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A method for the minimization of a function of several variables. It is based on the fact that each successive approximation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044710/g0447101.png" /> is obtained from the preceding one by a shift in the direction of the gradient of the function:
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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/g/g044/g044710/g0447102.png" /></td> </tr></table>
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The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044710/g0447103.png" /> can be obtained, e.g., from the condition of the magnitude
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A method for the minimization of a function of several variables. It is based on the fact that each successive approximation of the function  $  F $
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is obtained from the preceding one by a shift in the direction of the gradient of the function:
  
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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/g/g044/g044710/g0447104.png" /></td> </tr></table>
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$$
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\mathbf x ^ {n + 1 }  = \
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\mathbf x  ^ {n} - \delta _ {n} \
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\mathop{\rm grad}  F ( \mathbf x  ^ {n} ).
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$$
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The parameter  $  \delta _ {n} $
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can be obtained, e.g., from the condition of the magnitude
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$$
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F ( \mathbf x  ^ {n} - \delta _ {n} \
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\mathop{\rm grad}  F ( \mathbf x  ^ {n} )) \ \
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\textrm{ being  minimal  } .
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$$
  
 
See also [[Descent, method of|Descent, method of]]; [[Steepest descent, method of|Steepest descent, method of]].
 
See also [[Descent, method of|Descent, method of]]; [[Steepest descent, method of|Steepest descent, method of]].
  
 
====Comments====
 
====Comments====
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Dennis,  R.B. Schnabel,  "Numerical methods for unconstrained optimization and nonlinear equations" , Prentice-Hall  (1983)  {{MR|0702023}} {{ZBL|0579.65058}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Fletcher,  "Practical methods of optimization" , Wiley  (1980)  {{MR|0585160}} {{MR|0633058}} {{ZBL|0439.93001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.G. Luenberger,  "Linear and nonlinear programming" , Addison-Wesley  (1984)  {{MR|2423726}} {{MR|2012832}} {{ZBL|0571.90051}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Dennis,  R.B. Schnabel,  "Numerical methods for unconstrained optimization and nonlinear equations" , Prentice-Hall  (1983)  {{MR|0702023}} {{ZBL|0579.65058}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Fletcher,  "Practical methods of optimization" , Wiley  (1980)  {{MR|0585160}} {{MR|0633058}} {{ZBL|0439.93001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.G. Luenberger,  "Linear and nonlinear programming" , Addison-Wesley  (1984)  {{MR|2423726}} {{MR|2012832}} {{ZBL|0571.90051}} </TD></TR></table>

Latest revision as of 19:42, 5 June 2020


A method for the minimization of a function of several variables. It is based on the fact that each successive approximation of the function $ F $ is obtained from the preceding one by a shift in the direction of the gradient of the function:

$$ \mathbf x ^ {n + 1 } = \ \mathbf x ^ {n} - \delta _ {n} \ \mathop{\rm grad} F ( \mathbf x ^ {n} ). $$

The parameter $ \delta _ {n} $ can be obtained, e.g., from the condition of the magnitude

$$ F ( \mathbf x ^ {n} - \delta _ {n} \ \mathop{\rm grad} F ( \mathbf x ^ {n} )) \ \ \textrm{ being minimal } . $$

See also Descent, method of; Steepest descent, method of.

Comments

References

[a1] J.E. Dennis, R.B. Schnabel, "Numerical methods for unconstrained optimization and nonlinear equations" , Prentice-Hall (1983) MR0702023 Zbl 0579.65058
[a2] R. Fletcher, "Practical methods of optimization" , Wiley (1980) MR0585160 MR0633058 Zbl 0439.93001
[a3] D.G. Luenberger, "Linear and nonlinear programming" , Addison-Wesley (1984) MR2423726 MR2012832 Zbl 0571.90051
How to Cite This Entry:
Gradient method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gradient_method&oldid=47111
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