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A method for reducing constrained extremum problems to problems of unconstrained optimization. The method of penalty functions may be illustrated for problems in [[Mathematical programming|mathematical programming]]. Consider the problem of minimizing a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719901.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719902.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719903.png" />-dimensional Euclidean space. A penalty function, or penalty (for violating the restrictions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719905.png" />), is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719906.png" /> depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719907.png" /> and a numerical parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719908.png" /> with the following properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p0719909.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199013.png" /> be any point where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199014.png" /> takes an unconstrained (global) minimum, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199015.png" /> be the set of solutions of the original problem. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199016.png" /> is chosen such that the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199017.png" /> and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199018.png" /> tends to zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199019.png" />, or, if it is not possible to ensure this condition, such that the following relation holds:
+
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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/p/p071/p071990/p07199020.png" /></td> </tr></table>
+
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 +
{{TEX|done}}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199021.png" /> one often chooses the function
+
A method for reducing constrained extremum problems to problems of unconstrained optimization. The method of penalty functions may be illustrated for problems in [[Mathematical programming|mathematical programming]]. Consider the problem of minimizing a function  $  \phi ( x) $
 +
on a set  $  X = \{ {x } : {f _ {i} ( x) \geq  0,  i = 1 \dots m } \} $
 +
in an  $  n $-
 +
dimensional Euclidean space. A penalty function, or penalty (for violating the restrictions  $  f _ {i} ( x) \geq  0 $,
 +
$  i = 1 \dots m $),
 +
is a function  $  \psi ( x, \alpha ) $
 +
depending on  $  x $
 +
and a numerical parameter  $  \alpha $
 +
with the following properties: $  \psi ( x, \alpha ) = 0 $
 +
if  $  x \in X $
 +
and  $  \psi ( x, \alpha ) > 0 $
 +
if  $  x \notin X $.  
 +
Let  $  x( \alpha ) $
 +
be any point where the function  $  M( x, \alpha ) = \phi ( x) + \psi ( x, \alpha ) $
 +
takes an unconstrained (global) minimum, and let  $  X  ^  \star  $
 +
be the set of solutions of the original problem. The function  $  \psi ( x, \alpha ) $
 +
is chosen such that the distance between the points  $  x( \alpha ) $
 +
and the set  $  X  ^  \star  $
 +
tends to zero for  $  \alpha \rightarrow \infty $,
 +
or, if it is not possible to ensure this condition, such that the following relation holds:
  
<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/p/p071/p071990/p07199022.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\alpha \rightarrow \infty }  \phi ( x( \alpha ))  = \
 +
\inf _ {x \in X }  \phi ( x).
 +
$$
  
(frequently <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199023.png" />).
+
For  $  \psi ( x, \alpha ) $
 +
one often chooses the function
  
The choice of a particular form for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199024.png" /> is connected both with the problem of convergence of the method of penalty functions, and with problems arising in the unconstrained minimization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199025.png" />.
+
$$
 +
\psi ( x, \alpha )  = \alpha \sum _ { i= } 1 ^ { m }  | \min \{ f _ {i} ( x), 0 \} |
 +
^ {q} ,\ \
 +
q \geq  1
 +
$$
  
A more general statement of the method of penalty functions is based on reducing the problem of minimization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199026.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199027.png" /> to the problem of minimizing some parametric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199028.png" /> on a set of simpler structure (from the point of view of efficiency of applying numerical minimization methods) than the initial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199029.png" />.
+
(frequently  $  q = 2 $).
  
The following well-known general result shows that the method of penalty functions is universal. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199031.png" /> be reflexive Banach spaces (cf. [[Reflexive space|Reflexive space]]); let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199032.png" /> be the extended real line; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199033.png" /> be a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199034.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199035.png" /> that is weakly lower semi-continuous (cf. [[Semi-continuous function|Semi-continuous function]]); let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199037.png" />, be functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199038.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199039.png" /> that are continuous in the weak topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199040.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199042.png" />, be functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199043.png" />, with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199044.png" />, that are continuous in the weak topologies of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199046.png" /> (cf. [[Weak topology|Weak topology]]); and let the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199047.png" /> be non-empty. Consider the problem of finding those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199048.png" /> for which
+
The choice of a particular form for the function $  \psi ( x, \alpha ) $
 +
is connected both with the problem of convergence of the method of penalty functions, and with problems arising in the unconstrained minimization of $  M( x, \alpha ) $.
  
<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/p/p071/p071990/p07199049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
A more general statement of the method of penalty functions is based on reducing the problem of minimization of  $  \phi ( x) $
 +
on a set  $  X $
 +
to the problem of minimizing some parametric function  $  M( x, \alpha ) $
 +
on a set of simpler structure (from the point of view of efficiency of applying numerical minimization methods) than the initial set  $  X $.
 +
 
 +
The following well-known general result shows that the method of penalty functions is universal. Let  $  U $
 +
and  $  V $
 +
be reflexive Banach spaces (cf. [[Reflexive space|Reflexive space]]); let  $  \overline{\mathbf R}\; $
 +
be the extended real line; let  $  \phi $
 +
be a function defined on  $  U $
 +
with values in  $  \overline{\mathbf R}\; $
 +
that is weakly lower semi-continuous (cf. [[Semi-continuous function|Semi-continuous function]]); let  $  f _ {i} $,
 +
$  i = 1 \dots m $,
 +
be functions defined on  $  U $
 +
with values in  $  \overline{\mathbf R}\; $
 +
that are continuous in the weak topology of  $  U $;  
 +
let  $  h _ {j} $,
 +
$  j = 1 \dots n $,
 +
be functions defined on  $  U $,
 +
with values in  $  V $,
 +
that are continuous in the weak topologies of the spaces  $  U $
 +
and  $  V $(
 +
cf. [[Weak topology|Weak topology]]); and let the set  $  X = \{ {x } : {f _ {i} ( x) \geq  0,  i = 1 \dots m,  h _ {j} ( x) = 0,  j = 1 \dots n,  x \in U } \} $
 +
be non-empty. Consider the problem of finding those  $  x  ^  \star  \in U $
 +
for which
 +
 
 +
$$ \tag{* }
 +
\phi ( x  ^  \star  )  \leq  \phi ( x) \  \textrm{ for }  \textrm{ all }  x \in X.
 +
$$
  
 
For the function
 
For the function
  
<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/p/p071/p071990/p07199050.png" /></td> </tr></table>
+
$$
 +
M( x, y _ {1} \dots y _ {m} , \alpha )  = \phi ( x) +
 +
$$
  
<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/p/p071/p071990/p07199051.png" /></td> </tr></table>
+
$$
 +
+
 +
\alpha \left \{ \sum _ { i= } 1 ^ { m }  | f _ {i} ( x) - y _ {i} |  ^ {2}
 +
+ \sum _ { j= } 1 ^ { n }  \| h _ {j} ( x) \| _ {V}  ^ {2} \right \}
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199055.png" />, consider the problem of finding those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199058.png" />, for which
+
with $  \alpha > 0 $,  
 +
$  x \in U $,  
 +
$  y _ {i} \in \mathbf R $,  
 +
$  i = 1 \dots m $,  
 +
consider the problem of finding those $  x( \alpha ) \in U $
 +
and $  y _ {i} ( \alpha ) \geq  0 $,  
 +
$  i = 1 \dots m $,  
 +
for which
  
<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/p/p071/p071990/p07199059.png" /></td> </tr></table>
+
$$
 +
M( x( \alpha ), y _ {1} ( \alpha ) \dots y _ {m} ( \alpha ), \alpha ) \leq
 +
$$
  
<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/p/p071/p071990/p07199060.png" /></td> </tr></table>
+
$$
 +
\leq  \
 +
M( x, y _ {1} \dots y _ {m} , \alpha )
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199063.png" />. If
+
for all $  x \in U $,  
 +
$  y _ {i} \geq  0 $,  
 +
$  i = 1 \dots m $.  
 +
If
  
<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/p/p071/p071990/p07199064.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\| x \| \rightarrow \infty }  \phi ( x)  = + \infty ,
 +
$$
  
then any weak limit point of an arbitrary sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071990/p07199067.png" />, is a solution of the problem (*) and, moreover,
+
then any weak limit point of an arbitrary sequence $  \{ x( \alpha _ {k} ) \} $,  
 +
$  \alpha _ {k} \rightarrow \infty $,  
 +
$  k \rightarrow \infty $,  
 +
is a solution of the problem (*) and, moreover,
  
<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/p/p071/p071990/p07199068.png" /></td> </tr></table>
+
$$
 +
\lim\limits  \phi ( x( \alpha ))  = \phi ( x  ^  \star  ),\ \
 +
\alpha \rightarrow \infty .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Moiseev,  Yu.P. Ivanilov,  E.M. Stolyarova,  "Methods of optimization" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.P. Vasil'ev,  "Numerical methods for the solution of extremum problems" , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Fiacco,  G.P. MacCormick,  "Nonlinear programming: Sequential unconstrained minimization techniques" , Wiley  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Cea,  "Lectures on optimization: theory and algorithms" , Springer  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Moiseev,  Yu.P. Ivanilov,  E.M. Stolyarova,  "Methods of optimization" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.P. Vasil'ev,  "Numerical methods for the solution of extremum problems" , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Fiacco,  G.P. MacCormick,  "Nonlinear programming: Sequential unconstrained minimization techniques" , Wiley  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Cea,  "Lectures on optimization: theory and algorithms" , Springer  (1978)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:05, 6 June 2020


A method for reducing constrained extremum problems to problems of unconstrained optimization. The method of penalty functions may be illustrated for problems in mathematical programming. Consider the problem of minimizing a function $ \phi ( x) $ on a set $ X = \{ {x } : {f _ {i} ( x) \geq 0, i = 1 \dots m } \} $ in an $ n $- dimensional Euclidean space. A penalty function, or penalty (for violating the restrictions $ f _ {i} ( x) \geq 0 $, $ i = 1 \dots m $), is a function $ \psi ( x, \alpha ) $ depending on $ x $ and a numerical parameter $ \alpha $ with the following properties: $ \psi ( x, \alpha ) = 0 $ if $ x \in X $ and $ \psi ( x, \alpha ) > 0 $ if $ x \notin X $. Let $ x( \alpha ) $ be any point where the function $ M( x, \alpha ) = \phi ( x) + \psi ( x, \alpha ) $ takes an unconstrained (global) minimum, and let $ X ^ \star $ be the set of solutions of the original problem. The function $ \psi ( x, \alpha ) $ is chosen such that the distance between the points $ x( \alpha ) $ and the set $ X ^ \star $ tends to zero for $ \alpha \rightarrow \infty $, or, if it is not possible to ensure this condition, such that the following relation holds:

$$ \lim\limits _ {\alpha \rightarrow \infty } \phi ( x( \alpha )) = \ \inf _ {x \in X } \phi ( x). $$

For $ \psi ( x, \alpha ) $ one often chooses the function

$$ \psi ( x, \alpha ) = \alpha \sum _ { i= } 1 ^ { m } | \min \{ f _ {i} ( x), 0 \} | ^ {q} ,\ \ q \geq 1 $$

(frequently $ q = 2 $).

The choice of a particular form for the function $ \psi ( x, \alpha ) $ is connected both with the problem of convergence of the method of penalty functions, and with problems arising in the unconstrained minimization of $ M( x, \alpha ) $.

A more general statement of the method of penalty functions is based on reducing the problem of minimization of $ \phi ( x) $ on a set $ X $ to the problem of minimizing some parametric function $ M( x, \alpha ) $ on a set of simpler structure (from the point of view of efficiency of applying numerical minimization methods) than the initial set $ X $.

The following well-known general result shows that the method of penalty functions is universal. Let $ U $ and $ V $ be reflexive Banach spaces (cf. Reflexive space); let $ \overline{\mathbf R}\; $ be the extended real line; let $ \phi $ be a function defined on $ U $ with values in $ \overline{\mathbf R}\; $ that is weakly lower semi-continuous (cf. Semi-continuous function); let $ f _ {i} $, $ i = 1 \dots m $, be functions defined on $ U $ with values in $ \overline{\mathbf R}\; $ that are continuous in the weak topology of $ U $; let $ h _ {j} $, $ j = 1 \dots n $, be functions defined on $ U $, with values in $ V $, that are continuous in the weak topologies of the spaces $ U $ and $ V $( cf. Weak topology); and let the set $ X = \{ {x } : {f _ {i} ( x) \geq 0, i = 1 \dots m, h _ {j} ( x) = 0, j = 1 \dots n, x \in U } \} $ be non-empty. Consider the problem of finding those $ x ^ \star \in U $ for which

$$ \tag{* } \phi ( x ^ \star ) \leq \phi ( x) \ \textrm{ for } \textrm{ all } x \in X. $$

For the function

$$ M( x, y _ {1} \dots y _ {m} , \alpha ) = \phi ( x) + $$

$$ + \alpha \left \{ \sum _ { i= } 1 ^ { m } | f _ {i} ( x) - y _ {i} | ^ {2} + \sum _ { j= } 1 ^ { n } \| h _ {j} ( x) \| _ {V} ^ {2} \right \} $$

with $ \alpha > 0 $, $ x \in U $, $ y _ {i} \in \mathbf R $, $ i = 1 \dots m $, consider the problem of finding those $ x( \alpha ) \in U $ and $ y _ {i} ( \alpha ) \geq 0 $, $ i = 1 \dots m $, for which

$$ M( x( \alpha ), y _ {1} ( \alpha ) \dots y _ {m} ( \alpha ), \alpha ) \leq $$

$$ \leq \ M( x, y _ {1} \dots y _ {m} , \alpha ) $$

for all $ x \in U $, $ y _ {i} \geq 0 $, $ i = 1 \dots m $. If

$$ \lim\limits _ {\| x \| \rightarrow \infty } \phi ( x) = + \infty , $$

then any weak limit point of an arbitrary sequence $ \{ x( \alpha _ {k} ) \} $, $ \alpha _ {k} \rightarrow \infty $, $ k \rightarrow \infty $, is a solution of the problem (*) and, moreover,

$$ \lim\limits \phi ( x( \alpha )) = \phi ( x ^ \star ),\ \ \alpha \rightarrow \infty . $$

References

[1] N.N. Moiseev, Yu.P. Ivanilov, E.M. Stolyarova, "Methods of optimization" , Moscow (1978) (In Russian)
[2] F.P. Vasil'ev, "Numerical methods for the solution of extremum problems" , Moscow (1980) (In Russian)
[3] A.V. Fiacco, G.P. MacCormick, "Nonlinear programming: Sequential unconstrained minimization techniques" , Wiley (1968)
[4] J. Cea, "Lectures on optimization: theory and algorithms" , Springer (1978)

Comments

In the last two decades, new developments in the area of penalty function methods, namely multiplier penalty function methods (or augmented Lagrangian methods) and exact penalty function methods, have replaced for the most part the use of pure penalty function methods. See [a1].

References

[a1] R. Fletcher, "Practical methods of optimization" , Wiley (1987)
[a2] D.C. Luenberger, "Optimization by vector space methods" , Wiley (1969)
[a3] A.L. Peressini, F.E. Sullivan, J.J. Uhl, "The mathematics of nonlinear programming" , Springer (1988)
How to Cite This Entry:
Penalty functions, method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Penalty_functions,_method_of&oldid=13425
This article was adapted from an original article by V.G. Karmanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article