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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
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44
optimierung/Uebungsblatt3.tex
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@ -194,7 +194,7 @@ Stephan Niendorf (6242417)}
\multicolumn{10}{l}{\text{unter den Nebenbedingungen}} && \\
\; & &-& x_{1} &-& x_{2} &-& x_{0} &\leq & -4 \\
\; &&& 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
\end{alignat*}
@ -204,7 +204,7 @@ Stephan Niendorf (6242417)}
\begin{alignat*}{5}
x_{3} \,&=&\, -4 \,&-&\, x_{1} \,&+&\, x_{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}
\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) \\
&=&&\, 2 - x_{1} - 2x_{2} + 4 + x_{1} - x_{2} + x_{3} \\
&=&&\, 6 - 3x_{2} + x_{3} \\
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} \\
&=&&\, 5 + 2x_{1} - 2x_{2} + x_{3} \\
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} \\
&=&&\, 3 + 2x_{1} - 2x_{2} + x_{3} \\
w \,&=&&\, -\left(4 + x_{1} - x_{2} + x_{3}\right) \\
&=&&\, -4 - x_{1} + x_{2} - x_{3} \\
\end{alignat*}
@ -231,36 +231,36 @@ Stephan Niendorf (6242417)}
\begin{alignat*}{5}
x_{0} \,&=&\, 4 \,&+&\, x_{1} \,&-&\, x_{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}
\end{alignat*}
\underline{1. Iteration}:
Eingangsvariable: $x_{2}$ \\
Ausgangsvariable: $x_{4}$
Ausgangsvariable: $x_{5}$
Es folgt
\begin{alignat*}{2}
3x_{2} &=&& 6 + x_{3} - x_{4} \\
x_{2} &=&& 2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4} \\
x_{0} &=&& 4 + x_{1} - \left(2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4}\right) + x_{3} \\
&=&& 4 + x_{1} - 2 - \frac{1}{3}x_{3} + \frac{1}{3}x_{4} + x_{3}\\
&=&& 2 + x_{1} + \frac{2}{3}x_{3} + \frac{1}{3}x_{4} \\
x_{5} &=&& 5 + 2x_{1} - 2\left(2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4}\right) + x_{3} \\
&=&& 5 + 2x_{1} - 4 - \frac{2}{3}x_{3} + \frac{2}{3}x_{4} + x_{3} \\
&=&& 1 + 2x_{1} + \frac{1}{3}x_{3} + \frac{2}{3}x_{4} \\
w &=&& -2 - x_{1} + \left(2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4}\right) - x_{3} \\
&=&& -2 - x_{1} + 2 + \frac{1}{3}x_{3} - \frac{1}{3}x_{4} + x_{3} \\
&=&& 0 - x_{1} + \frac{4}{3}x_{3} - \frac{1}{3}x_{4}
2x_{2} &=&& 3 + 2x_{1} + x_{3} - x_{5} \\
x_{2} &=&& \frac{3}{2} + x_{1} + \frac{1}{2}x_{3} - \frac{1}{2}x_{5} \\
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} - \frac{3}{2} - x_{1} - \frac{1}{2}x_{3} + \frac{1}{2}x_{5} + x_{3}\\
&=&& \frac{5}{2} + \frac{1}{2}x_{3} + \frac{1}{2}x_{5} \\
x_{4} &=&& 6 - 3\left(\frac{3}{2} + x_{1} + \frac{1}{2}x_{3} - \frac{1}{2}x_{5}\right) + x_{3} \\
&=&& 6 - \frac{9}{2} - 3x_{1} + \frac{3}{2}x_{3} - \frac{3}{2}x_{5} + x_{3}\\
&=&& \frac{3}{2} - 3x_{1} + \frac{5}{2}x_{3} - \frac{3}{2}x_{5} \\
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} + \frac{3}{2} + x_{1} + \frac{1}{2}x_{3} - \frac{1}{2}x_{5} - x_{3} \\
&=&& \frac{1}{2} - \frac{1}{2}x_{3} - \frac{1}{2}x_{5}
\end{alignat*}
\underline{Ergebnis der 1. Iteration}:
\begin{alignat*}{5}
x_{2} \,&=&\, 2 \,&& &+&\, \frac{1}{3}x_{3} \,&-&\, \frac{1}{3}x_{4} \\
x_{0} \,&=&\, 2 \,&+&\, x_{1} \,&+&\, \frac{2}{3}x_{3} \,&+&\, \frac{1}{3}x_{4} \\
x_{5} \,&=&\, 1 \,&+&\, 2x_{1} \,&+&\, \frac{1}{3}x_{3} \,&+&\, \frac{2}{3}x_{4} \\ \cline{1 - 9}
w &=& 0 \,&-&\, x_{1} \,&+&\, \frac{4}{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} \,&=&\, \frac{5}{2} \,&& &+&\, \frac{1}{2}x_{3} \,&+&\, \frac{1}{2}x_{5} \\
x_{4} \,&=&\, \frac{3}{2} \,&-&\, 3x_{1} \,&+&\, \frac{5}{2}x_{3} \,&-&\, \frac{3}{2}x_{5} \\ \cline{1 - 9}
w &=& \frac{1}{2} \,&& &-&\, \frac{1}{2}x_{3} \,&-&\, \frac{1}{2}x_{5}
\end{alignat*}
Da das Hilfsproblem keine optimale Lösung besitzt, besitzt das ursprüngliche Problem keine zulässige Lösung und ist damit unlösbar.