2martens
/
uni
mirror of https://github.com/2martens/uni.git
1
0
Fork 0
Code Releases Activity

MATH2-Inf-3: 2a korrigiert.

This commit is contained in:
Jim Martens 2013-11-04 14:53:17 +01:00
parent 4211ad6c4d
commit 4da752beb0
1 changed files with 22 additions and 22 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

44
optimierung/Uebungsblatt3.tex
View File

@ -194,7 +194,7 @@ Stephan Niendorf (6242417)}
\multicolumn{10}{l}{\text{unter den Nebenbedingungen}} && \\ \multicolumn{10}{l}{\text{unter den Nebenbedingungen}} && \\
\; & &-& x_{1} &-& x_{2} &-& x_{0} &\leq & -4 \\ \; & &-& x_{1} &-& x_{2} &-& x_{0} &\leq & -4 \\
\; &&& x_{1} &+& 2x_{2} &-& x_{0} &\leq & 2 \\ \; &&& x_{1} &+& 2x_{2} &-& x_{0} &\leq & 2 \\
\; &&-&x_{1} &+& x_{2} &-& x_{0} &\leq & 1 \\ \; &&-&x_{1} &+& x_{2} &-& x_{0} &\leq & -1 \\
\multicolumn{8}{r}{$x_{0}, x_{1}, x_{2}$} \,&\geq &\, 0 \multicolumn{8}{r}{$x_{0}, x_{1}, x_{2}$} \,&\geq &\, 0
\end{alignat*} \end{alignat*}
@ -204,7 +204,7 @@ Stephan Niendorf (6242417)}
\begin{alignat*}{5} \begin{alignat*}{5}
x_{3} \,&=&\, -4 \,&-&\, x_{1} \,&+&\, x_{2} \,&+&\, x_{0} \\ x_{3} \,&=&\, -4 \,&-&\, x_{1} \,&+&\, x_{2} \,&+&\, x_{0} \\
x_{4} \,&=&\, 2 \,&-&\, x_{1} \,&-&\, 2x_{2} \,&+&\, x_{0} \\ x_{4} \,&=&\, 2 \,&-&\, x_{1} \,&-&\, 2x_{2} \,&+&\, x_{0} \\
x_{5} \,&=&\, 1 \,&+&\, x_{1} \,&-&\, x_{2} \,&+&\, x_{0} \\ \cline{1 - 9} x_{5} \,&=&\, -1 \,&+&\, x_{1} \,&-&\, x_{2} \,&+&\, x_{0} \\ \cline{1 - 9}
w &=& && && \,&-&\, x_{0} w &=& && && \,&-&\, x_{0}
\end{alignat*} \end{alignat*}
@ -220,9 +220,9 @@ Stephan Niendorf (6242417)}
x_{4} \,&=&&\, 2 - x_{1} - 2x_{2} + \left(4 + x_{1} - x_{2} + x_{3}\right) \\ x_{4} \,&=&&\, 2 - x_{1} - 2x_{2} + \left(4 + x_{1} - x_{2} + x_{3}\right) \\
&=&&\, 2 - x_{1} - 2x_{2} + 4 + x_{1} - x_{2} + x_{3} \\ &=&&\, 2 - x_{1} - 2x_{2} + 4 + x_{1} - x_{2} + x_{3} \\
&=&&\, 6 - 3x_{2} + x_{3} \\ &=&&\, 6 - 3x_{2} + x_{3} \\
x_{5} \,&=&&\, 1 + x_{1} - x_{2} + \left(4 + x_{1} - x_{2} + x_{3}\right) \\ x_{5} \,&=&&\, -1 + x_{1} - x_{2} + \left(4 + x_{1} - x_{2} + x_{3}\right) \\
&=&&\, 1 + x_{1} - x_{2} + 4 + x_{1} - x_{2} + x_{3} \\ &=&&\, -1 + x_{1} - x_{2} + 4 + x_{1} - x_{2} + x_{3} \\
&=&&\, 5 + 2x_{1} - 2x_{2} + x_{3} \\ &=&&\, 3 + 2x_{1} - 2x_{2} + x_{3} \\
w \,&=&&\, -\left(4 + x_{1} - x_{2} + x_{3}\right) \\ w \,&=&&\, -\left(4 + x_{1} - x_{2} + x_{3}\right) \\
&=&&\, -4 - x_{1} + x_{2} - x_{3} \\ &=&&\, -4 - x_{1} + x_{2} - x_{3} \\
\end{alignat*} \end{alignat*}
@ -231,36 +231,36 @@ Stephan Niendorf (6242417)}
\begin{alignat*}{5} \begin{alignat*}{5}
x_{0} \,&=&\, 4 \,&+&\, x_{1} \,&-&\, x_{2} \,&+&\, x_{3} \\ x_{0} \,&=&\, 4 \,&+&\, x_{1} \,&-&\, x_{2} \,&+&\, x_{3} \\
x_{4} \,&=&\, 6 \,&& &-&\, 3x_{2} \,&+&\, x_{3} \\ x_{4} \,&=&\, 6 \,&& &-&\, 3x_{2} \,&+&\, x_{3} \\
x_{5} \,&=&\, 5 \,&+&\, 2x_{1} \,&-&\, 2x_{2} \,&+&\, x_{3} \\ \cline{1 - 9} x_{5} \,&=&\, 3 \,&+&\, 2x_{1} \,&-&\, 2x_{2} \,&+&\, x_{3} \\ \cline{1 - 9}
w &=& -2 \,&-&\, x_{1} \,&+&\, x_{2} \,&-&\, x_{3} w &=& -2 \,&-&\, x_{1} \,&+&\, x_{2} \,&-&\, x_{3}
\end{alignat*} \end{alignat*}
\underline{1. Iteration}: \underline{1. Iteration}:
Eingangsvariable: $x_{2}$ \\ Eingangsvariable: $x_{2}$ \\
Ausgangsvariable: $x_{4}$ Ausgangsvariable: $x_{5}$
Es folgt Es folgt
\begin{alignat*}{2} \begin{alignat*}{2}
3x_{2} &=&& 6 + x_{3} - x_{4} \\ 2x_{2} &=&& 3 + 2x_{1} + x_{3} - x_{5} \\
x_{2} &=&& 2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4} \\ x_{2} &=&& \frac{3}{2} + x_{1} + \frac{1}{2}x_{3} - \frac{1}{2}x_{5} \\
x_{0} &=&& 4 + x_{1} - \left(2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4}\right) + x_{3} \\ x_{0} &=&& 4 + x_{1} - \left(\frac{3}{2} + x_{1} + \frac{1}{2}x_{3} - \frac{1}{2}x_{5}\right) + x_{3} \\
&=&& 4 + x_{1} - 2 - \frac{1}{3}x_{3} + \frac{1}{3}x_{4} + x_{3}\\ &=&& 4 + x_{1} - \frac{3}{2} - x_{1} - \frac{1}{2}x_{3} + \frac{1}{2}x_{5} + x_{3}\\
&=&& 2 + x_{1} + \frac{2}{3}x_{3} + \frac{1}{3}x_{4} \\ &=&& \frac{5}{2} + \frac{1}{2}x_{3} + \frac{1}{2}x_{5} \\
x_{5} &=&& 5 + 2x_{1} - 2\left(2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4}\right) + x_{3} \\ x_{4} &=&& 6 - 3\left(\frac{3}{2} + x_{1} + \frac{1}{2}x_{3} - \frac{1}{2}x_{5}\right) + x_{3} \\
&=&& 5 + 2x_{1} - 4 - \frac{2}{3}x_{3} + \frac{2}{3}x_{4} + x_{3} \\ &=&& 6 - \frac{9}{2} - 3x_{1} + \frac{3}{2}x_{3} - \frac{3}{2}x_{5} + x_{3}\\
&=&& 1 + 2x_{1} + \frac{1}{3}x_{3} + \frac{2}{3}x_{4} \\ &=&& \frac{3}{2} - 3x_{1} + \frac{5}{2}x_{3} - \frac{3}{2}x_{5} \\
w &=&& -2 - x_{1} + \left(2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4}\right) - x_{3} \\ w &=&& -2 - x_{1} + \left(\frac{3}{2} + x_{1} + \frac{1}{2}x_{3} - \frac{1}{2}x_{5}\right) - x_{3} \\
&=&& -2 - x_{1} + 2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4} + x_{3} \\ &=&& -2 - x_{1} + \frac{3}{2} + x_{1} + \frac{1}{2}x_{3} - \frac{1}{2}x_{5} - x_{3} \\
&=&& 0 - x_{1} + \frac{4}{3}x_{3} - \frac{1}{3}x_{4} &=&& \frac{1}{2} - \frac{1}{2}x_{3} - \frac{1}{2}x_{5}
\end{alignat*} \end{alignat*}
\underline{Ergebnis der 1. Iteration}: \underline{Ergebnis der 1. Iteration}:
\begin{alignat*}{5} \begin{alignat*}{5}
x_{2} \,&=&\, 2 \,&& &+&\, \frac{1}{3}x_{3} \,&-&\, \frac{1}{3}x_{4} \\ x_{2} \,&=&\, \frac{3}{2} \,&+&\, x_{1} \,&+&\, \frac{1}{2}x_{3} \,&-&\, \frac{1}{2}x_{5} \\
x_{0} \,&=&\, 2 \,&+&\, x_{1} \,&+&\, \frac{2}{3}x_{3} \,&+&\, \frac{1}{3}x_{4} \\ x_{0} \,&=&\, \frac{5}{2} \,&& &+&\, \frac{1}{2}x_{3} \,&+&\, \frac{1}{2}x_{5} \\
x_{5} \,&=&\, 1 \,&+&\, 2x_{1} \,&+&\, \frac{1}{3}x_{3} \,&+&\, \frac{2}{3}x_{4} \\ \cline{1 - 9} x_{4} \,&=&\, \frac{3}{2} \,&-&\, 3x_{1} \,&+&\, \frac{5}{2}x_{3} \,&-&\, \frac{3}{2}x_{5} \\ \cline{1 - 9}
w &=& 0 \,&-&\, x_{1} \,&+&\, \frac{4}{3}x_{3} \,&-&\, \frac{1}{3}x_{4} w &=& \frac{1}{2} \,&& &-&\, \frac{1}{2}x_{3} \,&-&\, \frac{1}{2}x_{5}
\end{alignat*} \end{alignat*}
Da das Hilfsproblem keine optimale Lösung besitzt, besitzt das ursprüngliche Problem keine zulässige Lösung und ist damit unlösbar. Da das Hilfsproblem keine optimale Lösung besitzt, besitzt das ursprüngliche Problem keine zulässige Lösung und ist damit unlösbar.