a multi-extremum problem. For the solution of practical variational problems a strong relative minimum can be found approximately, using the numerical methods of variational calculus (see Variational calculus, numerical methods of).

For problems in which a strong relative minimum is unique, the necessary conditions for optimality of a strong relative minimum are simultaneously sufficient conditions for an absolute minimum. This situation holds, for example, in the theory of optimal control of linear problems of time-optimal control (see Time-optimal control problem), and also for certain other classes of problems in variational calculus.

References

[1] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)
[2] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)

Comments

References

[a1] L.E. [L.E. El'sgol'ts] Elsgolc, "Calculus of variations" , Pergamon (1961) (Translated from Russian)
[a2] L. Cesari, "Optimization - Theory and applications" , Springer (1983)
How to Cite This Entry:
Strong relative minimum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_relative_minimum&oldid=48877
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article