Nonlinear Opt.: Quadratic Problems 2: Unterschied zwischen den Versionen
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| − | There is a business that sells insulation stuff. Therefore they use two materials row wood x | + | |
| + | There is a business that sells insulation stuff. Therefore they use two materials row wood <math>x_1</math> and recycled wood wool <math>x_2</math>. The costs of recycling old furniture and building timber is higher then the ram wood. The business wants to minimize their costs. | ||
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| + | <math>c(x_1)=x_1</math> | ||
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| + | <math>c(x_2)=-2x_1+2x_2</math> | ||
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| + | Total costs: <math>c_t (x_1,x_2)= x_1^2 -2x_1x_2 +2x_2^2</math> | ||
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| + | First we transform the problem in the quadratic form: <math>F(x)= c^Tx +\frac{1}{2} x^TCx</math> | ||
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| + | <math>c(x_1,x_2)= x_1^2 -2x_1x_2 +2x_2^2</math> | ||
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| + | <math>\Rightarrow c(x_1,x_2)= (x_1 -2x_2)x_1)+(2x_2)x_2</math> | ||
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| + | <math>\Rightarrow c(x_1,x_2)= (x_1 -x_2)x_1)+(-x_1+x_2)x_2</math> | ||
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| + | <math>\Rightarrow c(x_1,x_2)= \frac{1}{2} [(2x_1 -2x_2)x_1)+(-2x_1+4x_2)x_2]</math> | ||
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| + | <math>c=\frac{1}{2} \begin{pmatrix} 2&-2 \\ -2&4 \end{pmatrix} </math> | ||
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| + | <math>c(x_1,x_2)=\begin{pmatrix} 0 & 0 \end{pmatrix} | ||
| + | \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} | ||
| + | +\frac{1}{2}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} | ||
| + | \begin{pmatrix} 2&-2 \\ -2&4 \end{pmatrix} | ||
| + | \begin{pmatrix} x_1 & x_2 \end{pmatrix} </math> | ||
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| + | Secondly we must proof if it´s a konvex or koncav funktion thus if it´s positiv or negativ definite. It should be positiv definite to represent a global minimum costs. | ||
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| + | To test the form of the graph, we draw up the Hesse matrix. | ||
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| + | <math>H(x_1,x_2)=\begin{pmatrix} \frac{d^2f}{dx_1dx_1}(x) & \frac{d^2f}{dx_1dx_2}(x) \\ | ||
| + | \frac{d^2f}{dx_2dx_1}(x) & \frac{d^2f}{dx_2dx_2}(x) \end{pmatrix} </math> | ||
| + | |||
| + | <math>\Rightarrow H(x_1,x_2)= \begin{pmatrix} 2&-2 \\ -2&4 \end{pmatrix} </math> | ||
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| + | If the main mintors are bigger (or equal) zero, then the matrix is positiv definit (positiv semidefinit) and is is konvex. | ||
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| + | <math> det \begin{vmatrix} 2 \end{vmatrix} =2 </math> | ||
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| + | <math> det \begin{vmatrix} 2&-2 \\ -2&4 \end{vmatrix} =4 </math> | ||
Version vom 27. Juni 2013, 11:41 Uhr
Example
There is a business that sells insulation stuff. Therefore they use two materials row wood and recycled wood wool . The costs of recycling old furniture and building timber is higher then the ram wood. The business wants to minimize their costs.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c(x_2)=-2x_1+2x_2
Total costs: Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c_t (x_1,x_2)= x_1^2 -2x_1x_2 +2x_2^2
First we transform the problem in the quadratic form:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c(x_1,x_2)= x_1^2 -2x_1x_2 +2x_2^2
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \Rightarrow c(x_1,x_2)= (x_1 -2x_2)x_1)+(2x_2)x_2
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \Rightarrow c(x_1,x_2)= (x_1 -x_2)x_1)+(-x_1+x_2)x_2
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \Rightarrow c(x_1,x_2)= \frac{1}{2} [(2x_1 -2x_2)x_1)+(-2x_1+4x_2)x_2]
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c=\frac{1}{2} \begin{pmatrix} 2&-2 \\ -2&4 \end{pmatrix}
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c(x_1,x_2)=\begin{pmatrix} 0 & 0 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} +\frac{1}{2}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \begin{pmatrix} 2&-2 \\ -2&4 \end{pmatrix} \begin{pmatrix} x_1 & x_2 \end{pmatrix}
Secondly we must proof if it´s a konvex or koncav funktion thus if it´s positiv or negativ definite. It should be positiv definite to represent a global minimum costs.
To test the form of the graph, we draw up the Hesse matrix.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \Rightarrow H(x_1,x_2)= \begin{pmatrix} 2&-2 \\ -2&4 \end{pmatrix}
If the main mintors are bigger (or equal) zero, then the matrix is positiv definit (positiv semidefinit) and is is konvex.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): det \begin{vmatrix} 2&-2 \\ -2&4 \end{vmatrix} =4
