Nonlinear Opt.: Lagrangian condition 4: Unterschied zwischen den Versionen

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Aktuelle Version vom 6. Juli 2013, 10:47 Uhr

Inhaltsverzeichnis

1 Theory
2 Problem
3 Procedure
4 Example
5 Sources

Theory

The Lagrangian method is part of the non-linear optiization theory. It is used to solve an objective function with constraints. The application area of this methed is very wide-ranging.

Problem

Objective function $f(x_{1},...,x_{n})=min!\quad or\quad max!$

subject to $g_{k}(x_{1},...,x_{n})=c$ element R

The objective function $f(x_{1},...,x_{n})$ and $g_{k}(x_{1},...,x_{n})$ must be continuosly differentiable!

Procedure

First you have to rewrite the constraints to the form of

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): g_k(x_1,...,x_n)-c = 0

$k=1,...,m$

Then you set up the Lagrangian function:

$L(x_{1},...,x_{n};\lambda _{1},...,\lambda _{k})=f(x_{1},...,x_{n})+\lambda _{1}g_{1}(x_{1},...,x_{n})+...+\lambda _{m}g_{m}(x_{1},...,x_{n})$

Third you must derive the Lagrangian function with respect to all variables and set these derivatives to zero .

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial}{\partial x_i}L(x_1,...,x_n; \lambda_1,...,\lambda_m) = 0

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial}{\partial \lambda_k}L(x_1,...,x_n;\lambda_1,...,\lambda_m)= 0

$i=1,...,n$

$k=1...,m$

Last you have to solve all the equations.

Example

Optimise the objective function $30x_{1}x_{2}$ subject to the constraint $x_{1}+25x_{2}=500$

First

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1+25x_2-500= 0

Second

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): L=30x_1x_2+\lambda(500-x_1-25x_2)

Third

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial L}{\partial x_1} =30x_2-\lambda=0

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial L}{\partial x_2} =30x_1-25\lambda=0

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial L}{\partial \lambda}=500-x_1-25x_2=0

Fourth

$30x_{1}=25\lambda (2)$

=> $6/5x_{1}=\lambda$

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_2=20-x_1/25 (3)

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 30x_2-\lambda=0 (1)

=> Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 30(20-x_1/25)-6/5x_1=0

=> Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 600-6/5x_1-6/5x_1=0

=> Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 600-12/5x_1=0

=> $x_{1}=250$

Sources

Kurt Meyberg, Peter Vachenauer Höhere Mathematik 1 (6. Auflage)

Hal R. Varian Intermediate Microeconomics (8th Edition)

Script of Operations Research

Hans Corsten, Ralf Gössinger Produktionswirtschaft; Einführung in das industrielle Produktionsmanagement(13. Auflage)

@@ Zeile 2: / Zeile 2: @@
 == Theory ==
-The Lagrangian method is part of the non-linear optiization theory. It is used to solve an objective function with constraints. The application area of this mothed is very wide-ranging.
+The Lagrangian method is part of the non-linear optiization theory. It is used to solve an objective function with constraints. The application area of this methed is very wide-ranging.
+== Problem ==
+'''Objective function'''      <math>f(x_1,...,x_n)= min!   \quad   or \quad max!</math>
+'''subject to '''  <math>g_k(x_1,...,x_n) = c </math>  element R
-== Problem ==
-'''Objective function'''      <math>f(x_1,...,x_n)= min!   \quad   or \quad max!</math>
+The objective function <math>f(x_1,...,x_n)</math> and <math>g_k(x_1,...,x_n)</math> must be continuosly differentiable!
+== Procedure ==
-'''subject to '''  <math>g_k(x_1,...,x_n) = c </math> e element R
+First you have to rewrite the constraints to the form of
-The objective function <math>f(x_1,...,x_n)</math> and <math>g_k(x_1,...,x_n)</math> must be continuosly differentiable!
+<math>g_k(x_1,...,x_n)-c = 0 </math>
-== Procedure ==
+<math>k=1,...,m</math>
-First you have to rewrite the constraints to the form of <math>g_k(x_1,...,x_n)-c = 0 </math>
-Then you set up the Lagrangian function: <math>L(x_1,...,x_n;\lambda_1,...,\lambda_k = f(x_1,...,x_n) +\lambda_1 g_1(x_1,...,x_n) +...+\lambda_k g_k(x_1,...,x_n</math>
+Then you set up the Lagrangian function:
-Third you must derive the Lagrangian function with respect to all variables and set These derivatives to Zero .
-<math>\frac{\partial}{\partial x_i}L(x_1,...,x_n; \lambda_1,...,\lambda_k = 0 </math>
+<math>L(x_1,...,x_n;\lambda_1,...,\lambda_k) = f(x_1,...,x_n) +\lambda_1 g_1(x_1,...,x_n) +...+\lambda_m g_m(x_1,...,x_n)</math>
-<math>\frac{\partial}{\partial \lambda_k}L(x_1,...,x_n;\lambda_1,...,\lambda_m</math>
-Last you have to solve all the equations.
+Third you must derive the Lagrangian function with respect to all variables and set these derivatives to zero .
+<math>\frac{\partial}{\partial x_i}L(x_1,...,x_n; \lambda_1,...,\lambda_m) = 0 </math>
+<math>\frac{\partial}{\partial \lambda_k}L(x_1,...,x_n;\lambda_1,...,\lambda_m)= 0</math>
+<math>i=1,...,n</math>
+<math>k=1...,m</math>
+Last you have to solve all the equations.
 == Example ==
@@ Zeile 42: / Zeile 54: @@
-'''First'''   <math>x_1+25x_2-500= 0</math>
+'''First'''
+ <math>x_1+25x_2-500= 0</math>
-----
-'''Second'''  <math>L=30x_1x_2+\lambda(500-x_1-25x_2)</math>
+'''Second'''
-----
-'''Third'''<math>\</math> <math> \frac{\partial L}{\partial x_1}
+<math>L=30x_1x_2+\lambda(500-x_1-25x_2)</math>
-=30x_2-\lambda=0</math><math>\\</math>
+'''Third'''
+<math> \frac{\partial L}{\partial x_1}
+=30x_2-\lambda=0</math>
 <math> \frac{\partial L}{\partial x_2}
-=30x_1-25\lambda=0</math><math>\</math>
+=30x_1-25\lambda=0</math>
 <math> \frac{\partial L}{\partial \lambda}=500-x_1-25x_2=0</math>
-'''Fourth''' <math>\newline</math>
+'''Fourth'''
-<math>30x_1=25\lambda(2)</math><math>\newline</math>
-so <math>6/5x_1=\lambda</math><math>\newline</math>
-<math>\\</math>
-<math>x_2=20-x_1/25</math>
+<math>30x_1=25\lambda(2)</math>
-<math>\newline</math>
+=> <nowiki></nowiki>    <math>6/5x_1=\lambda</math>
+<math>x_2=20-x_1/25</math>(3)
+<math>30x_2-\lambda=0</math>(1)
+=>  <math>30(20-x_1/25)-6/5x_1=0</math>
+=>  <math>600-6/5x_1-6/5x_1=0</math>
+=> <math>600-12/5x_1=0</math>
+=> <math>x_1=250</math>
+== Sources ==
+Kurt Meyberg, Peter Vachenauer Höhere Mathematik 1 (6. Auflage)
+Hal R. Varian Intermediate Microeconomics (8th Edition)
+Script of Operations Research
+Hans Corsten, Ralf Gössinger Produktionswirtschaft; Einführung in das industrielle Produktionsmanagement(13. Auflage)

Nonlinear Opt.: Lagrangian condition 4: Unterschied zwischen den Versionen

Aktuelle Version vom 6. Juli 2013, 10:47 Uhr

Inhaltsverzeichnis

Theory

Problem

Procedure

Example

Sources

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